Inverse Trigonometric Functions Questions

We provide inverse trigonometric functions practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on inverse trigonometric functions skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of inverse trigonometric functions Questions

Question No Questions Class
1 +
43. The number of integer x satisfying sin-‘ x – 2
cos-‘(1 – 13 – x D = is
b. 2
h
d
a.
c.
1
3
d. 4
12
2 29. sin-‘(sin 5) > x2 – 4x holds if
a. x=2 – 19 – 21
b. x=2 + 19 – 2 1
c. x>2 + 19 – 21
d. xe (2 – 49 – 2 7, 2+ 9 – 2 )
12
3 Range of ( boldsymbol{f}(boldsymbol{x})=sin ^{-1} boldsymbol{x}+tan ^{-1} boldsymbol{x}+ )
( cos ^{-1} x ) is
( mathbf{A} cdot[0, pi] )
В. ( left[frac{pi}{4}, frac{3 pi}{4}right] )
с. ( [-pi, 2 pi] )
D. None of these
12
4 88. The solution set of the equation
sin-
1- x2

1 – x² + cos’x = cot-1 V
–sin-‘x is
a. [-1, 1]- {0}
c. [-1,0) U {1}
b. (0, 1] U {-1}
d. [-1,1]
12
5 The value of ( tan ^{-1}left(frac{x}{y}right)- )
( tan ^{-1}left(frac{x-y}{x+y}right), x, y>0 ) is
( A cdot frac{pi}{4} )
B. ( -frac{pi}{4} )
( c cdot frac{pi}{2} )
D. ( -frac{pi}{2} )
12
6 If ( boldsymbol{alpha}=mathbf{3} sin ^{-1} frac{mathbf{6}}{mathbf{1 1}} ) and ( boldsymbol{beta}=mathbf{3} cos ^{-1} frac{mathbf{4}}{mathbf{9}} )
where the inverse trigonometric functions take only the principal values then the correct option(s) is(are) This question has multiple correct options
( mathbf{A} cdot cos beta>0 )
B. ( sin beta0 )
D. ( cos alpha<0 )
12
7 The principle value of ( cos ^{-1}left(frac{-1}{2}right) ) is
A ( cdot frac{-pi}{3} )
в. ( frac{2 pi}{3} )
c. ( frac{4 pi}{3} )
D.
12
8 Solve ( : boldsymbol{y}=sin ^{-1}(sec boldsymbol{x}) ) 12
9 Prove that: ( sin ^{-1}left(frac{3}{5}right)+cos ^{-1}left(frac{12}{13}right)= )
( sin ^{-1}left(frac{56}{65}right) )
12
10 The equation ( sin ^{-1} x-cos ^{-1} x= )
( cos ^{-1}left(frac{sqrt{3}}{2}right) ) has
A. No solution
B. Unique solution
c. Infinite solution
D. None of these
12
11 If the non-zero numbers ( x, y, z ) are ( A P )
and ( tan ^{-1} x, tan ^{-1} y, tan ^{-1} z ) are also in
( A P, ) then
A. ( x y=y z )
B ( cdot z^{2}=x y )
c. ( x=y=z )
D . ( x^{2}=y z )
12
12 Find the projection of the vector ( hat{mathbf{i}}-widehat{boldsymbol{j}} ) on the vector ( hat{mathbf{i}}+widehat{boldsymbol{j}} ) 12
13 Solve:
( operatorname{cosec}^{-1}(cos x) ) is real ( , ) if
A. ( x in[-1,1] )
в. ( x in R )
c. ( x ) is an odd multiple of ( frac{pi}{2} )
D. x is a multiple of ( pi )
12
14 Solve
( cot ^{-1} cot left(frac{5 pi}{4}right) )
12
15 The value of ( cos left(2 cos ^{-1} 0.8right) ) is
A . 0.48
B. 0.96
( c .0 .6 )
D. 0.28
12
16 50. The least and the greatest values of (sin x)² + (cos x)3
-13
13
a.
I a
22
8²8
c. 32 8
d. none of these
12
17 Show that
( tan ^{-1}left(frac{1}{2}right)+tan ^{-1}left(frac{1}{3}right)=frac{pi}{4} )
12
18 Find the number of values of ( x ) of the
form ( 6 n, ) where ( n ) is an integer, in the
domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( boldsymbol{x} ln |boldsymbol{x}-1|+frac{sqrt{mathbf{6 4}-boldsymbol{x}^{2}}}{sin boldsymbol{x}} )
12
19 The number of real solutions of
( tan ^{-1}(sqrt{x(x+1)}+ )
( sin ^{-1} sqrt{left(x^{2}+x+1right)}=frac{pi}{2} ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. infinite
12
20 Prove that ( sin ^{-1}left(frac{8}{17}right)+sin ^{-1}left(frac{3}{5}right)= )
( cos ^{-1}left(frac{36}{85}right) )
12
21 Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
begin{array}{c}
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \
x y1
end{array}
]
(a) ( tan ^{-1} frac{1}{4}+2 tan ^{-1} frac{1}{5}+tan ^{-1} frac{1}{6}+ )
[
tan ^{-1} frac{1}{x}=frac{pi}{4}
]
(b) ( tan ^{-1}(x-1)+tan ^{-1} x+ )
[
tan ^{-1}(x+1)=tan ^{-1} 3 x
]
12
22 If ( 6 operatorname{Sin}^{-1}left(x^{2}-6 x+12right)=2 pi, ) then the
value of ( x, ) is
12
23 Inverse circular functions,Principal
values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
begin{array}{c}
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \
x y1 \
operatorname{atan}^{-1}left(sqrt{frac{a-b}{a+b}} tan frac{theta}{2}right)= \
cos ^{-1} frac{a cos theta+b}{a+b cos theta}
end{array}
]
12
24 If ( f(x)=2 tan ^{-1} x+ )
( sin ^{-1}left(frac{2 x}{1+x^{2}}right), x>1, ) then ( f(5) ) is
equal to:
A . ( pi )
в.
( mathbf{c} cdot 4 tan ^{1}(5) )
D. ( tan ^{-1}left(frac{64}{155}right) )
12
25 Find the value of ( x ) which satisfy equation ( cos left(2 sin ^{-1} xright)=frac{1}{3} )
A ( cdot x=frac{1}{sqrt{5}} ) and ( x=frac{-1}{sqrt{5}} )
B . ( x=frac{1}{sqrt{3}} ) and ( x=frac{-1}{sqrt{3}} )
c. ( x=frac{1}{sqrt{2}} ) and ( x=frac{-1}{sqrt{2}} )
D. None of these
12
26 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2}, ) the
value of ( quad x^{100}+y^{100}+z^{100}- )
( frac{9}{x^{101}+y^{101}+z^{101}} ) is
( mathbf{A} cdot mathbf{0} )
B. 1
c. 2
D. 3
12
27 86. Which of the following is the solution set of the equation
2cos-‘x= cot” 23″-1
a. (0,1)
c. (-1,0)
b. (-1,1) – {0}
d. [-1,1]
12
28 13
59. The value of a such that sinsinho, sin’ cara
Ta are
59. The value of a such that sin
the angles of a triangle is
VIO
– 1
a. T
12
29 Find the value of ( x )
If ( , sin ^{-1} x+sin ^{-1} 2 x=frac{pi}{3} )
12
30 ( fleft(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1 ) then find
the value of ( x )
12
31 Find the value of ( boldsymbol{x} )
If ( , tan ^{-1}(x-1)+tan ^{-1} x+tan ^{-1}(x+ )
1) ( =tan ^{-1} 3 x )
12
32 Illustration 5.67
Solve sin-‘ x + sin- (1 – x) = cos-‘x.
12
33 For the principal value:
( cos ^{-1}left(frac{1}{2}right)-2 sin ^{-1}left(-frac{1}{2}right) )
12
34 Find the principle value of ( tan ^{-1}(-sqrt{3}) )
( mathbf{A} cdot pi / 3 )
в. ( -pi / 3 )
c. ( pi / 6 )
D. ( -pi / 6 )
12
35 Prove ( tan ^{-1} frac{3}{4}+tan ^{-1} frac{3}{5}-tan ^{-1} frac{8}{19}= )
( frac{pi}{4} )
12
36 ( f tan ^{-1} frac{x-1}{x-2}+tan ^{-1} frac{x+1}{x+2}=frac{pi}{4}, ) then
find the value of ( x )
12
37 If ( -1<x<0, ) then ( cos ^{-1} x ) is equal to
This question has multiple correct options
A ( cdot sec ^{-1} frac{1}{x} )
B . ( pi-sin ^{-1} sqrt{1-x^{2}} )
( pi+tan ^{-1} frac{sqrt{1-x^{2}}}{x} )
D. ( cot ^{-1} frac{x}{sqrt{1-x^{2}}} )
12
38 ( sec ^{-1} 2 ) 12
39 If ( sin ^{-1} x=frac{pi}{5} ) for ( operatorname{somex} in[-1,1] ) then
find the value of ( cos ^{-1} x )
12
40 The value of ( cos left[frac{1}{2} cos ^{-1} cos left(-frac{14 pi}{5}right)right] )
is
This question has multiple correct options
( ^{A} cdot cos left(-frac{7 pi}{5}right) )
B cdot ( sin left(frac{pi}{10}right) )
c. ( cos left(frac{2 pi}{5}right) )
D. ( -cos left(frac{3 pi}{5}right) )
12
41 The domain of ( boldsymbol{f}(boldsymbol{x})=frac{sin ^{-1} boldsymbol{x}}{boldsymbol{x}} ) is
( mathbf{A} cdot[-1,1] )
B. {0}
( c cdot[-1,0) )
D. None of these
12
42 If ( left[sin ^{-1} cos ^{-1} sin ^{-1} tan ^{-1} thetaright]=1, ) where
[.] denotes the greatest integer function, the ( theta ) lies in the interval
A. [tan sin cos ( 1, text { sin tan } cos sin 1] )
B. [sin tan cos ( 1, text { tan } sin cos sin 1] )
c. ( [tan sin cos 1, tan sin cos sin 1] )
D. None of these
12
43 24. The value of sin-(cos(cos(cosx) + sin-‘(sin x))), where
Xe
is equal to
Bla
b. – 1
B
dond. -**
12
44 The value of ( k ) if the equation ( k x+ ) ( sin ^{-1}left(x^{2}-8 x+17right)+cos ^{-1}left(x^{2}-right. )
( 8 x+17)=frac{9 pi}{2} ) has atleast one solution
is
( mathbf{A} cdot 2 pi )
в. ( pi )
( c cdot 1 )
D.
12
45 Show that: ( cos ^{-1} frac{4}{5}+cos ^{-1} frac{12}{13}= )
( cos ^{-1} frac{33}{65} )
12
46 Illustration 5.22 Find the number of solutions of
2tan-‘tan x) = 6 – X.
12
47 SE
4. Find the sum cosec. V10 + cosec- 50 + cosec – 7170
+ … + cosec Vln? +1) (x2 + 2n +2).
12
48 13. Which of the following pairs of function/functions has
same graph?
a. y= tan (cos-‘x); y=V1-
b. y = tan (cot- x); y =
c. y = sin ( tan “x); y=
+
d. y = cos(tan-“x); y = sin(cot-‘x)
12
49 Solve ( : sin ^{-1}left(frac{2 pi}{4}right) ) 12
50 prove that
[
begin{array}{l}
2 tan ^{-1}left[tan frac{alpha}{2} tan left(frac{pi}{4}-frac{beta}{2}right)right]= \
tan ^{-1} frac{sin alpha cos beta}{cos alpha+sin beta}
end{array}
]
12
51 ; g(x) = sin ‘ x + cos
x are
46. f(x) = tan-x+tan
identical functions if
a. XER
c. x + [-1, 1]
b. x > 0
d. x 6 (0,1]
12
52 Find the principal value of:
( sin ^{-1}left(frac{sqrt{mathbf{3}}+1}{2 sqrt{2}}right) )
12
53 ( sin ^{-1}(1-x)-2 sin ^{-1} x=frac{pi}{2}, ) then ( x ) is
equal to:
12
54 If ( boldsymbol{y}=boldsymbol{s} boldsymbol{e} boldsymbol{c}^{-1}left[frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right]+boldsymbol{s i n}^{-1}left[frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right] )
( z=operatorname{cosec}^{-1}left[frac{2 x+3}{3 x+2}right]+ )
( cos ^{-1}left[frac{3 x+2}{2 x+3}right] ) then
This question has multiple correct options
A. ( y=pi / 2 )
2
B. ( z=pi / 2 )
( mathbf{c} cdot y+z=pi )
D. ( y+z=pi / 2 )
12
55 41. If sin ‘ x | + |cos + x1 =
, then x e
a. R
c. [0, 1]
b. [-1, 1]
d. 0
12
56 The function ( boldsymbol{f}:left[-frac{mathbf{1}}{mathbf{2}}, frac{mathbf{1}}{mathbf{2}}right] rightarrowleft[-frac{boldsymbol{pi}}{mathbf{2}}, frac{boldsymbol{pi}}{mathbf{2}}right] )
defined by ( sin ^{-1}left(3 x-4 x^{3}right) ) is
A. both one-one onto
B. onto but not one-one
c. one-one but not onto
D. niether one-one nor onto
12
57 Let ( cos ^{-1}left(4 x^{3}-3 xright)=a+b cos ^{-1} x )
( x inleft(frac{1}{2}, 1right], ) then the value of ( lim _{y rightarrow a} b cos (y) )
is
A. ( -frac{1}{3} )
в. -3
( c cdot frac{1}{3} )
D. 3
12
58 Prove:
( sin ^{-1}left(frac{1}{x}right)=operatorname{cosec}^{-1} x, forall x geq 1 ) or ( x leq )
-1
( cos ^{-1}left(frac{1}{x}right)=sec ^{-1} x, forall x geq 1 ) or ( x leq )
-1
( tan ^{-1}left(frac{1}{x}right)=cot ^{-1} x, quad forall x>0 )
12
59 If ( sec ^{-1} frac{1}{sqrt{1-x^{2}}}+cot ^{-1}left(frac{sqrt{1-x^{2}}}{x}right)= )
( sin ^{-1}(k) ) then ( k= )
B. ( 2 x sqrt{1-x^{2}} )
c. ( sqrt{1-x^{2}} )
D. ( 2 x )
12
60 3. Find the range of f(x) = cot- (2x – x?). 12
61 14. The value of lim cos (tan-‘(sin(tan-? x))) is equal to
(
xo
a. -1
d.
12
62 14. If 0<a, <a2 <… <an, then prove that
17-
12
63 Prove that ( 2 tan ^{-1} frac{1}{2}-tan ^{-1} frac{1}{7}=frac{pi}{4} )
Prove that ( 3 sin ^{-1} x=sin ^{-1}(3 x- )
( left.4 x^{3}right), x inleft[frac{-1}{2}, frac{1}{2}right] )
12
64 тл
87. The number of solution of equation sin ‘x+n sin ‘(1 – x)
= ****, where n > 0, m = 0, is
2
a. 3
b. 1
c. 2
d. None of these
12
65 Find the value of ( x, ) if ( tan ^{-1}left(frac{2 x}{1-x^{2}}right)+cot ^{-1}left(frac{1-x^{2}}{2 x}right)= )
( frac{2 pi}{3}, x>0 )
12
66 ( cos left(tan ^{-1} frac{3}{4}right) ) 12
67 If ( tan ^{-1} 2 x+tan ^{-1} 3 x=frac{pi}{4}, ) Then ( x ) is
equal to
A. -1
в. ( frac{1}{6} )
c. ( _{-1, frac{1}{6}} )
D.
12
68 Assertion
If ( a^{2}+b^{2}=c^{2}, c neq, a b neq 0 ) then the non
zero solution of the equation ( sin ^{-1} frac{boldsymbol{a} boldsymbol{x}}{boldsymbol{c}}+sin ^{-1} frac{boldsymbol{b} boldsymbol{x}}{boldsymbol{c}}=boldsymbol{operatorname { s i n }}^{-1} boldsymbol{x} ) is ( pm )
Reason
( sin ^{-1} x+sin ^{-1} y=sin ^{-1}(x+y) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
69 Find the principle value of :
( tan ^{-1}left(frac{1}{sqrt{3}}right) )
12
70 Find the principal value of:
( cot ^{-1}left(-frac{1}{sqrt{3}}right) )
12
71 The value of ( 3 tan ^{-1} frac{1}{2}+2 tan ^{-1} frac{1}{5}+ )
( sin ^{-1} frac{142}{65 sqrt{5}} ) is
A ( cdot frac{pi}{4} )
B. ( frac{pi}{2} )
c. ( pi )
D. none of these
12
72 Prove that ( frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right)=tan ^{-1} sqrt{x} ) 12
73 Range of ( sin ^{-1} x-cos ^{-1} x ) is
( ^{mathbf{A}} cdotleft[frac{-3 pi}{2}, frac{pi}{2}right] )
В ( cdotleft[frac{-5 pi}{3}, frac{pi}{3}right. )
( ^{mathbf{c}} cdotleft[frac{-3 pi}{2}, piright] )
D. ( [0, pi] )
12
74 If the range for ( boldsymbol{y}= )
( left(cot ^{-1} xright)left(cot ^{-1}(-x)right) ) is
( mathbf{0}<boldsymbol{y} leq frac{boldsymbol{pi}^{boldsymbol{a}}}{boldsymbol{b}} )
Find the value of ( a+b )
A . 2
в. 4
c. 5
D. 6
12
75 Find the value of ( tan ^{2}left(frac{1}{2} sin ^{-1} frac{2}{3}right) ) 12
76 (JEE Adv. 2013) 12
77 Find the principal value of the following
( cot left(tan ^{-1} x+cot ^{-1} xright) )
12
78 ( sum_{m=1}^{n} tan ^{-1}left(frac{2 m}{m^{4}+m^{2}+2}right) ) is equal
to
A ( cdot tan ^{-1}left(n^{2}+n+1right)-frac{pi}{4} )
B cdot ( tan ^{-1}left(n^{2}+n+1right)+frac{pi}{4} )
c. ( tan ^{-1}left(n^{2}+n-1right)-frac{pi}{4} )
D ( cdot tan ^{-1}left(n^{2}-n-1right)-frac{pi}{4} )
12
79 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{cos ^{-1} sin (boldsymbol{x}+boldsymbol{pi} / mathbf{3})} ) Then
This question has multiple correct options
( mathbf{A} cdot fleft(frac{5 pi}{9}right)=e^{5 pi / 18} )
B ( cdot fleft(frac{8 pi}{9}right)=e^{13 pi / 18} )
( ^{mathbf{C}} cdot fleft(-frac{7 pi}{4}right)=e^{pi / 12} )
D ( quad fleft(-frac{7 pi}{4}right)=e^{11 pi / 12} )
12
80 The inequality ( sin ^{-1} x>cos ^{-1} x ) vholds
for
A. all values of ( x )
B ( cdot x inleft(0, frac{1}{sqrt{2}}right) )
( ^{c} cdot_{x inleft(frac{1}{sqrt{2}}, 1right)} )
D. no value of ( x )
12
81 Зл
а. Тоr — Зл
12
82 Find the range of f(x) = sin- x + tan-‘x+
Illustration 5.45
cos-‘x.
12
83 Inverse circular functions,Principal
values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
begin{array}{l}
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \
x y1
end{array}
]
(a) Find whether ( x=2 ) satisfies the
equation
[
begin{array}{c}
tan ^{-1} frac{x+1}{x-1}+tan ^{-1} frac{x-1}{x}= \
tan ^{-1}(-7)
end{array}
]
If not, then how should the equation be re-written?
(b) ( tan ^{-1} frac{4}{3}+tan ^{-1} frac{5}{6}+tan ^{-1} frac{39}{2}- )
( pi=dots )
(c) If ( x_{1}, x_{2}, x_{3}, x_{4} ) are roots of equation
[
x^{4}-x^{3} sin 2 beta+x^{2} cos 2 beta-x cos beta-
]
( sin beta=0, ) then prove that
[
sum_{i=1}^{4} tan ^{-1} x_{1}=frac{pi}{2}-beta
]
12
84 If ( sin left(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then ( x ) is
equal to
A .
B. 0
( c cdot frac{4}{5} )
D.
12
85 ( cos ^{-1}left[cos left(left(-frac{17}{15}right) piright)right] ) is equal to
A ( cdot frac{17 pi}{15} )
в. ( frac{13 pi}{15} )
( c cdot frac{3 pi}{15} )
D. ( -frac{17 pi}{15} )
12
86 If ( sin ^{-1} frac{3}{x}+sin ^{-1} frac{4}{x}=frac{pi}{2}, ) then ( x ) is
equal to
( A cdot 3 )
B. 5
( c cdot 7 )
D. 11
12
87 f ( 2 tan ^{-1} x=cos ^{-1}(3 / sqrt{13}) ) then the
value of ( 60 x^{4}-540 x^{2}+360 x+9261 )
is equal to
12
88 ( int frac{x}{left(x^{2}+4right) sqrt{x^{2}+1}} d x= )
( frac{1}{sqrt{k}} tan ^{-1} sqrt{frac{x^{2}+1}{3}}+c . ) what is ( k ? )
12
89 The value of ( sin ^{-1}left(sin 5 frac{pi}{3}right)= )
A ( cdot-frac{pi}{3} )
B .
( mathbf{c} cdot frac{4 pi}{3} )
D. ( frac{3 pi}{3 pi} )
12
90 Write the value of ( tan ^{-1}left(frac{1}{x}right) ) for ( x<0 ) in terms of ( cot ^{-1}(x) ) 12
91 If ( x ) takes negative permissible value,
then ( sin ^{-1} x= )
B. ( -cos ^{-1} sqrt{1-x^{2}} )
( mathrm{c} cdot cos ^{-1} sqrt{x^{2}-1} )
D・ ( pi-cos ^{-1} sqrt{1-x^{2}} )
12
92 If two angles of a triangle are ( tan ^{-1}(2) )
and ( tan ^{-1}(3), ) then the third angle is
( ^{A} cdot frac{pi}{4} )
в.
( c cdot frac{pi}{3} )
D.
12
93 2x
2x
Hlustration 5.15 if sin ‘ = tan 2, then find the
If sin
, then find the
tan-1_
Illustration 5.75
value of x.
1+12
I-r2, then fin
12
94 Illustration 5.18 Evaluate the following:
i. sin-‘(sin 10)
ii. sin-‘(sin 5)
iii. cos(cos 10) iv. tan-‘(tan(-6))
12
95 10
44. If tan-x + 2 cot-‘x = ***, then x is equal to
13 –
b. 3
d. 2
c. √3
12
96 Solve :
( cos ^{-1}left(log _{2} xright)=0 )
12
97 The value of ( sin left(2 sin ^{-1} mathbf{0 . 8}right) )
A ( cdot frac{1}{25} )
в. ( frac{25}{24} )
c. ( frac{24}{25} )
D. none
12
98 60. The number of solutions of the equation tan-‘(1 + x) +
tan-‘(1 – x) =
a. 2
b. 3
c. 1
an
d. 0
12
99 73. If
sin
3 sin 28
= tan-‘x, then x =
5 + 4 cos 20
a. tan 30
c. (1/3) tano
b. 3 tano
d. 3 cote
12
100 If ( cot ^{-1}left(frac{1}{x+1}right)+cot ^{-1}left(frac{1}{x-1}right)= )
( tan ^{-1} 3 x-tan ^{-1} x )
then ( boldsymbol{x}= )
A. ( pm 1 / 2 )
B. ( -1, pm 1 / 3 )
c. 2,±1
D. ( -1 . pm 1 / 2 )
12
101 Illustration 5.2
Solve sin-‘x>-1.
12
102 ( sin ^{-1} x+sin ^{-1} frac{1}{x}+cos ^{-1} x+ )
( cos ^{-1} frac{1}{x}= )
A . ( pi )
в.
c. ( frac{3 pi}{2} )
D. None of these
12
103 ( sin left[2 cos ^{-1} cot left(2 tan ^{-1} xright)right]=0 ) if
This question has multiple correct options
( mathbf{A} cdot x=-1-sqrt{2} )
B. ( x=1+sqrt{2} )
( mathbf{c} cdot x=1-sqrt{2} )
D. ( x=sqrt{2}-1 )
12
104 The solution set of the equation ( tan ^{-1} x-cot ^{-1} x=cos ^{-1}(2-x) ) is
A ( .(0,1) )
в. (-1,1)
( c cdot[1,3) )
D. (1,3)
12
105 The value of ( x ) where ( x>0 )
( tan left(sec ^{-1} frac{1}{x}right)=sin left(tan ^{-1} 2right) ) is
A ( cdot sqrt{5} )
в. ( frac{sqrt{5}}{3} )
c.
D.
12
106 If ( x=n pi-tan ^{-1} 3 ) is a solution of the
equation ( 12 tan 2 x+frac{sqrt{10}}{cos x}+1=0 )
then
A. ( n ) is any integer
B. n is an even integer
c. ( n ) is a positive integer
D. ( n ) is an odd integer
12
107 Find the value of ( tan ^{-1} sqrt{3}-sec ^{-1}(-2) )
is equal to
( A )
B. ( -frac{pi}{3} )
c.
D. ( frac{2 pi}{3} )
12
108 Prove that ( : tan ^{-1}left[frac{6 x-8 x^{3}}{1-12 x^{2}}right]- )
( tan ^{-1}left[frac{4 x}{1-4 x^{2}}right]=tan ^{-1} 2 x,|2 x|<frac{1}{sqrt{3}} )
12
109 Write the principal value of ( cos ^{-1}left(frac{1}{2}right)- ) ( 2 sin ^{-1}left(-frac{1}{2}right) ) 12
110 If ( 4 sin ^{-1} x+cos ^{-1} x=pi, ) then find the
value of ( boldsymbol{x} )
12
111 Match the following 12
112 Two angles of a triangle are ( cot ^{-1} 2 ) and
( cot ^{-1} 3 . ) Then the third angle
A.
в. ( frac{3 pi}{4} )
c.
D.
12
113 The value of ( sin ^{-1}left(frac{3}{5}right)+tan ^{-1}left(frac{1}{7}right) )
( A cdot 0 )
в.
( c cdot frac{pi}{3} )
D.
12
114 Which of the following quantities is/are positive?
This question has multiple correct options
A ( cdot cos left(tan ^{-1}(tan 4)right) )
B. ( sin left(cot ^{-1}(cot 4)right) )
c. ( tan left(cos ^{-1}(cos 5)right) )
D. ( cot left(sin ^{-1}(sin 4)right) )
12
115 37. If sin la + sin ‘ b + sin c = 1, then the value of
a (1-a?) +b/(1-6?) +c/(1-c?) will be
a. 2abc
b. abc
nh
d. – abc
c.
– abc
12
116 If ( cos ^{-1} x+cos ^{-1} y=2 pi ) then
( sin ^{-1} x+sin ^{-1} y= )
( A cdot pi )
в. ( -pi )
( c cdot frac{pi}{2} )
D. None of these
12
117 If ( cos ^{-1}left(frac{1}{x}right)=theta ) then ( tan theta= )
A ( frac{1}{sqrt{x^{2}-1}} )
B. ( sqrt{x^{2}+1} )
c. ( sqrt{1-x^{2}} )
D. ( sqrt{x^{2}-1} )
12
118 Find the value of ( x .left(tan ^{-1} xright)^{2}+ )
( left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} )
12
119 Write the value of ( cot ^{-1}(-x) ) for all ( x epsilon R )
in terms of ( cot ^{-1} x )
12
120 Evaluate ( cos ^{-1} x+ )
( cos ^{-1}left{frac{x}{2}+frac{1}{2} sqrt{3-3 x^{2}}right}, x epsilonleft[frac{1}{2}, 1right] )
A ( cdot-frac{pi}{6} )
B. ( +frac{pi}{6} )
( c cdot-frac{pi}{3} )
D. ( +frac{pi}{3} )
12
121 If ( x ) and ( y ) are positive and ( x y>1 ), then
what is ( tan ^{-1} x+tan ^{-1} y ) equal to?
A ( cdot tan ^{-1}left(frac{x+y}{1-x y}right) )
в. ( pi+tan ^{-1}left(frac{x+y}{1-x y}right) )
c. ( pi-tan ^{-1}left(frac{x+y}{1-x y}right) )
D. ( tan ^{-1}left(frac{x-y}{1+x y}right) )
12
122 Solve for ( x ) :
( tan ^{-1} x=frac{1}{2} cot ^{-1} x )
12
123 If value of ( mathbf{x} ) which satisfy equation
( left(cot ^{-1} xright)^{2}-3left(cot ^{-1} xright)+2>0 ) is ( xcot b )
Find the value of ( a+b )
A . 1
B. 2
( c .3 )
D. 4
12
124 If ( sin ^{-1}(1-x)-2 sin ^{-1} x=pi / 2, ) then
( x ) equals-
в. ( _{0, frac{1}{2}} )
c. 0
D. None of these
12
125 If ( sec ^{-1} x+sec ^{-1} y+sec ^{-1} z=3 pi )
then ( boldsymbol{x} boldsymbol{y}+boldsymbol{y} boldsymbol{z}+boldsymbol{z} boldsymbol{x}= )
( mathbf{A} cdot mathbf{0} )
B. –
( c .3 )
( D )
12
126 The value of ( cos ^{-1}left(cos frac{5 pi}{4}right) ) is?
A ( cdot frac{-3 pi}{4} )
в. ( frac{3 pi}{4} )
c. ( frac{-5 pi}{4} )
D. ( frac{5 pi}{4} )
12
127 The domain of the function ( f(x)= ) ( sqrt{cos ^{-1}left(frac{1-|x|}{2}right)} )
A ( cdot(-3,3) )
в. [-3,3]
C ( cdot(-infty,-3) cup(-3, infty) )
D. ( (-infty,-3) cup(3, infty) )
12
128 20. The sum of the solutions of the equation
2 sin-‘ Vx2 +x+1 +cos” Vx2 + x = 31 is
a. 0
b. – 1
c. 1
d. 2
12
129 ( operatorname{Let} cos ^{-1}(x)+cos ^{-1}(2 x)+cos ^{-1}(3 x) )
be ( pi ) If ( x ) satisfies the equation ( a x^{3}+ ) ( b x^{2}+c x-1=0, ) then the value of
( (b-a-c) ) is
12
130 Illustration 5.63
Find the value of 4 tan-
tan 1
S
99
12
131 ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3}, ) then the
value of ( x ) is-
A ( cdot sqrt{2} )
B. 3
( c cdot sqrt{3} )
D. ( frac{sqrt{3}-1}{sqrt{3}+1} )
12
132 Solve for ( boldsymbol{x} )
( 2 tan ^{-1}(cos x)=tan ^{-1}(2 operatorname{cosec} x) )
12
133 The ascending order of ( boldsymbol{A}= ) ( sin ^{-1}left(log _{3} 2right), B=cos ^{-1}left(log _{3}left(frac{1}{2}right)right) )
and ( C=tan ^{-1}left(log _{1 / 3} 2right) ) is
A. ( mathrm{C}, mathrm{B}, mathrm{A} )
в. В, А, С
c. ( mathrm{c}, mathrm{A}, mathrm{B} )
D. B, C, A
12
134 The value of ( tan left{frac{1}{2} cos ^{-1}left(frac{sqrt{5}}{3}right)right} ) is
A ( cdot frac{3+sqrt{5}}{2} )
B. ( 3+sqrt{5} )
c. ( frac{1}{2}(3-sqrt{5}) )
D. None of these
12
135 Find the value of :
[
begin{array}{l}
cos left[frac{pi}{6}+2 tan ^{-1}(1)right]+ \
sin left[3 sin ^{-1}left(frac{1}{2}right)+2 cos ^{-1}left(frac{1}{2}right)right]
end{array}
]
12
136 Which of the following is the solution set of the equation ( 2 cos ^{-1}(x)= )
( cot ^{-1}left(frac{2 x^{2}-1}{2 x sqrt{1-x^{2}}}right) ? )
( mathbf{A} cdot(0,1) )
B . ( (-1,1)-{0} )
c. (-1,0)
D cdot [-1,1]
12
137 Evaluate:
[
begin{array}{l}
tan ^{-1}left(-frac{1}{sqrt{3}}right)+tan ^{-1}(-sqrt{3})+ \
tan ^{-1}left(sin left(-frac{pi}{2}right)right)
end{array}
]
12
138 The value of ( tan left(2 tan ^{-1} 1 / 5-pi / 4right) ) is?
A. ( -7 / 17 )
в. ( +7 / 17 )
c. ( -12 / 17 )
D. ( -+2 / 17 )
12
139 f ( x y+y z+z x=1 ) then find the value
of ( tan ^{-1} x+tan ^{-1} y+tan ^{-1} z )
12
140 The number of integral values of ( k ) for
which the equation ( sin ^{-1} x+ )
( tan ^{-1} x=2 k+1 ) has a solution is
A . 1
B. 2
( c .3 )
D. 4
12
141 If ( boldsymbol{x} boldsymbol{epsilon}[-1,0), ) then find the value of
( cos ^{-1}left(2 x^{2}-1right)-2 sin ^{-1} x )
( mathbf{A} cdot-pi / 2 )
в. ( +pi / 2 )
c. ( -pi )
D. ( +pi )
12
142 Solve: ( sin left(tan ^{-1} xright),|x|<1 ) is equal to
A ( cdot frac{x}{sqrt{1-x^{2}}} )
в. ( frac{1}{sqrt{1-x^{2}}} )
c. ( frac{1}{sqrt{1+x^{2}}} )
D. ( frac{x}{sqrt{1+x^{2}}} )
12
143 Simplify ( tan ^{-1}left(frac{6 x}{1-8 x^{2}}right) )
( A cdot tan ^{-1} 2 x+tan ^{-1} 4 x )
B. ( tan ^{-1} 2 x-tan ^{-1} 4 x )
c. ( -tan ^{-1} 2 x-tan ^{-1} 4 x )
D. ( 2 tan ^{-1} 2 x-tan ^{-1} 4 x )
12
144 Find the principle value of ( cos ^{-1}left[cos left(frac{7 pi}{3}right)right] ) 12
145 If ( cos ^{-1}left(4 x^{3}-3 xright)=2 pi-3 cos ^{-1} x )
then ( x ) lies in interval
A ( cdotleft[-1,-frac{1}{2}right] )
в. ( |x|<frac{1}{2} )
( mathbf{c} cdotleft[frac{1}{2}, 1right. )
D. None of these
12
146 If ( sin ^{-1} x+4 cos ^{-1} x=pi, ) then ( x= )
A. ( 1 / 2 )
в. ( frac{1}{sqrt{2}} )
c. ( frac{sqrt{3}}{2} )
D.
12
147 The principal value of ( sin ^{-1}left(frac{-1}{2}right) ) is
A ( cdot frac{-pi}{6} )
В. ( frac{5 pi}{6} )
c. ( frac{7 pi}{6} )
D. none of these
12
148 Prove that
( operatorname{cosec}left(tan ^{-1}left(cos left(cot ^{-1}left(sec left(sin ^{-1} aright)right)right)right)right) )
( sqrt{mathbf{3}-boldsymbol{a}^{2}}, ) where ( boldsymbol{a} in[mathbf{0}, mathbf{1}] )
12
149 ( cos ^{-1}(x)=cot ^{-1}left(frac{x}{sqrt{1-x^{2}}}right) ) where
is in the common domain of the
functions.
A. True
B. False
12
150 Solve: ( tan ^{-1} 2 x+tan ^{-1} 3 x=frac{pi}{4} ) 12
151 Find the principal value of the following:
( operatorname{cosec}^{-1}(-sqrt{2}) )
12
152 If ( sin left(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then find
the value of ( x )
A . -1
B.
( c cdot frac{1}{5} )
D.
12
153 The domain of ( sin ^{-1}left[log _{2}left(x^{2} / 2right)right] ) is
A . [2,1]
в. [1,2]
c. [-2,-1]( cup[1,2] )
D. [-2,0]
12
154 Which of the following is/are the value of ( cos left[frac{1}{2} cos ^{-1}left(cos left(-frac{14 pi}{5}right)right)right] ? )
This question has multiple correct options
A ( cdot cos left(-frac{7 pi}{5}right) )
B cdot ( sin left(frac{pi}{10}right) )
c. ( cos left(frac{2 pi}{5}right) )
D. ( cos left(-frac{3 pi}{5}right) )
12
155 The range of ( a r c sin x+a r c cos x+ )
( arctan x ) is
12
156 The range of ( tan ^{-1} x )
A ( cdot(-pi, pi) )
B.
D. ( left(-frac{pi}{2}, frac{pi}{2}right) )
12
157 ( sum_{r=1}^{n} tan ^{-1}left(frac{2^{r-1}}{1+2^{2 r-1}}right) ) is equal to:
( mathbf{A} cdot tan ^{-1}left(2^{n}right) )
B cdot ( tan ^{-1}left(2^{n}right)-frac{pi}{4} )
c. ( tan ^{-1}left(2^{n+1}right) )
( mathbf{D} cdot tan ^{-1}left(2^{n+1}right)-frac{pi}{4} )
12
158 The domain of the function
( sin ^{-1}left(log _{2}left(frac{x}{3}right)right) ) is
A ( cdot frac{1}{2}, 3 )
B. ( frac{1}{2}, 4 )
c. ( frac{3}{2}, 6 )
D. ( frac{1}{2}, 2 )
12
159 42. If (sin x)2 – (cos- ‘x)2 = an? then find the range of a.
dc. 1-1, 1]
d. -1,
12
160 1. The principal value of sin “sin 25) is (1986- 2 Marks)
(a) 20 (6) 24 ( 47 (a) none
12
161 Write the following in the simplest form:
( tan ^{-1}left{frac{sqrt{1+x^{2}}-1}{x}right}, x neq 0 )
12
162 83. If sin ta+sin-‘b + sin c = , then av1-a? +bv1 – b
+cV1-c? is equal to
a. a + b + c
b. a-b2c2
c. 2abc
d. 4abc
12
163 ( cos ^{-1}left{cos left(frac{5 pi}{4}right)right} ) is given by
( ^{text {A }} cdot frac{5 pi}{4} )
в. ( frac{3 pi}{4} )
c. ( frac{-pi}{4} )
D. none of these
12
164 33. The value of
water cours to slevo
.
a. (a – B) (02 +B2)
c. (a+B) (o? +B)
b. (a+B) (02-B)
d. none of these
12
165 If the domain of the function ( f(x)= ) ( sqrt{3 cos ^{-1}(4 x)-pi} ) is [a,b] then the
value of ( (4 a+64 b) ) is
12
166 12. The value of k (k > 0) such that the length of the longest
interval in which the function f(x) = sinsin kx +
cos(cos kx) is constant is te/4 is/are
a. 8
b. 4
c. 12
d. 16
12
167 Write the principal value of ( cos ^{-1}left(frac{1}{2}right)-2 sin ^{-1}left(-frac{1}{2}right) ) 12
168 Evaluate ( sin left(frac{pi}{6}+cos ^{-1} frac{1}{4}right) ) 12
169 If ( x>0 ) and ( cos ^{-1}left(frac{12}{x}right)+ )
( cos ^{-1}left(frac{35}{x}right)=frac{pi}{2}, ) then ( x ) is
( A cdot 7 )
B. 39
c. 37
D. -37
12
170 If ( 3 tan ^{-1} x+cot ^{-1} x=pi, ) then ( x )
equals:
( mathbf{A} cdot mathbf{0} )
B.
c. -1
D.
12
171 The value of ( cos left{cos ^{-1}left(-frac{sqrt{3}}{2}right)+frac{pi}{6}right} )
is-
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D.
12
172 The value of
( sin ^{-1}left(cot left(sin ^{-1} sqrt{frac{2-sqrt{3}}{4}}+cos ^{-1} frac{sqrt{2}}{2}right)right. )
is
A . 0
B. ( frac{pi}{2} )
( c cdot frac{pi}{3} )
D. none of these
12
173 The number of values of ( x ) for which
[
begin{array}{l}
sin ^{-1}left(x^{2}-frac{x^{4}}{3}+frac{x^{6}}{9} dotsright)+ \
cos ^{-1}left(x^{4}-frac{x^{8}}{3}+frac{x^{12}}{9} dotsright)=frac{pi}{2}, text { where } \
0 leq|x|<sqrt{3}, text { is }
end{array}
]
12
174 Simplify ( tan ^{-1}left[frac{boldsymbol{a} cos boldsymbol{x}-boldsymbol{b} sin boldsymbol{x}}{boldsymbol{b} cos boldsymbol{x}+boldsymbol{a} sin boldsymbol{x}}right], ) if
( frac{boldsymbol{a}}{boldsymbol{b}} tan boldsymbol{x}>-1 )
12
175 15. If cos’x + cos’y + cos’z = 1, then
a. x² + y2 + x2 + 2xyz = 1
b. 2(sin “x+sin ‘y + sin ‘z) = cos ‘x+cos ly+ cos’z
c. xy + yz + zx = x + y + z-1
d.
X+ –
+
y + –
+ z + – 126
12
176 ( M C Q: sin left(sin ^{-1} frac{5 pi}{6}right)+ )
( cos ^{-1}left(cos frac{5 pi}{3}right)+tan ^{-1}left(tan frac{7 pi}{3}right) )
( A cdot frac{5 pi}{6} )
в. ( frac{pi}{3} )
c. ( frac{7 pi}{6} )
( D cdot frac{29 pi pi pi}{6} )
12
177 If ( sin ^{-1}left(frac{x}{5}right)+operatorname{cosec}^{-1}left(frac{5}{4}right)=frac{pi}{2}, ) then
the value of ( x )
( A cdot 3 )
B . 2
c. 1
( D )
12
178 ( cos left(tan ^{-1} frac{3}{4}right)=? )
( A cdot frac{3}{5} )
( B cdot frac{4}{5} )
( c cdot frac{4}{9} )
D. none of these
12
179 38. If a sinx – b cos= x=c, then a sin’x + b cos ‘x is equal
a. O
Ttab +c(b-a)
atb
Tab + c(a – b)
atba
12
180 ( cot ^{-1}(sqrt{cos alpha})-tan ^{-1}(sqrt{cos alpha})=x )
then ( sin x ) is equal to
( A cdot tan ^{2} frac{alpha}{2} )
B ( cdot cot ^{2} frac{alpha}{2} )
c. ( tan alpha )
D. ( cot frac{alpha}{2} )
12
181 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) and
( boldsymbol{f}(mathbf{1})=mathbf{1}, boldsymbol{f}(boldsymbol{p}+boldsymbol{q})= )
( boldsymbol{f}(boldsymbol{p}) cdot boldsymbol{f}(boldsymbol{q}) quad forall boldsymbol{p}, boldsymbol{q} in boldsymbol{R} operatorname{th} operatorname{en} boldsymbol{x}^{f(1)}+ )
( boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(boldsymbol{3})}-frac{boldsymbol{x}+boldsymbol{y}+boldsymbol{z}}{boldsymbol{x}^{f(1)}+boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(3)}}= )
( A cdot O )
в.
( c cdot 2 )
( D )
12
182 If ( cot ^{-1} x+tan ^{-1} 3=frac{pi}{2} ) then ( x= )
A ( -frac{1}{3} )
в. ( frac{1}{4} )
( c cdot 3 )
D.
12
183 ( fleft(sin ^{-1} xright)^{2}+left(cos ^{-1} xright)^{2}=frac{17 pi^{2}}{36}, ) find
( boldsymbol{x} )
12
184 5. If cot-1
,ne N, then the maximum value of n is
a. 6
c. 5
b. 7
d. none of these
12
185 If ( M ) denotes the maximum value of
( left(1+sec ^{-1} xright)left(1+cos ^{-1} xright) & m )
denotes the maximum value of
( left(1+operatorname{cosec}^{-1} xright)left(1+sin ^{-1} xright), ) then
( left[frac{M}{m}right] ) is (where [.] denotes greatest integer function)
12
186 Prove that ( 3 sin ^{-1}=sin ^{-1}(3 x-1) )
( left.4 x^{3}right), x epsilonleft[frac{-1}{2}, frac{1}{2}right] )
12
187 The number of triplets ( (x, y, z) ) satisfies
the equation ( boldsymbol{f}(boldsymbol{x}, boldsymbol{y}, boldsymbol{z})=sin ^{-1} boldsymbol{x}+ )
( sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) is
( mathbf{A} cdot mathbf{1} )
B. 2
c. 0
D. Infinite
12
188 The set of values of ( x ) for which
( tan ^{-1} frac{x}{sqrt{1-x^{2}}}=sin ^{-1} x ) holds is
A. ( R )
в. (-1,1)
( mathbf{c} cdot[0,1] )
D. [-1,0]
12
189 Find the principal value of the following
( cos ^{-1}left(cos frac{7 pi}{6}right) )
12
190 Find the value of: ( sin left(2 tan ^{-1} frac{1}{4}right)+ )
( cos left(tan ^{-1} 2 sqrt{2}right) )
12
191 Assertion
STATEMENT 1: Let ( boldsymbol{f}(boldsymbol{x})= ) ( sin ^{-1}left(frac{2 x}{1+x^{2}}right), f^{prime}(2)=-frac{2}{5} )
Reason
STATEMENT
( 2: sin ^{-1}left(frac{2 x}{1+x^{2}}right)=pi )
( 2 tan ^{-1} x forall x>1 )
A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT
B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE
12
192 for 0<x< 12, then x equals
(a) 12 (b) I (c) -1/2
(20015)
(d) 1
12
193 Solve the equation ( 3 sin ^{-1}left(frac{2 x}{1+x^{2}}right)- )
( 4 cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right)+2 tan ^{-1}left(frac{2 x}{1-x^{2}}right)=frac{pi}{3} )
12
194 Illustration 5.61
II aj, az, az, …,a,, is an A.P. with common
Ifaj, az, az, …, an is an A
difference d, then prove that
d
(n-1)d
+ tan-1
1+ an-1an
It aan
12
195 If ( tan ^{-1} frac{sqrt{1+x^{2}}-sqrt{1-x^{2}}}{sqrt{1+x^{2}}+sqrt{1-x^{2}}}=alpha, ) then
( x^{2}= )
( mathbf{A} cdot cos 2 alpha )
B. ( sin 2 alpha )
( c cdot tan 2 alpha )
D. ( cot 2 alpha )
12
196 The value of
( sin ^{-1}left[cot left[sin ^{-1}(sqrt{frac{2-sqrt{3}}{4}})+cos ^{-1}right.right. )
is
12
197 Inverse circular functions,Principal
values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y}
]
( x y1
]
Prove
(a) ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+sin ^{-1} frac{16}{65}=frac{pi}{2} )
(b) ( sin ^{-1} frac{3}{5}+sin ^{-1} frac{8}{17}=cos ^{-1} frac{36}{85} )
( (c) sin ^{-1} frac{3}{5}+cos ^{-1} frac{12}{13}=cos ^{-1} frac{33}{65} )
12
198 Solve ( : cos ^{-1} sqrt{frac{1+cos x}{2}} ) 12
199 The value of p for which system has a solution is
A . 1
B. 2
c. 0
D. –
12
200 Find ( x, ) If ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} )
A . -1
B.
( c cdot 0 )
( D )
12
201 Solve ( boldsymbol{y}= )
( tan ^{-1}left(frac{3 x-x^{3}}{1-3 x^{2}}right),-frac{1}{sqrt{3}}<x<frac{1}{sqrt{3}} )
12
202 Evaluate:
( sin ^{-1}left(sin frac{5 pi}{6}right) )
12
203 ( sin ^{-1} mathbf{6} boldsymbol{x}+boldsymbol{operatorname { s i n }}^{-1} boldsymbol{6} sqrt{mathbf{3}} boldsymbol{x}=-boldsymbol{pi} / 2 ) if ( mathbf{x} ) is
equal to
A . – ( -1 / 12 )
B. 1/6
( c cdot 1 / 12 )
D. -1/6
12
204 Which of the following is the solution
set of the equation ( 2 cos ^{-1} x= ) ( cot ^{-1}left(frac{2 x^{2}-1}{2 x sqrt{1-x^{2}}}right) )
A ( .(0,1) )
B . ( (-1,1)-{0} )
c. (-1,0)
D. [-1,1]
12
205 Write the value of ( 2 sin ^{-1} frac{1}{2}+ )
( cos ^{1}left(-frac{1}{2}right) )
12
206 Find the principal value of:
( sec ^{-1}(sqrt{2})+2 operatorname{cosec}^{-1}(-sqrt{2}) )
12
207 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{tan ^{-1} frac{boldsymbol{x}}{mathbf{1}+boldsymbol{x}^{2}}+tan ^{-1} frac{mathbf{1}+boldsymbol{x}^{2}}{boldsymbol{x}}right}= )
A . 0
B.
( c cdot frac{1}{2} )
( D )
12
208 If ( 4 sin ^{-1} x+cos ^{-1} x=pi, ) then ( x ) is
equal to:
A ( cdot frac{1}{2} )
B. 2
c. 1
D.
12
209 3. Which of the following is/are the value of
COS
cos(cos” (cos(-1947)]
(211
d. -cos
C. COS
12
210 Solve
( sin ^{-1}(cos x) )
12
211 ( sin ^{-1} frac{3}{5}+sin ^{-1} frac{4}{5} ) is equal to
A ( cdot frac{pi}{2} )
в.
( c cdot frac{pi}{4} )
D.
12
212 Evaluate the following:
( cos ^{-1}(cos 12) )
12
213 Write the principal value of :
( left[cos ^{-1} frac{sqrt{3}}{2}+cos ^{-1}left(-frac{1}{2}right)right] )
12
214 Illustration 5.7 Find the range of f(x) = 13 tan ‘x- cos’O
– cos(-1).
12
215 f ( sum_{i=1}^{2 n} sin ^{-1} x_{i}=n pi, ) then ( sum_{i=1}^{2 n} x_{i} ) is
equal to
( mathbf{A} cdot n / 2 )
B. ( 2 n )
c. ( frac{n(n+1)}{2} )
D. none of these
12
216 If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}+dots dots dots dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots dots dots inftyright)=frac{pi}{2} )
and ( 0<x<sqrt{2} ) then ( x= )
A ( cdot frac{1}{2} )
B.
( c cdot-frac{1}{2} )
D. –
12
217 The principle value of ( tan ^{-1}(-sqrt{3}) ) is
A ( cdot frac{2 pi}{3} )
в. ( frac{4 pi}{3} )
c. ( frac{-pi}{3} )
D. none of these
12
218 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} )
then the value of ( x^{9}+y^{9}+z^{9}-frac{1}{x^{9} y^{9} z^{9}} )
is equal to
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D.
12
219 Illustration 5.33
Prove that
12
220 If ( tan ^{-1} 4=4 tan ^{-1} x, ) then ( x^{5}-7 x^{3}+ )
( 5 x^{2}+2 x ) is equal to
12
221 If ( cos ^{-1} x+cos ^{-1} y=frac{pi}{2} ) then prove that
( cos ^{-1} x=sin ^{-1} y )
12
222 Solution of the equation ( tan left(cos ^{-1} xright)=sin left(cot ^{-1} frac{1}{2}right) )
A ( cdot x=pm frac{sqrt{7}}{3} )
B. ( x=pm frac{sqrt{5}}{3} )
c. ( x=pm frac{3 sqrt{5}}{2} )
D. None of these
12
223 The value of
( sec left[sin ^{-1}left(sin frac{50 pi}{9}right)+cos ^{-1} cos left(frac{31 tau}{9}right.right. )
is equal to
A ( cdot sec frac{10 pi}{9} )
в. ( sec 9 pi )
c. -1
D.
12
224 Illustration 5.64
If (x – 1) (x² + 1) > 0, then find the value
tan
12
225 Prove that:
( 2 tan ^{-1}left(frac{1}{5}right)+sec ^{-1}left(frac{5 sqrt{2}}{7}right)+ )
( 2 tan ^{-1}left(frac{1}{8}right)=frac{pi}{4} )
12
226 Evaluate the following:
( cos ^{-1}(cos 4) )
12
227 The value of ( sin ^{-1}left(cos frac{53 pi}{5}right) ) is
A ( cdot frac{3 pi}{5} )
в. ( frac{-3 pi}{5} )
c. ( frac{pi}{10} )
D. ( frac{-pi}{10} )
12
228 ( tan ^{-1}left(frac{x+1}{x-1}right)+tan ^{-1}left(frac{x+1}{x}right)= )
( tan ^{-1}(2)+pi )
12
229 Find the principal value of ( cos ^{-1}left(cos frac{7 pi}{6}right) ) 12
230 Write the principal value of ( sin ^{-1}left(-frac{1}{2}right) ) ( ? ) 12
231 Solve ( cos left[tan ^{-1}left[sin left(cot ^{-1} xright)right]right] )
A ( cdot sqrt{frac{x^{2}+2}{x^{2}+3}} )
B. ( sqrt{frac{x^{2}+2}{x^{2}+1}} )
c. ( sqrt{frac{x^{2}+1}{x^{2}+2}} )
D. None of these
12
232 The domain of the function ( sin ^{-1} 2 x ) is:
( mathbf{A} cdot[0,1] )
B . [-1,1]
c. [-2,2]
D. ( left[frac{-1}{2}, frac{1}{2}right] )
12
233 The value of ( cos ^{-1}left(-frac{1}{2}right)+ )
( sin ^{-1}left(-frac{sqrt{mathbf{3}}}{mathbf{2}}right) ) is
( A cdot frac{pi}{3} )
B.
c. ( frac{2 pi}{3} )
D. none of these
12
234 The number of integer ( boldsymbol{x} ) satisfying ( sin ^{-1}|x-2|+cos ^{-1}(1-|3-x|)=frac{pi}{2} )
is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 3 )
( D )
12
235 For ( tan ^{-1}left(frac{1-x}{1+x}right), 0 leq x leq 1 )
What is the sum of the smallest and the
largest values of function.
A ( cdot frac{pi}{4} )
в. ( frac{pi}{2} )
c. ( frac{3 pi}{4} )
D. ( frac{3 pi}{2} )
12
236 Evaluate ( cos ^{-1}left(cos left(frac{pi}{4}right)right) )
A.
B. ( -frac{pi}{4} )
c. ( frac{3 pi}{4} )
D. ( -frac{3 pi}{4} )
12
237 If ( f:left(-frac{pi}{2}, frac{pi}{2}right) rightarrow(-infty, infty) ) is defined
by ( f(x)=tan x, ) then ( f^{-1}(2+sqrt{3})= )
( A cdot frac{pi}{12} )
в.
c. ( frac{5 pi}{12} )
D.
12
238 The value of ( tan left{2 tan ^{-1} frac{1}{5}-frac{pi}{4}right} ) is
A .
B.
( c cdot frac{7}{17} )
D. none of these
12
239 Prove that
( cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)= )
( frac{x}{2} ; x inleft(0, frac{pi}{4}right) )
12
240 Prove the following:
( cos ^{-1}left(frac{12}{13}right)+sin ^{-1}left(frac{3}{5}right)= )
( sin ^{-1}left(frac{56}{65}right) )
12
241 f ( sin left{sin ^{-1} frac{1}{5}+cos ^{-1} xright}=1, ) then ( x )
is equal to
( A cdot 1 )
B.
( c cdot frac{4}{5} )
D.
12
242 The domain of ( sin ^{-1}[x], ) where ( [x] ) is greatest integer function, given by
A ( cdot[-1,1] )
B . [-1,2)
( mathbf{c} cdot{-1,0,1} )
D. None of these
12
243 From the mast head of a ship the angle of depression of a boat is ( tan ^{-1}left(frac{5}{12}right) ) If the mast head is 100 metres. The
distance of the boat from the ship is
( mathbf{A} cdot 120 m )
B. ( 180 m )
c. ( 240 m )
D. None of these
12
244 The principal value of ( sin ^{-1}left{sin frac{5 pi}{6}right} ) is
A ( cdot frac{pi}{6} )
в. ( frac{5 pi}{6} )
c. ( frac{7 pi}{6} )
D. none of these
12
245 Show that:
( sin ^{-1}left(frac{12}{13}right)+cos ^{-1}left(frac{4}{5}right)+ )
( tan ^{-1}left(frac{63}{16}right)=pi )
12
246 17. If tan-‘(x²+3[x] – 4) + cot -‘ (4:+ sin ‘ sin 14) = -, then
the value of sin ‘sin 2x is
a. 6-21
b. 21-6
c. 1-3
c. 3 – 1
12
247 i) Solve for ( x: tan ^{-1}(x-1)+ )
( tan ^{-1} x+tan ^{-1}(x+1)=tan ^{-1} 3 x )
ii) Prove that ( tan ^{-1}left(frac{6 x-8 x^{3}}{1-12 x^{2}}right)- )
( tan ^{-1}left(frac{4 x}{1-4 x^{2}}right)=tan ^{-1} 2 x ;|2 x|< )
( frac{1}{sqrt{3}} )
12
248 ( sin ^{-1}(sin (4))=? ) 12
249 If ( 0 leq x leq 1, ) then ( sin left{tan ^{-1} frac{1-x^{2}}{2 x}+cos ^{-1} frac{1-x^{2}}{1+x^{2}}right} )
equal to
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. none of these
12
250 The formula ( cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right)= )
( 2 tan ^{-1} x ) holds only for
( mathbf{A} cdot x in R )
B . | ( x mid leq 1 )
c. ( x in(-1,1) )
D・ ( x in(0, infty) )
12
251 If ( theta=sin ^{-1} x+cos ^{-1} x-tan ^{-1} x, x geq )
( 0, ) then the smallest interval in which ( theta )
lies is-
( ^{mathrm{A}} cdot frac{pi}{2} leq theta leq frac{3 pi}{4} )
В. ( quad 0 leq theta leq frac{pi}{4} )
( ^{mathrm{c}}-frac{pi}{4} leq theta leq 0 )
( stackrel{pi}{4} leq theta leq frac{pi}{2} )
12
252 Find
( int tan ^{-1} frac{x}{sqrt{a^{2}-x^{2}}} d x ;|x|langle a )
12
253 If ( a x+bleft(sec left(tan ^{-1} xright)right)=c ) and ( a y+ )
( bleft(sec left(tan ^{-1} yright)right)=c, ) then the value of
( frac{boldsymbol{x}+boldsymbol{y}}{mathbf{1}-boldsymbol{x} boldsymbol{y}} ) is,
A ( cdot frac{2 a b}{a^{2}-c^{2}} )
в. ( frac{2 a c}{a^{2}-c^{2}} )
c. ( frac{c^{2}-b^{2}}{a^{2}+b^{2}} )
D. none of these
12
254 2.
The trigonometric equation sin – x=2 sina
has a solution for
[2003]
(a) Jelz te (b)}<lakte
(©) all real values of a (d) lal<
12
255 Solve:
( sin ^{-1}(cos x) )
12
256 Write the principal values of the
following: ( sin ^{-1}left(-frac{1}{2}right)+cos ^{-1}left(-frac{1}{2}right) )
12
257 If ( tan ^{-1} x+tan ^{-1} y=frac{2 pi}{3}, ) then
( cot ^{-1} x+cot ^{-1} y ) is equal to
( ^{A} cdot frac{pi}{2} )
в.
c.
D. ( frac{sqrt{3}}{2} )
E . ( pi )
12
258 6x

Illustration 5.77 If cos-1
find the values of x.
1+ 9×2 –
5 + 2 tan-‘3x, then
12
259 If ( tan alpha=frac{m}{m+1} ) and ( tan beta=frac{1}{2 m+1} )
find the possible values of ( (boldsymbol{alpha}+boldsymbol{beta}) )
( A cdot 30 )
B. 90
( c cdot 60 )
D. ( 45^{circ} )
12
260 Find the value of ( sin ^{-1}left(cos frac{33 pi}{5}right) ) 12
261 Write ( tan ^{-1}left[frac{sqrt{1+x^{2}}-1}{x}right], x neq 0 )
the simplest form.
12
262 1+r2
45. The number of solutions of the equation cos
– cos x = + sin ‘x is
b. 1
c. 2
d. 3
a. 0
12
263 Show that ( sin ^{-1} frac{12}{13}+cos ^{-1} frac{4}{5}+ )
( cot ^{-1} frac{63}{16}=frac{pi}{2} )
12
264 Assertion
If ( boldsymbol{x}^{2}-boldsymbol{p} boldsymbol{x}+boldsymbol{q}=mathbf{0} ) where ( boldsymbol{p} ) is twice the
tangent of the arithmetic mean of
( sin ^{-1} x ) and ( cos ^{-1} x ; ) q is the geometric
mean of ( tan ^{-1} x ) and ( cot ^{-1} x ) then ( x=1 )
Reason
( tan left(sin ^{-1} x+cos ^{-1} xright)=1 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
265 The following graph represents:
( mathbf{A} cdot cot ^{-1}(x+1) )
( B cdot sin ^{-1}left(x^{4}+1right) )
( mathbf{c} cdot tan ^{-1}left(x^{3}+1right) )
( D cdot cos ^{-1}left(x^{2}+1right) )
12
266 Find ( y=left(sin ^{-1} xright)^{x^{2}}, ) then ( y^{prime}(0)=? ) 12
267 If the equation ( sin ^{-1}left(x^{2}+x+1right)+ )
( cos ^{-1}(a x+1)=frac{pi}{2} ) has exactly two
distinct solutions then value of ( a )
could not be

This question has multiple correct options
( A cdot-1 )
B.
( c )
( D )

12
268 ( cos left[2 sin ^{-1} sqrt{frac{1-x}{2}}right]= )
( A cdot x )
B. ( frac{1}{x} )
( c cdot 2 x )
D. 3x
12
269 Find the value of ( sin left(cot ^{-1} xright) ) 12
270 ( sec ^{2}left(tan ^{-1} 2right)+operatorname{cosec}^{2}left(cot ^{-1} 3right) ) is equal
to
( mathbf{A} cdot mathbf{5} )
B. 13
c. 15
D. 6
12
271 Calculate ( (192-214) )
( sin ^{-1}+2 tan ^{-1}(-sqrt{3}) )
12
272 If the equation ( sin ^{-1}left(x^{2}+x+1right)+ )
( cos ^{-1}(lambda x+1)=frac{pi}{2} ) has exactly two
solutions for ( lambda epsilon[a, b) ) then the value of
( (a+b) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
273 Find the value of ( cos ^{-1}left(cos frac{2 pi}{3}right)+ )
( sin ^{-1}left(sin frac{2 pi}{3}right) )
12
274 Find ( sin ^{-1}left(frac{sqrt{mathbf{3}}+1}{2 sqrt{2}}right)= ) 12
275 Let ( cos ^{-1}left(4 x^{3}-3 xright)=a+b cos ^{-1} x )
( x epsilonleft[-frac{1}{2},-1right) ) then the value of ( a+b pi ) is
A ( .2 pi )
в. ( 3 pi )
( c )
D. ( -2 pi )
12
276 Write the value of ( 2 sin ^{-1} frac{1}{2}+ )
( cos ^{1}left(-frac{1}{2}right) )
12
277 Solve for ( boldsymbol{x}: cos ^{-1}left(frac{boldsymbol{x}^{2}-mathbf{1}}{boldsymbol{x}^{2}+mathbf{1}}right)+ )
( frac{mathbf{1}}{mathbf{2}} tan ^{-1} frac{mathbf{2} boldsymbol{x}}{mathbf{1}-boldsymbol{x}^{2}}=frac{mathbf{2} boldsymbol{pi}}{mathbf{3}} )
12
278 Illustration 5.34
Prove that
T
sinu {vl 45.7 -5) 5.5 * 0<x<1
sin-1
+
sin-' x
1,0<x<1
2
12
279 Solve the equation ( tan ^{-1}left[frac{1-x}{1+x}right]= )
( frac{1}{2} tan ^{-1} x,(x>0) )
12
280 15. Let f (x) = sin x + cos x + tan x + sin- ‘ x + cos- ‘ x +
tan- ‘x. Then find the maximum and minimum values of
f(x)
12
281 Find the value of ( x ) which satisfy
euqation ( : tan ^{-1} 2 x+tan ^{-1} 3 x=pi / 4 )
A. ( x=-1 / 6 )
В. ( x=+1 / 6 )
c. ( x=-1 )
D. ( x=+1 )
12
282 If the equation ( sin ^{-1}left(x^{2}+x+1right)+ )
( cos ^{-1}(lambda x+1)=frac{pi}{2} ) has exactly two
solutions, then ( lambda ) can not have the
integral value(s) This question has multiple correct options
A . -1
B. 0
c. 1
D. 2
12
283 Solve:
( cos ^{-1}left(frac{x-x^{-1}}{x+x^{-1}}right) )
12
284 If ( sin ^{-1}left(frac{2 a}{1+a^{2}}right)-cos ^{1}left(frac{1-b^{2}}{1+b^{2}}right)= )
( tan ^{-1}left(frac{2 x}{1-x^{2}}right), ) then what is the value
of ( x ? )
( A cdot frac{a}{b} )
B. ( a b )
( c cdot frac{b}{a} )
D. ( frac{a-b}{1+a b} )
12
285 Find the value of ( x ) for which;
( operatorname{cosec}^{-1}(cos x) ) is real
A. ( x=-pi )
B . ( x=pi )
c. ( x=2 pi )
D. All of the above
12
286 ( cos ^{-1}left(frac{pi}{3}+sec ^{-1}(-2)right)= )
( A cdot-1 )
B.
( c cdot 0 )
D. None of these
12
287 The value of
( sin ^{-1}left[cot left(sin ^{-1} sqrt{left(frac{2-sqrt{3}}{4}right)}right)+cos ^{-1}right. )
is
A . 0
в. ( frac{pi}{4} )
( c cdot frac{pi}{6} )
D. ( frac{pi}{2} )
12
288 62. If cot-x + cot ‘y + cot’z = , x, y, z > 0 and xy < 1,
then x + y + z is also equal to
b. XYZ
1 1 1
a. -+-+-
X Y Z
c. xy + yz + zx
d. none of these
12
289 Prove that ( sin ^{-1}(2 x sqrt{1-x^{2}})= )
( 2 cos ^{-1} x, frac{1}{sqrt{2}} leq x leq 1 )
12
290 ( tan ^{-1}(2)+tan ^{-1}(3)= ) 12
291 Find the principal value of:
( cos ^{-1}left(sin frac{4 pi}{3}right) )
12
292 64. If tan- x + tan ‘y + tan-‘z = “, then
N
a. x + y + z – xyz = 0
c. xy + yz + zx + 1 = 0
to b. x+y+z + xyz = 0 18
d. xy + yz + zx – 1 = 0
12
293 If ( frac{1}{sqrt{2}}<x<1, ) then ( cos ^{-1} x+ )
( cos ^{-1}left(frac{x+sqrt{1-x^{2}}}{sqrt{2}}right) ) is equal to
A ( cdot 2 cos x^{-1} )
B. ( 2 cos ^{-1} x )
( c cdot frac{pi}{4} )
( D )
12
294 5. Which of the following quantities is/are positive?
a. cos(tan-‘(tan 4)) b. sin(cot-‘(cot 4))
c. tan(cos(cos 5)) d. cot(sin-‘(sin 4))
1)
12
295 Illustration 5.53 If x;€ [0, 1] Vi = 1, 2, 3, …, 28 then find
the maximum value of
Vsin x ſcos x2 + ſsin x2 cos xz.
+/sin- xz ſcos x4 + … +& sin ‘ x 28 /costx.
12
296 If ( sin ^{-1} x=frac{pi}{5}, ) for some ( x in(-1,1) )
then find the value of ( cos ^{-1} x )
12
297 7. Let tan “y= tan “x + tan” (1232)
hen a value of y is:
[JEE M 2015)
who
3x – X3
3x + x
1+3×2
1+ 3×2
3x + x
3x – x3
1-
32
m
(d) 1 – 3x²
12
298 2.
Find all the solution of 4 cos xsin x – 2 sinx = 3 sin x
(1983 – 2 Marks
12
299 If ( frac{1}{2} sin ^{-1}left[frac{3 sin 2 theta}{5+4 cos 2 theta}right]=tan ^{-1} x )
then ( boldsymbol{x}= )
( mathbf{A} cdot tan 3 theta )
B. ( 3 tan theta )
c. ( (1 / 3) tan theta )
D. ( 3 cot theta )
12
300 90. The equation 3 cos-“x – Ax-* = 0 has
a. one negative solution
b. one positive solution
c. no solution
d. more than one solution
12
301 16. If sin-la-
+ … + cos'(1 + b + b2 + …) =
then
2a-3
a.
b =
3a-2
b. b=
3a
2a
c. a=2-31
c.
a=
d.
d. a=3–26
2-36
a=
3-26
12
302 65. If x2 + y2 + z2 = r2, then tan-“|
+ tan
(ar)
+ tan-” ) is equal to
To
a.
b.
c. O
d. none of these
12
303 Find the value of ( x ) if
( sin left{sin ^{-1} frac{1}{5}+cos ^{-1} xright}=1 )
12
304 Find the value of ( sin ^{-1}left[sin left(-frac{17 pi}{8}right)right] ) 12
305 The solution of ( sin ^{-1}left(frac{2 a}{1+a^{2}}right)-cos ^{-1}left(frac{1-b^{2}}{1+b^{2}}right)= )
( 2 tan ^{-1} x )
A ( frac{a-b}{1-a b} )
B. ( frac{1+a b}{a-b} )
c. ( frac{a b-1}{a+b} )
D. ( frac{a-b}{1+a b} )
12
306 ( sin ^{-1}left(sin left(frac{2 x^{2}+4}{1+x^{2}}right)right)<pi-3 ) if
A . ( -1 leq x leq 0 )
в. ( 0 leq x leq 1 )
c. ( -1<x1 )
12
307 ( operatorname{Let} f(x)=cos left(tan ^{-1} 2 xright) )
( sin left{tan ^{-1}left(frac{1}{2 x+1}right)right} ) and
( boldsymbol{a}=cos left(tan ^{-1}left(sin left(cot ^{-1} 2 xright)right)right) ) and ( boldsymbol{b}= )
( cos left(frac{pi}{2}+cos ^{-1} 2 xright) )
The value of ( x ) for which ( f(x)=0 ) is
( A cdot-frac{1}{4} )
B. ( frac{1}{4} )
c. 0
D. ( frac{1}{2} )
12
308 Evaluate the following:
( tan ^{-1}(tan 2) )
12
309 Value of ( tan ^{-1}left{frac{sin 2-1}{cos 2}right} ) is
A ( cdot frac{pi}{2}-1 )
B cdot ( 1-frac{pi}{4} )
c. ( 2-frac{pi}{2} )
D. ( frac{pi}{4}-1 )
12
310 ( frac{1}{2} tan ^{-1} x=cos ^{1}left{frac{1+sqrt{1+x^{2}}}{2 sqrt{1+x^{2}}}right}^{frac{1}{2}} ) 12
311 The number of real solution of the
equation ( tan ^{-1} sqrt{x^{2}-3 x+2}+ )
( cos ^{-1} sqrt{4 x-x^{2}-3}=pi ) is
( A )
B. 2
( c cdot c )
D. infinite
12
312 Consider the following:
1. ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{3}{5}=frac{pi}{2} )
2. ( tan ^{-1} sqrt{3}+tan ^{-1} 1=-tan ^{-1}(2+ )
( sqrt{mathbf{3}}) )
Which of the above is/are correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
313 If ( cos ^{-1}left(frac{1}{x}right)=theta ) then the value of ( tan theta )
is
A ( cdot frac{1}{sqrt{x^{2}-1}} )
B. ( sqrt{x^{2}-1} )
c. ( sqrt{1-x^{2}} )
D. ( sqrt{1+x^{2}} )
12
314 Evaluate :
( tan left(2 tan ^{-1} frac{1}{5}right) )
A ( cdot frac{5}{6} )
в. ( frac{5}{12} )
c. ( frac{7}{12} )
D. none of these
12
315 The number of solutions of the equation ( tan ^{-1}left(frac{1}{2 x+1}right)+tan ^{-1}left(frac{1}{4 x+1}right)= )
( tan ^{-1}left(frac{2}{x^{2}}right) ) is
( A )
B.
( c cdot 2 )
D. 3
12
316 Find the value of ( cos ^{-1}left(cos frac{13 pi}{6}right) ) 12
317 Solve:
( tan ^{-1}left(frac{2 x}{1-x^{2}}right) )
12
318 If ( boldsymbol{x}<mathbf{0}, ) then ( tan ^{-1} boldsymbol{x} ) is equal to
This question has multiple correct options
A ( cdot-pi+cot ^{-1} frac{1}{x} )
B. ( sin ^{-1} frac{x}{sqrt{1+x^{2}}} )
c. ( -cos ^{-1} frac{1}{sqrt{1+x^{2}}} )
D. ( -operatorname{cosec}^{-1} frac{sqrt{1+x^{2}}}{x} )
12
319 Evaluate: ( tan ^{-1}(1)+cos ^{-1}left(frac{1}{2}right)+ )
( sin ^{-1}left(frac{1}{2}right) ) which lies in the interval
( [mathbf{0}, boldsymbol{pi}] )
12
320 tan-‘ x tan+ 2x tan- 3x
85. Let tan- 3x tan-‘x tan- 2×1 = 0, then the number
tan 2x tan- 3x tan- x
of values of x satisfying the equation is
a. 1
b. 2
c. 3
d. 4
12
321 Find the value of ( cos ^{-1}left(cos frac{5 pi}{3}right)+ )
( sin ^{-1}left(sin frac{5 pi}{3}right) )
12
322 ( tan left[2 tan ^{-1} frac{1}{5}-frac{pi}{4}right]=? )
( A cdot frac{7}{17} )
в. ( frac{-7}{17} )
( c cdot frac{7}{12} )
D. ( frac{-7}{12} )
12
323 Solve the equation for ( x ) ( sin ^{-1} x+sin ^{-1}(1-x)=cos ^{-1} x, x neq 0 ) 12
324 For the principal value:
( tan ^{-1}left{2 sin left(4 cos ^{-1} frac{sqrt{3}}{2}right)right} )
12
325 If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then find the
value of ( cos ^{-1} x+cos ^{-1} y )
12
326 If ( boldsymbol{alpha} boldsymbol{epsilon}left(-frac{boldsymbol{pi}}{2}, boldsymbol{0}right), ) then find the value of
( tan ^{-1}(cot alpha)-cot ^{-1}(tan alpha) )
12
327 ♡i
69. The sum of series sec-‘ V2 + sec 1 V10 + sec ! V50
(n? + 1)(n? – 2n+2) is
V (n? – n+1)
+…..+ sec-1
a. tan-1
c. tan-‘(n+1)
b. tan ‘n
d. tan-‘(n-1)
12
328 Find the domain of the following
function:
( boldsymbol{f}(boldsymbol{x})=cos ^{-1} sqrt{log [boldsymbol{x}] frac{|boldsymbol{x}|}{boldsymbol{x}}}, ) where, ( [cdot] )
denotes the greatest integer function.
12
329 Solve the following:
( tan left(frac{1}{2} sin ^{-1} frac{3}{4}right) )
12
330 20. 2 tan(tan-‘(x) + tan-‘(x)), where x e R-{-1,1), is
1. is equal
to
2x
a.
b. tan(2 tan-‘x)
od 8
c. tan (cot-‘(-x) – cot-‘(x))o
d. tan(2 cot-‘ x)
.
8
mo
d
12
331 57. If y=tan –+tan-+b, (0<b<1) and 0<ys, then
the maximum value of b is
a. 1/2
b. 1/3
d. 2/3
c.
1/4
12
332 Find the value of ( cos ^{-1}left(cos frac{5 pi}{3}right) ) 12
333 3.
The number of real solutions of
tan x(x+1)+sin – Vx2 + x +1 = n/2 is
(1999 – 2 Marks)
(a) zero (b) one (c) two (d) infinite
12
334 The set of values of ‘ ( x^{prime} ) for which the
formula ( 2 sin ^{-1} x=sin ^{-1}(2 x sqrt{1-x^{2}}) )
is true, is
A. (-1,0)
B. [0,1]
c. ( left[-frac{sqrt{3}}{2}, frac{sqrt{3}}{2}right] )
D. ( left[-frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right] )
12
335 ( cos ^{-1}left(frac{3+5 cos x}{5+3 cos x}right)= )
A ( cdot tan ^{-1}left(frac{1}{2} tan frac{x}{2}right) )
B ( cdot 2 tan ^{-1}left(-frac{1}{2} tan frac{x}{2}right) )
c. ( frac{1}{2} tan ^{-1}left(2 tan frac{x}{2}right) )
D. ( 2 tan ^{-1}left(frac{1}{2} tan frac{x}{2}right) )
12
336 Show that ( 2 tan ^{-1} frac{3}{5}=tan ^{-1} frac{15}{8} ) 12
337 Assertion
STATEMENT 1: Domain of ( tan ^{-1} x ) and
( cot ^{-1} x ) is ( R )
Reason
STATEMENT 2: ( boldsymbol{f}(boldsymbol{x})=tan boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})= )
( cot x ) are unbounded function
A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT1
B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT1
C. STATEMENT1 is TRUE and STATEMENT 2 is FALSE
D. STATEMENT1 is FALSE and STATEMENT 2 is TRUE
12
338 Assertion
Consider ( boldsymbol{f}(boldsymbol{x})=sin ^{-1}left(sec left(tan ^{-1} boldsymbol{x}right)+right. )
( cos ^{-1}left(operatorname{cosec}left(cot ^{-1} xright)right. )
Statement-1: Domain of ( f(x) ) is a
singleton.
Reason
Statement-2: Range of the function ( boldsymbol{f}(boldsymbol{x}) )
is a singleton.
A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
c. Statement- lis true, Statement-2 is false.
D. Statement-1 is false, Statement-2 is true
12
339 The solution set of the equation ( sin ^{-1} sqrt{1-x^{2}}+cos ^{-1} x= )
( cot ^{-1} frac{sqrt{1-x^{2}}}{x}-sin ^{-1} x ) is?
( mathbf{A} cdot[-1,1]-{0} )
в. (0,1]( cup{-1} )
c. [-1,0)( cup{1} )
D. [-1,1]
12
340 If ( cot ^{-1}left(frac{1}{x+1}right)+cot ^{-1}left(frac{1}{x-1}right)= )
( tan ^{-1} 3 x-tan ^{-1} x )
then ( boldsymbol{x}= )
A. ( pm 1 / 2 )
B. ( -1, pm 1 / 3 )
c. 2,±1
D. ( -1 . pm 1 / 2 )
12
341 Assertion ( (A) ) If ( 0<x<frac{pi}{2} ) then
( sin ^{-1}(cos x)+cos ^{-1}(sin x)=pi-2 x )
Reason
( (mathrm{R}) cos ^{-1} x=frac{pi}{2}-sin ^{-1} x forall x in )
( [mathbf{0}, mathbf{1}] )
A. Both ( A ) and ( R ) are true and ( R ) is the correct explanation of ( A )
B. Both A and R are true but R is not correct explanation of ( A )
C. ( A ) is true but Ris false
D. A is false but ( R ) is true
12
342 If ( f(x)=sin ^{-1}left{frac{sqrt{3}}{2} x-frac{1}{2} sqrt{1-x^{2}}right} )
( -frac{1}{2} leq x leq 1, ) then ( f(x) ) is equal to :
( ^{mathbf{A}} cdot sin ^{-1}left(frac{1}{2}right)-sin ^{-1}(x) )
B ( cdot sin ^{-1} x-frac{pi}{6} )
( c cdot sin ^{-1} x+frac{pi}{6} )
D. none of these
12
343 Assertion
( mathrm{f} sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2}, ) then
( frac{3 sum_{r=1}^{2008}left(x^{r}+y^{r}right)}{2 sum_{r=1}^{2008}left(x^{r} y^{r}right)}=3 )
Reason
( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) is
possible only if ( boldsymbol{x}=boldsymbol{y}=boldsymbol{z}=mathbf{1} )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
12
344 Evaluate the following:
( cos ^{-1}(cos 5) )
12
345 Find the principal values of ( sin ^{-1}left(-frac{1}{sqrt{2}}right) ) 12
346 Illustration 5.29
Simplify sin cot’ tan cos’x, x > 0.
12
347 f ( 0 leq x leq 1, ) then ( tan left{frac{1}{2} sin ^{-1} frac{2 x}{1+x^{2}}+frac{1}{2} cos ^{-1} frac{2 x}{1+x^{2}}right} )
( A )
B.
c. ( frac{2 x}{1+x^{2}} )
( D )
12
348 The sum of the solution of the equation ( 2 sin ^{-1} sqrt{x^{2}+x+1}+ )
( cos ^{-1} sqrt{x^{2}+x}=frac{3 pi}{2} ) is
( mathbf{A} cdot mathbf{0} )
B. – 1
( c cdot 1 )
( D )
12
349 Find the principal value of:
( sin ^{-1}left(-frac{sqrt{3}}{2}right)-2 sec ^{-1}left(2 tan frac{pi}{6}right) )
12
350 ( cos ^{-1}(44 / 125) ) is equal to
A ( .2 alpha )
в. ( 3 alpha )
c. ( pi-3 alpha )
D. ( pi-2 alpha )
12
351 1. Solve 2 cos + s = sinº (23 W1–?)
1. Solve
= sin
12
352 Find the principal value:
( tan ^{-1}left(-frac{1}{sqrt{3}}right) )
12
353 Illustration 5.9 If sin-‘(x2 – 4x + 5) + cos='(02- 2y + 2) =
then find the value of x and y.
12
354 Evaluate the following:
( sin ^{-1}(sin 2) )
12
355 35. The value of 2 tan-‘ (cosec tan ‘x – tan cot ‘x) is equal to
a. cot’ x
b. cot!!
c. tan-‘x
d. none of these
12
356 The number of solutions of the equation
( tan ^{-1}(x-1)+tan ^{-1}(x)+tan ^{-1}(x+ )
1) ( =tan ^{-1}(3 x) ) is :
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. 4
12
357 llustration 5.20
Solve cos'(cos x) > sin
(sin x),x € [0,21].
12
358 Let ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}+cos boldsymbol{x}+tan boldsymbol{x}+ )
( arcsin x+arccos x+arctan x . ) If ( mathrm{M} ) and
( mathrm{m} ) are maximum and minimum values
of ( f(x) ) then their arithmetic mean is
equal to
A ( cdot frac{pi}{2}+cos 1 )
B . ( frac{pi}{2}+sin 1 )
C ( cdot frac{pi}{4}+tan 1+cos 1 )
D. ( frac{pi}{4}+tan 1+sin 1 )
12
359 ff ( quad A=tan ^{-1}left(frac{x sqrt{3}}{2 k-x}right) ) and ( B= )
( tan ^{-1}left(frac{2 x-k}{k sqrt{3}}right) . ) Then, ( A-B ) is equal
to
A ( cdot frac{pi}{2} )
в.
c.
D. None of these
12
360 The value ( csc left(cos ^{-1}(-12 / 13)right) ) is?
( mathbf{A} cdot+12 / 5 )
в. ( -12 / 5 )
( mathbf{c} cdot+13 / 5 )
D. ( -13 / 5 )
12
361 Illustration 5.54
Prove that cos– + cos!
13
-cos-1 33
5
12
362 Consider the function ( boldsymbol{y}= )
( log _{a}(x+sqrt{x^{2}+1}), a>0, a neq 1 . ) The
inverse of the function
A. does not exist
B cdot is ( x=log _{a}(y+sqrt{y^{2}+1}) )
( mathbf{c} cdot ) is ( x=sin (y ln a) )
D ( quad ) is ( x=cosh left(-y ln frac{1}{a}right) )
12
363 The value of ( cos ^{-1}left(cos frac{5 pi}{3}right)+ )
( sin ^{-1}left(sin frac{5 pi}{3}right) ) is
A ( cdot frac{pi}{2} )
В. ( frac{5 pi}{2} )
( c cdot frac{10 pi}{2} )
D.
12
364 Let ( boldsymbol{f}:[mathbf{0}, boldsymbol{4} boldsymbol{pi}] rightarrow[mathbf{0}, boldsymbol{pi}] ) be defined by
( f(x)=cos ^{-1}(cos x) . ) The number of
points ( boldsymbol{x} in[mathbf{0}, mathbf{4} boldsymbol{pi}] ) satisfying the
equation ( f(x)=frac{10-x}{10} ) is
12
365 80. If the equation x3 + bx2 + cx + 1 = 0, (b<c), has only
one real root a, then the value of 2 tan-' (cosec a) +
tan-' (2 sina sec-a) is
a. -T
b. – – –
52
odd. a com
12
366 12. If sin-‘x = 0 + B and sin-y= 0-B, then 1 + xy is equal
to
a. sin? e + sin?
B b . sin? 8+ cos2 B3
c. cos? 0+ cos? O d. cos² + sin? ß
12
367 1 + x
cos- x
Prove that cos-
Illustration 5.32
-1<x< 1.
12
368 Solve:( tan ^{-1}left(frac{6 x}{1-8 x^{2}}right) ) 12
369 Write the function in the simplest form:
( tan ^{-1} frac{1}{sqrt{x^{2}-1}},|x|>1 )
12
370 The value of ( 2 tan ^{-1}(-2) ) is equal to
This question has multiple correct options
( ^{mathbf{A}} cdot sin ^{-1}left(-frac{4}{5}right) )
B. ( -sin ^{-1}left(frac{4}{5}right) )
( mathbf{C} cdot sin ^{-1}left(frac{4}{5}right)-pi )
D ( -cos ^{-1}left(frac{4}{5}right)-frac{pi}{2} )
12
371 Solve the equation for ( x: sin ^{-1} frac{5}{x}+ ) ( sin ^{-1} frac{12}{x}=frac{pi}{2}, x neq 0 ) 12
372 Find the principal value of:
( sec ^{-1}(2) )
12
373 The range of the function ( f(x)= )
( sin ^{-1}left(x^{2}-2 x+2right) )
( A cdot phi )
B. ( left[-frac{pi}{2}, frac{pi}{2}right] )
c. ( frac{pi}{2} )
D. none of these
12
374 ( fleft(frac{1}{2 i^{2}}right)=t, ) then ( tan t ) equals 12
375 The set for which ( 2 cos ^{-1} x= )
( cos ^{-1}left(2 x^{2}-1right) ) is valid is
( mathbf{A} cdot x in[0,1] )
B ( cdot x in(0,1) )
( mathbf{c} cdot x in[0,1) )
D. ( x in(0,1] )
12
376 The value of ( sin left(tan ^{-1} x+cot ^{-1} xright) ) is 12
377 The value of ( sin ^{-1}left(sin 2010^{0}right)+ )
( cos ^{-1}left(cos 2010^{0}right)+tan ^{-1}left(tan 2010^{0}right) ) is
( A cdot frac{pi}{6} )
B .
( mathbf{c} cdot frac{2 pi}{3} )
D. ( frac{5 pi}{6} )
12
378 Illustration 5.49 If sec !x= cosecly, then find the value of
cos –
– + cos-
12
379 Value of ( x ) for which ( cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right)= )
( 2 tan ^{-1} x ) satisfied is ( x epsilon[a, infty) )
Find the value of ( a )
A ( . a=-infty )
B . ( a=-1 )
( mathbf{c} cdot a=0 )
D. ( a=1 )
12
380 The value of
( lim _{|x| rightarrow infty} cos left(tan ^{-1}left(sin left(tan ^{-1} xright)right)right) ) is
equal to
A . -1
B. ( sqrt{2} )
c. ( -frac{1}{sqrt{2}} )
D. ( frac{1}{sqrt{2}} )
12
381 Illustration 5.11 If cos’2+ cos’u + cos-‘ y=31, then find
the value of λμ + μγ+ γλ.
12
382 If ( sin ^{-1} frac{x}{5}+operatorname{cosec}^{-1} frac{5}{4}=frac{pi}{2}, ) then ( x ) is
equal to:
( A )
B. 4
( c cdot 3 )
D. 5
12
383 Illustration 5.69
Ifx e
then show that
cos” (3 (1 + cos2x) + Vísinºx – 48 cos?r) sin x)
= x – cos-‘(7 cos x)
12
384 Write the following in simplest form:
( tan ^{-1}left(frac{sqrt{left(1+x^{2}right)}-1}{x}right) )
12
385 Find the principal value of:
( sin ^{-1}left(cos frac{3 pi}{4}right) )
12
386 The principal value of ( tan ^{-1}left(cot frac{43 pi}{4}right) ) is
A ( cdot-frac{3 pi}{4} )
B. ( frac{3 pi}{4} )
( c cdot-frac{pi}{4} )
( D cdot frac{pi}{4} )
12
387 Find the real solution of the equation ( tan ^{-1} sqrt{x(x+1)}+sin ^{-1} sqrt{x^{2}+x+1}= ) 12
388 Itsin-1 26
anx, then x is
76. If sin-11
– 2a)
(1+a?)
equal to [a, b e (0, 1)]
(1+6²
o ab
b. ltab
1 + ab
1+ ab
b
atb
1- ab
1- ab
12
389 36. If tan-1 V1 + x2 – 1
– = 4°, then
a. x= tan 2°
c. x =tan(1/4)
b. x = tan 4°
d. x = tan 8°
12
390 If cos(2 sin-‘x) = -, then find the values
Illustration 5.23
of x.
12
391 Solve ( : tan ^{-1}left(frac{x-1}{x-2}right)+ )
( tan ^{-1}left(frac{x+1}{x+2}right)=frac{pi}{4} )
12
392 For all values of ( x, ) the values of ( 3- ) ( cos x+cos left(x+frac{pi}{3}right) ) lie in the interval
A ( .[-2,3] )
B. [-2,1]
c. [2,4]
D. [1,5]
12
393 9. For the equation cos-x + cos2x + =0, the number of
real solution is
Toita. 1 dl olb. 2 10 9gan de
c. O
d. ) ) 200
12
394 21. Complete solution set of tan”(sin’x) > 1 is
– (-a) 6) (Ta’ tal-c03
c. (-1,1) – {0}
d. None of these
12
395 The range of values of p for which the
equation ( sin cos ^{-1}left(cos left(tan ^{-1} xright)right)=p )
has a solution is
( A cdotleft(-frac{1}{sqrt{2}}, frac{2}{sqrt{2}}right) )
в. [0,1)
c. ( left(frac{1}{sqrt{21}}right) )
D. (-1,1)
12
396 If ( tan ^{-1}(x+1)+tan ^{-1}(x-1)= )
( tan ^{-1}left(frac{8}{31}right), ) then ( x ) is equal
( A cdot frac{1}{2} )
B. ( -frac{1}{2} )
( c cdot frac{1}{4} )
D.
12
397 If ( cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)=frac{x}{m}, x in )
( left(0, frac{pi}{4}right) ).Find ( m )
12
398 Solve:
( cos ^{-1}left(sin frac{4 pi}{3}right) )
A. ( -frac{5 pi}{6} )
в.
( c cdot frac{7 pi}{6} )
D. ( frac{5 pi}{6} )
12
399 The number of integral values of k for which the equation ( sin ^{-1} x+ )
( tan ^{-1} x=2 k+1 ) has a solutions is:
A . 1
B. 2
( c .3 )
D. 4
12
400 Number of real value of ( x ) satisfying the equation, arctan ( sqrt{x(x+1)}+ ) ( arcsin sqrt{x(x+1)+1}=frac{pi}{2} ) is
A .
B.
( c cdot 2 )
D. more than 2
12
401 f ( tan ^{-1}left(frac{x-1}{x-2}right)+tan ^{-1}left(frac{x+1}{x+2}right)= )
4
then ( x ) is
12
402 The number of real solutions of the
equation ( tan ^{-1} sqrt{x(x+1)}+ )
( sin ^{-1} sqrt{x^{2}+x+1}=frac{pi}{2} ) is
A. One
B. Four
c. Two
D. Infinitely many
12
403 ( sin ^{-1} sin 15+cos ^{-1} cos 20+ )
( tan ^{-1} tan 25=? )
A . 1.04719754
в. 11.04719754
c. 111.04719754
D. 1111.04719754
12
404 Simplify ( tan ^{-1} sqrt{2}-cot ^{-1}(1 / sqrt{2}) ) 12
405 If ( tan left(2 tan ^{-1}left(frac{1}{5}right)-frac{pi}{4}right)=-frac{lambda}{17}, ) then ( lambda ) is
equal to
12
406 Illustration 5.65
Find the value of cot- – + sin
13
12
407 The principal value of ( cos ^{-1}left(-sin frac{7 pi}{6}right. )
( A cdot frac{5 pi}{3 pi} )
B. ( frac{7 pi}{6} )
( c cdot frac{pi}{3} )
D. none of these
12
408 The value of sin
( sin left{tan ^{-1}left(tan frac{7 pi}{6}right)+cos ^{-1}left(cos frac{7 pi}{3}right)right} ) is
A .
в.
( c cdot-1 )
D. None of these
12
409 The number of real solutions of the
equation
[
begin{array}{l}
sin ^{-1}left(sum_{i=1}^{infty} x^{i+1}-x sum_{i=1}^{infty}left(frac{x}{2}right)^{i}right)=frac{pi}{2}- \
cos ^{-1}left(sum_{i=1}^{infty}left(-frac{x}{2}right)^{i}-sum_{i=1}^{infty}(-x)^{i}right) text { lying in }
end{array}
]
the interval ( left(-frac{1}{2}, frac{1}{2}right) ) is
12
410 Illustration 5.62 Find the value of
tan
(l+rtph
r=0
12
411 Find the principal value:
( tan ^{-1}left(frac{1}{sqrt{3}}right) )
12
412 Illustration 5.1 Find the principal value of the following:
(i) cosec-‘(2) (ii) tan-‘ (-13)
(ii) cos( – – –
12
413 Find tan-
in terms of sin,
Illustration 5.28
where x e (0, a).
ſa² – x
12
414 Illustration 5.47
Solve sin- –
+ sin
Nia
12
415 Illustration 5.44
Solve
12
416 Illustration 5.39
Find the range of y = (cot- x)(cot ‘(-x)).
12
417 illustration 5.76 ut sin (za) + 2 tun (13)
Illustration 5.76
If sin-
+ 2 tan-
N18
x² + 4
independent of x, find the values of x.
12
418 For the principal value:
( sin ^{-1}left(-frac{1}{2}right)+2 cos ^{-1}left(-frac{sqrt{3}}{2}right) )
12
419 The value of
( cos left[frac{1}{2} cos ^{-1}left[cos left[-frac{14 pi}{5}right]right]right] ) is/are –
This question has multiple correct options
( ^{A} cdot cos left[-frac{7 pi}{5}right] )
B. ( sin left[frac{pi}{10}right. )
( ^{mathbf{c}} cdot cos left[frac{2 pi}{5}right] )
D. ( -cos left[frac{3 pi}{5}right] )
12
420 Prove that ( tan ^{-1} frac{63}{16}=sin ^{-1} frac{5}{13}+ )
( cos ^{-1} frac{3}{5} )
12
421 Write the following into simple test form;
(1) ( sin left{2 tan ^{-1} sqrt{frac{1-n}{1+n}}right} )
12
422 If ( cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=pi ), then
prove that ( x^{2}+y^{2}+z^{2}+2 x y z=1 )
12
423 Illustration 5.4
Solve for x if (cot- x)2 – 3 (cot-‘x) +2>0.
12
424 Show that:
[
tan ^{-1}left(frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}right)=frac{pi}{4}
]
( frac{1}{2} cos ^{-1} x,-frac{1}{sqrt{2}} leq x leq 1 )
12
425 ( tan left(2 cos ^{-1} frac{3}{5}right)= )
( A cdot frac{8}{3} )
B . ( frac{24}{25} )
( c cdot frac{7}{25} )
( D cdot frac{-24}{7} )
12
426 31. The value of
tan(sin-‘(cos(sin-+ x))) tan(cos-‘(sin (cosx))),
where x € (0, 1), is equal to
a. 0 Obt n b. 1
c. -1
d. none of these
12
427 Illustration 5.56
Find the value of tan-1
tan 2A
+
I
tan (cot A) + tan-‘(cot? A), for 0 <A<*.
12
428 The value of ( sec ^{-1}left(sec frac{8 pi}{5}right) ) is
A ( cdot frac{2 pi}{5} )
в. ( frac{3 pi}{5} )
( c cdot frac{8 pi}{5} )
D. none of these
12
429 The value of ( tan ^{-1}left(frac{1}{2} tan 2 Aright)+ )
( tan ^{-1}(cot A)+tan ^{-1}left(cot ^{3} Aright) ) for ( 0< )
( boldsymbol{A}<frac{boldsymbol{pi}}{boldsymbol{4}} ) is?
( mathbf{A} cdot 4 tan ^{-1} 1 )
B. ( 2 tan ^{-1} 2 )
( c cdot 0 )
D. None
12
430 Illustration 5.27 Solve sin-‘(1 – x) – 2 sin-‘x = 12
431 8. If cos” ()+cos (23) = (**), then
OS
x is equal to:
JEEM 2019-9 Jan (M)
(6) 1145
12
(0) VIAG
12
(a) Vi45
11
12
432 The set of values of parameter ( a ) so that the equation ( left(sin ^{-1} xright)^{3}+left(cos ^{-1} xright)^{3}= )
( a pi^{3} ) has a solution.
( mathbf{A} cdotleft[frac{-1}{32}, frac{7}{8}right] )
в. ( left[frac{1}{32},, frac{9}{8}right] )
( ^{mathbf{c}} cdotleft[0, frac{7}{8}right] )
D. ( left[frac{1}{32}, frac{7}{8}right] )
12
433 Find the value of
( tan ^{-1}left[2 cos left(2 sin ^{-1} frac{1}{2}right)right] )
12
434 ( tan left(cot ^{-1} xright)=cot left(tan ^{-1} xright) ) 12
435 78. Ifx = 2 tang 4 xy=sin” –
where xe (0, 0),
then xı + x2 is equal to
a. 0
c.
b. 21
d. none of these
12
436 ( tan ^{-1}left(frac{5-x}{6 x^{2}-5 x-3}right) ) 12
437 2. 2 tan- ‘(- 2) is equal to
a. –
b. – 1+ cos-1
5
c.
– –
2
+ tan-1
d. – T + cot-
Cot-1
12
438 Evaluate:
( sum_{r=1}^{infty} tan ^{-1}left(frac{2}{1+(2 r+1)(2 r-1)}right) )
12
439 Evaluate ( tan left[frac{1}{2} cos ^{-1}left(frac{3}{sqrt{11}}right)right] ) 12
440 If ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8}, ) then ( x )
A . -1
B.
c. 1
D. 2
12
441 The value of
( lim _{|x| rightarrow infty} cos left(tan ^{-1}left(sin left(tan ^{-1} xright)right)right) ) is equal
to
A . -1
B. ( sqrt{2} )
( c cdot-frac{1}{sqrt{2}} )
D. ( frac{1}{sqrt{2}} )
12
442 ( tan ^{-1}left[frac{cos x}{1+sin x}right] ) is equal to
A ( cdot frac{pi}{4}-frac{x}{2}, ) for ( x epsilonleft(-frac{pi}{2}, frac{3 pi}{2}right) )
B. ( frac{pi}{4}-frac{x}{2}, ) for ( x inleft(-frac{pi}{2}, frac{pi}{2}right) )
( ^{mathbf{C}} cdot frac{pi}{4}-frac{x}{2}, ) for ( x inleft(frac{3 pi}{2}, frac{5 pi}{2}right) )
D ( cdot frac{pi}{4}-frac{x}{2}, ) for ( x inleft(-frac{3 pi}{2},-frac{3 pi}{2}right) )
12
443 Find the principle value of: ( cos ^{-1}left(-frac{1}{2}right) ) 12
444 10. If p > q> 0 and pr <- l< qr, then find the value of
tan-1 P-9 + tan-1 9-" + tan-1 " – p.
1 + pq
1 +gr
1 +rp
12
445 10
10
Illustration 5.52
Find the value of
tan ºr
r=1s=1
12
446 The value of ( sin ^{-1}left(cos frac{33 pi}{5}right) ) is
A ( cdot frac{3 pi}{5} )
в. ( frac{7 pi}{5} )
c. ( frac{pi}{10} )
D. ( -frac{pi}{10} )
12
447 If ( sin ^{-1} x+sin ^{-1} y=frac{pi}{2} ) and ( sin 2 x= )
( cos 2 y, ) then
This question has multiple correct options
A ( cdot x=frac{pi}{8}+sqrt{frac{1}{2}-frac{pi^{2}}{64}} )
B. ( y=sqrt{frac{1}{2}-frac{pi^{2}}{64}}-frac{pi}{12} )
c. ( _{x}=frac{pi}{12}+sqrt{frac{1}{2}-frac{pi^{2}}{64}} )
D. ( y=sqrt{frac{1}{2}-frac{pi^{2}}{64}}-frac{pi}{8} )
12
448 6. Ifx <0, then tan x is equal to
lon
a. – 1 + cot-1
I
a.
b.
– T + cot
x
sin-
v1 + x
–1
d.
– cosec
12
449 If ( frac{(x+1)^{2}}{x^{3}+x}=frac{A}{x}+frac{B x+C}{x^{2}+1}, ) then
( sin ^{-1} A+tan ^{-1} B+sec ^{-1} C= )
A ( cdot frac{pi}{2} )
B.
( c cdot c )
D. ( frac{5 pi}{6} )
12
450 Prove that:
( tan ^{-1} frac{1}{4}+tan ^{-1} frac{2}{9}=frac{1}{2} cos ^{-1} frac{3}{5} )
12
451 Given that ( 0 leq x leq frac{1}{2} ) the value of
( tan left[sin ^{-1}left(frac{x}{sqrt{2}}+sqrt{frac{1-x^{2}}{2}}right)-sin ^{-1} xright. )
is
( A )
B.
( c cdot 1 sqrt{3} )
D. ( sqrt{3} )
12
452 5. Which of the following quantities is/are positive?
a. cos(tan-‘(tan 4)) b. sin(cot-‘(cot 4))
c. tan(cos(cos 5) d. cot(sin-‘(sin 4))
12
453 Solve :
( cot ^{-1}left(frac{sqrt{1-sin x}+sqrt{1+sin x}}{sqrt{1-sin x}-sqrt{1+sin x}}right) )
A ( cdot_{pi}-frac{x}{2} )
в. ( _{pi+frac{x}{2}} )
c. ( frac{x}{2} )
D . ( 2 pi )
12
454 Find the value of ( x ) which satisfy
equation ( : 2 tan ^{-1}(cos x)= )
( tan ^{-1}(2 csc x) )
A ( cdot x=n pi+frac{pi}{4} )
B cdot ( x=n pi-frac{pi}{4} )
c. ( x=n pi+frac{pi}{2} )
D. ( x=n pi-frac{pi}{2} )
12
455 If ( theta=sin ^{-1} x+cos ^{-1} x-tan ^{-1} x, 1 leq )
( x<infty, ) the smallest interval in which ( theta )
lies is
12
456 If ( cos ^{-1} x-sin ^{-1} x=0, ) then ( x ) is equal
to-
( ^{mathrm{A}} pm frac{1}{sqrt{2}} )
B. 1
( ^{mathrm{c}} pm frac{1}{sqrt{3}} )
D. ( frac{1}{sqrt{2}} )
12
457 ( mathbf{s} frac{sin ^{-1}}{cos ^{-1}}=tan ^{-1} ) a valid relation? 12
458 1 COS
The value of tam cor” ()un (1) –
n (1) Bosch (a) none
(1983-1 Mark)
12
459 Illustration 5.6 Find the values of
(i) sin’ (2) (ii) cos-1 Vx– x+1
(iii) tan-
(iv) sec
1+x4
(m) tant
(iv) see “(x+)
X
12
460 If ( cos ^{-1}left(frac{x}{2}right)+cos ^{-1}left(frac{y}{3}right)=theta, ) if ( 9 x^{2}- )
( 12 x y cos theta+4 y^{2}=m sin ^{2} theta . ) Find ( m . )
12
461 Write the simplest form of ( tan ^{-1}left(frac{cos x-sin x}{cos x+sin x}right), 0<x<frac{pi}{2} ) 12
462 Find the range of ( boldsymbol{f}(boldsymbol{x})= ) ( cos ^{-1} sqrt{log _{[x]} frac{|x|}{x}}, ) where [.]denotes the
greatest integer.
A ( cdotleft{frac{pi}{2}right} )
в. ( left{frac{pi}{4}right} )
( c cdotleft{frac{pi}{6}right} )
D. ( left{frac{pi}{8}right} )
12
463 There is flag-staff at the top of 10 metres high tower. If the flag-staff
makes an angle ( tan ^{-1}(1 / 8) ) at a point
24 metres away from the tower, then the
height of the flag staff in metres is
A ( .26 / 7 )
B. 27/8
c. ( 27 / 6 )
D. 26 /3
12
464 Match the column 12
465 If ( tan ^{-1} y=4 tan ^{-1} x, ) then ( frac{1}{y} ) is zero for
A ( . x=1 pm sqrt{2} )
B. ( x=sqrt{2} pm sqrt{3} )
c. ( x=3 pm 2 sqrt{2} )
D. all value of ( x )
12
466 Find the principal value of:
( tan ^{-1}(-1)+cos ^{-1}left(-frac{1}{sqrt{2}}right) )
12
467 Solve ( sin ^{-1}left{frac{sin x+cos x}{sqrt{2}}right},-frac{3 pi}{4}< )
( x<frac{pi}{4} )
12
468 Find the principal value of the following:
( sec ^{-1}(-2) )
12
469 5.
The value of cot( cosec
t an
12
470 A value of ( x ) satisfying the equation
( sin left[cot ^{-1}(1+x)right]=cos left[tan ^{-1} xright], ) is.
A. ( -frac{1}{2} )
B. ( frac{1}{2} )
c. -1
( D )
12
471 56. = tan’ (2 tan’) – tan-
– tan 0 then tan
=
a. 2
c. 2/3
b. -1
d. 2
12
472 Show that ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is
constant for ( x geq 1 . ) Also find that
constant.
12
473 If ( cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=3 pi )
then ( x y+y x+z x ) is equal to
A . 1
B.
( c .-3 )
D. 3
12
474 Number of solutions of the equation ( sin left(frac{1}{3} cos ^{-1} xright)=1 ) are
A. only one
B. no solution
c. only three
D. at least two
12
475 If ( left(sin ^{-1} xright)^{2}+left(sin ^{-1} yright)^{2}+left(sin ^{-1} zright)^{2}= )
( frac{3 pi^{2}}{4}, ) then find the minimum value of
( boldsymbol{x}+boldsymbol{y}+boldsymbol{z} )
12
476 Illustration 5.14 Find x satisfying (tan-‘x] + [cot–x] = 2,
where [:] represents the greatest integer function.
12
477 There exists a positive real number ( x )
satisfying ( cos left(tan ^{-1} xright)=x ). The value of ( cos ^{-1}left(frac{x^{2}}{2}right) ) is
A ( cdot frac{pi}{10} )
в.
c. ( frac{2 pi}{5} )
D. ( frac{4 pi}{5} )
12
478 The solution of the equation ( 2 cos ^{-1} x=sin ^{-1}(2 x sqrt{1-x^{2}}) )
A ( cdot[-1,0] )
B ( cdot[0,1] )
c. [-1,1]
D. ( left[frac{1}{sqrt{2}}, 1right] )
12
479 Write the given trigonometric expression in its simplest form. ( tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right), a>0 ; frac{-a}{sqrt{3}} leq )
( boldsymbol{x} leq frac{boldsymbol{a}}{sqrt{mathbf{3}}} )
12
480 Prove: ( 2 sin ^{-1} frac{3}{5}=tan ^{-1} frac{24}{7} ) 12
481 If sin sin-
Illustration 5.43
value of x.
???
+ cos-‘x = 1, then find the
12
482 Prove that ( : sin ^{-1}left(frac{5}{13}right)+ )
( cos ^{-1}left(frac{4}{5}right)=frac{1}{2} sin ^{-1}left(frac{3696}{4225}right) )
12
483 If ( tan ^{-1}(1+x)+tan ^{-1}(1-x)=frac{pi}{2} )
then ( x=? )
A .
B. –
c.
D.
12
484 A tower stands at the top of a hill whose height is three times the height of the tower. The tower is found to subtend an
angle of ( mid tan ^{-1}(1 / 7) ) at a point ( 2 k m ) away on the horizontal throught the foot of the hill. Then the height of the tower is
( ^{mathbf{A}} cdot frac{1}{2} k m ) or ( frac{1}{3} k m )
B. ( frac{1}{3} k m ) or ( frac{2}{3} k m )
c. ( frac{2}{3} k m ) or ( frac{1}{2} k m )
D. ( frac{3}{4} k m ) or ( frac{1}{2} k m )
12
485 4:(**”)un jo sguvą 51 12
486 The value of
( left{s i n^{-1}left[x^{2}+frac{1}{2}right]+cos ^{-1}left[x^{2}-frac{1}{2}right]right} )
where {} and ( [.] ) denote fractional part
function and greatest integer function respectively
( mathbf{A} cdot pi-3 )
B. ( 4-pi )
( c cdot 2-frac{pi}{2} )
D. o
12
487 Find the number of terms of the AP
( 121,117,113, dots,-3 ? )
A . 32
B. 30
c. 28
D . 26
12
488 Illustration 5.72 If f(x) = sin-‘ x then prove that
lim f(3x – 4x?) = 1 – 3 lim sin-‘ x.
12
489 If ( a<frac{1}{32}, ) then the number of solutions
of ( left(sin ^{-1} xright)^{3}+left(cos ^{-1} xright)^{3}=a pi^{3}, ) is
( A cdot 0 )
B.
( c cdot 2 )
D. infinite
12
490 Find the inverse of the following
functions:
( boldsymbol{f}(boldsymbol{x})=sin ^{-1}left(frac{boldsymbol{x}}{mathbf{3}}right), boldsymbol{x} in[-mathbf{3}, mathbf{3}] )
( operatorname{then} boldsymbol{f}^{-1}(boldsymbol{pi} / mathbf{2}) )
12
491 Consider the function ( boldsymbol{f}(boldsymbol{x})= )
( sin left(sin ^{-1} 2 xright)+sec left(sec ^{-1} 3 xright)+ )
( tan left(tan ^{-1} 4 xright) ) and ( g(x)=9 x, ) then
which of the following is correct? This question has multiple correct options
A ( cdot ) fog ( (x) ) and ( g o f(x) ) are equal function
B . ( f(x) ) is an odd function
C. Number of integers in range of ( f(x) ) are 4
D. Number of integers in domain of ( f(x) ) are 2
12
492 If ( sin ^{-1}(1-x)-2 sin ^{-1} x=frac{pi}{2}, ) then ( x )
is equal to
A ( cdot 0, frac{1}{2} )
в. ( _{1, frac{1}{2}} )
( c cdot frac{1}{2} )
D.
12
493 If ( x=cos ^{2}left(tan ^{-1}left(sin left(cot ^{-1} 3right)right)right), ) then
( 1331 x^{3}-3630 x^{2}+3300 x+7369= )
( m ) then find the sum of the second and
third digits of ( boldsymbol{m} )
12
494 Solve the equation: ( tan ^{-1} x+ ) ( 2 cot ^{-1} x=frac{2 pi}{3} ) 12
495 What is ( tan ^{-1}left(frac{1}{2}right)+tan ^{-1}left(frac{1}{3}right) ) equa
to?
A ( cdot frac{pi}{2} )
в. ( frac{pi}{3} )
c. ( frac{pi}{4} )
D.
12
496 (2008)
6. If0<x<1, then
VI+x? [{x cos (cot" x) + sin (cot 1 x)}2 – 1]1/2 =
(2) vita
(b) x
(c) x V1 + x²
(d) V1 + x²
12
497 If we consider only the principle values of the inverse
trigonometric functions then the value of
tan cos
SUD
(1994)
12
498 Find the principal value of:
( operatorname{cosec}^{-1}left(2 tan frac{11 pi}{6}right) )
12
499 Illustration 5.57 Simplify
3 sin 2a
| 5 + 3 cos 2a
tan a

+ tan-1
1, where –
I < a < 1 2
12
500 Which one of the following quantities is negative?
A ( cdot cos left(tan ^{-1}(tan 4)right) )
B. ( sin left(cot ^{-1}(cot 4)right) )
c. ( tan left(cos ^{-1}(cos 5)right) )
D・cot ( left(sin ^{-1}(sin 4)right) )
12
501 Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y}
]
( x y1 )
(a) If ( tan ^{-1} frac{sqrt{1+x^{2}}-sqrt{1-x^{2}}}{sqrt{left(1+x^{2}right)}+sqrt{left(1+x^{2}right)}}= )
( alpha . ) then prove that ( x^{2}=sin 2 alpha )
(b) If ( frac{operatorname{mtan}(alpha-theta)}{cos ^{2} theta}=frac{n tan theta}{cos ^{2}(alpha-theta)} )
then prove that ( boldsymbol{theta}= ) ( frac{1}{2}left[alpha-tan ^{-1}left(frac{n-m}{n+m} tan alpharight)right] )
(c) ( cos ^{-1} frac{cos alpha+cos beta}{1+cos alpha cos beta}= )
( 2 tan ^{-1}left(tan frac{alpha}{2} tan frac{beta}{2}right) )
12
502 Evaluate ( sin ^{-1}(cos x) ) 12
503 If value of ( x ) which satisfy equation ( sin ^{-1} frac{2 x}{1+x^{2}}=tan ^{-1} frac{2 x}{1-x^{2}}, ) is ( -a< )
( boldsymbol{x}<boldsymbol{b} )
Find the value of ( a+b )
A . -1
B.
c. 1
D.
12
504 If the sum of maximum and minimum
values of ( boldsymbol{E}=left(sin ^{-1} xright)^{2}+ )
( 2 pi cos ^{-1} x+pi^{2} ) is ( frac{a pi^{2}}{b}, ) where ( a ) and ( b )
are coprime, then the value of ( (a-b) ) is
12
505 ( $ $ operatorname{lint}|,| operatorname{left}left[x sin ^{wedge}{-1} x mid, text { lleft } mid{1right. )
( left|operatorname{sqrt}left{left(1|,-|, mathbf{x}^{wedge} 2right}|,| r i g h t midright} wedge{-1}right| )
|right ( ] wedge mid, $ $ d x )
12
506 f ( tan ^{-1}left(frac{x-1}{x-2}right)+tan ^{-1}left(frac{x+1}{x+2}right)= )
find the value of ( x )
( overline{mathbf{4}} )
12
507 Prove that ( frac{pi}{2}-sin ^{-1} x= )
( operatorname{cosec}^{-1} frac{1}{sqrt{1-x^{2}}} )
12
508 ff ( y=left(tan ^{-1} xright)^{2} ) then show
( left(x^{2}+1right)^{2} y_{2}+2 xleft(x^{2}+1right) y_{1}=2 )
12
509 Illustration 5.68
Solve
12
510 If the value of ( tan ^{-1}left(tan frac{3 pi}{4}right) ) is ( -frac{pi}{k} )
then ( k ) is
12
511 Simplify: ( sin ^{-1}(2 x sqrt{1-x^{2}})= )
( dots . .left(|x|<frac{1}{sqrt{2}}right) )
12
512 47. The value of a for which ax? + sin (x2 – 2x + 2) +
cos (r? – 2x + 2) = 0 has a real solution is
12
513 84. If f (x) = sin-
is equal to
SxS 1, then f(x)
c. sin x +*
c. sin-2x +
d. none of these
12
514 If the number ( 93215 x 2 ) is completely divisible by ( 11, ) then ( x ) is equal to
( A cdot 2 )
B. 3
( c cdot 1 )
D. 4
12
515 Solve:
( cot ^{-1} x+cot ^{-1} 2=frac{pi}{2} )
12
516 If ( quad x inleft(frac{3 pi}{2}, 2 piright), ) then the value of the expression
( sin ^{-1}left[cos left{cos ^{-1}(cos x)+sin ^{-1}(sin x)right}right. )
( ^{A} cdot-frac{pi}{2} )
в.
c. 0
D.
12
517 ( f(x)=cos ^{-1}left(frac{sqrt{2 x^{2}+1}}{x^{2}+1}right), ) then
range of ( boldsymbol{f}(boldsymbol{x}) ) is
( mathbf{A} cdot[0, pi] )
B cdot ( left(0, frac{pi}{4}right. )
( c cdotleft(0, frac{pi}{3}right. )
D ( cdotleft[0, frac{pi}{2}right) )
12
518 ( tan left(3 tan ^{-1} 3right)+cos left(3 cos ^{-1}(1 / 3)right)+1 )
is equal to
( A cdot 1 )
B. 9/13
c. ( 4 / 27 )
D. 295/351
12
519 Prove that:
(i) ( 2 tan ^{-1} sqrt{frac{b}{a}}=cos ^{-1}left(frac{a-b}{a+b}right) )
(ii) Find the principal value of ( cos ^{-1}left(-frac{1}{sqrt{2}}right) )
12
520 The number of solutions of the
equation ( sin ^{-1}left(frac{1+x^{2}}{2 x}right)=frac{pi}{2} sec (x- )
1) is
12
521 Number of value ( x ) satisfying the equation ( sin ^{-1}left(frac{5}{x}right)+sin ^{-1}left(frac{12}{x}right)= )
( frac{pi}{2} ) is
A .
B.
( c cdot 2 )
D. more than 2
12
522 ( operatorname{Let} cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=pi )
then prove that ( x^{2}+y^{2}+z^{2}+2 x y z= )
1
12
523 The number of solution of the equation ( tan ^{-1}(x-1)+tan ^{-1}(x)+tan ^{-1}(x+ )
1) ( =tan ^{-1}(3 x) ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. 4
12
524 Itxt
2. Find the domain for f(x) = sin
2x
12
525 If ( sin ^{-1}(6 x)+sin ^{-1}(6 sqrt{3} x)=-frac{pi}{2} )
then the value of ( x ) is
A ( cdot frac{1}{12} )
B. ( -frac{1}{12} )
c. ( -frac{1}{4 sqrt{3}} )
D. ( frac{1}{4 sqrt{3}} )
12
526 Illustration 5.66 Solve sin-‘x + sin 2x = 12
527 Illustration 5.15 Find the number of solutions of the
equation cos(cos=’x) = cosec(cosec-‘x).
12
528 Find the number of solutions of the
equation ( cos left(cos ^{-1} xright)= )
( operatorname{cosec}left(operatorname{cosec}^{-1} xright) )
A.
B.
( c cdot 2 )
D.
12
529 The value of
( tan left(sin ^{-1}left(cos left(sin ^{-1} xright)right)right) tan left(cos ^{-1}(sin right. )
where ( boldsymbol{x} boldsymbol{epsilon}(boldsymbol{0}, boldsymbol{1}), ) is equal to
( mathbf{A} cdot mathbf{0} )
B.
( c cdot-1 )
D. none of these
12
530 If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then
( cos ^{-1} x+cos ^{-1} y ) is equal to
A ( cdot frac{2 pi}{3} )
в.
c.
D.
12
531 Find the principal value of ( sin ^{-1}left(-frac{1}{2}right) ) 12
532 Indicate the relation which can hold in
their respective domain for infinite
values of ( x )
This question has multiple correct options
( mathbf{A} cdot tan left|tan ^{-1} xright|=|x| )
B cdot ( cot left|cot ^{-1} xright|=|x| )
C ( cdot tan ^{-1}|tan x|=|x| )
D ( cdot sin left|sin ^{-1} xright|=|x| )
12
533 Evaluate ( cos left[cos ^{-1}left(-frac{sqrt{3}}{2}right)+frac{pi}{6}right] ) 12
534 Find the principal value of ( tan ^{-1}(-sqrt{3}) ) 12
535 ( sin ^{-1}left(cos left(sin ^{-1} xright)right)+ )
( cos ^{-1}left(sin left(cos ^{-1} xright)right) ) is equal to
A.
в. ( frac{pi}{2} )
c. ( frac{3 pi}{4} )
D.
12
536 n
Illustration 5.71 Find the value
-1/1+ /(k – 1)k(k+1)(k + 2)
lim
k(k+1)
COS
no
k=2
12
537 Number of solution of equation ( left|sin ^{-1} x-cos ^{-1} xright|=x+2 ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
538 The value of ( sin h^{-1}(3) )
( A cdot log (1+sqrt{2}) )
B ( cdot log (2+sqrt{3}) )
( mathbf{c} cdot log (3+sqrt{5}) )
D ( cdot log (3+sqrt{10}) )
12
539 If ( cot ^{-1}left(frac{1}{a}right)+cot ^{-1}left(frac{1}{b}right)+ )
( cot ^{-1}left(frac{1}{c}right)=frac{pi}{2}, ) then
( mathbf{A} cdot a+b+c=a b c )
B. ( a b+b c+c a=1 )
c. ( a b+b c+c a=a b c )
D. none of these
12
540 Exhaustive set of values of parameter ( boldsymbol{a} )
so that ( sin ^{-1} x-tan ^{-1} x=a quad ) has a
solution is
A. ( left[-frac{pi}{6}, frac{pi}{6}right] )
B. ( left[-frac{pi}{4}, frac{pi}{4}right] )
( mathbf{c} cdotleft[-frac{pi}{2}, frac{pi}{2}right] )
D. none of these
12
541 Prove that:
[
tan ^{-1}left[frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}right]=frac{pi}{4}-
]
( frac{1}{2} cos ^{-1} x,-frac{1}{sqrt{2}} leq x leq 1 )
12
542 . If a = tan-1/ 4x – 4.83 )
ut (1-672
2 ,
= 2 sin-1
lain-1 2x
(+72) and
tan – = k, then
a. a+=for x € 1,
b. a= ß for x € (-k, k)
c. a+B=- = for x e 1,)
d. a+B=0 for x € (-k, k)
12
543 Show that ( tan ^{-1} 1 / 2+tan ^{-1} 2 / 11+ )
( tan ^{-1} 4 / 3=pi / 2 )
12
544 Write the value of ( cos ^{-1}left(cos frac{5 pi}{4}right) ) 12
545 Find the set of values of parameter a so that the equation
(sin-2x)3 + (cos’x)’ = at has a solution.
12
546 If ( tan ^{-1} x=frac{pi}{10} ) for some ( x in R, ) then
find the value of ( cot ^{-1} x )
12
547 12. Solve the equation tan ” * +1+tan **= tan” (-7).
x
-1
12
548 T 1
-+-COS
4 2
X
+ tan
cos’x, x = 0, is equal
4
2
b. 2x
d. none of these
12
549 If ( cos ^{-1} lambda+cos ^{-1} mu+cos ^{-1} gamma=3 pi )
then find the value of ( lambda mu+mu gamma+gamma lambda )
A . 1
B . 2
( c .3 )
D. 4
12
550 83. If sin ‘x+ sin ‘y + sin- z = T, then x4 + y4 + 24 + 4x?y?z?
= K (x+ y2 + y22 + 2?x?), where K is equal to
a. 1
b. 2
c. 4
d. none of these
12
551 If value of ( x ) which satisfy ( sin ^{-1} x leq ) ( cos ^{-1} x ) is ( x epsilonleft[a, frac{b}{sqrt{c}}right] )
Find the value of ( a+b+c )
A.
B.
( c cdot 2 )
D.
12
552 Find the value of ( x ) if ( sin (arcsin x)= )
( frac{sqrt{2}}{4} )
A ( cdot frac{sqrt{2}}{4} )
B. ( frac{sqrt{7}}{7} )
c. ( frac{sqrt{2}}{2} )
D. ( frac{2 sqrt{2}}{3} )
12
553 The value of the expression ( 2 sec ^{-1} 2+ ) ( sin ^{-1} frac{1}{2} ) is
A ( cdot frac{pi}{6} )
в. ( frac{5 pi}{6} )
c. ( frac{7 pi}{6} )
D.
12
554 The value of ( sin ^{-1}(sin 10) ) is
A . 10
в. ( 10-3 pi )
( c .3 pi-10 )
D. none of these
12
555 Find the minimum value the function
( f(x)=frac{pi^{2}}{16 cot ^{-1}(-x)}-cot ^{-1} x )
( ^{A} cdot-frac{pi}{4} )
B. ( -frac{pi}{2} )
c.
D.
12
556 Illustration 5.36 If x € (-1, 0), then find the value of
cos-(2×2 – 1)-2 sin-‘x.
12
557 Assertion
Statement 1 Range of ( boldsymbol{f}(boldsymbol{x})=tan ^{-1} boldsymbol{x}+ )
( sin ^{-1} x+cos ^{-1} x ) is ( (0, pi) )
Reason
Statement ( 2 f(x)=tan ^{-1} x+ )
( sin ^{-1} x+cos ^{-1} x=frac{pi}{2}+tan ^{-1} x ) for
( boldsymbol{x} boldsymbol{epsilon}(-mathbf{1}, mathbf{1}] )
A. Both the statements are TRUE and STATEMENT 2 is the
correct explanation of STATEMENT 1
B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT
C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE
12
558 ( tan ^{-1}(-2) ) is equal to
A ( cdot-cos ^{-1}left(frac{-3}{5}right) )
в. ( _{pi}+cos ^{-1} frac{3}{5} )
c. ( -frac{pi}{2}+tan ^{-1} xleft(-frac{3}{5}right) )
D. none of these
12
559 Solve the following equations. ( sin ^{-1} frac{5}{x}+sin ^{-1} frac{x}{2}=frac{pi}{2} ) 12
560 2x
8. The maximum value of f(x) = tan
osa. 18°
c. 22.5°
sus b. 36° ismert..
d. 15°
2
in se
12
561 27. If f(x)= x11 + x – x? + x + 1 and f(sin-‘(sin 8)) = a, a
is a constant, then f (tan-(tan 8)) is equal to ..
a. a
b. a- 2
c. a + 2
d. 2-a
.
12
562 f ( sin ^{-1}left(frac{x}{5}right)+operatorname{cosec}^{-1}left(frac{5}{4}right)=frac{pi}{2} ) then
( a ) value of ( x ) is:
( A )
B. 3
( c cdot 4 )
D. 5
12
563 f ( sin ^{-1}left(tan frac{17 pi}{4}right)-sin ^{-1}(sqrt{frac{3}{x}}) )
( left(frac{pi}{6}right)=0, ) then ( x ) is a root of the
equation
A ( cdot x^{2}-x-6=0 )
B . ( x^{2}+x-6=0 )
c. ( x^{2}-x-12=0 )
D. ( x^{2}+x-12=0 )
12
564 The solution set of the equation
( sin ^{-1} x=2 tan ^{-1} x )
( mathbf{A} cdot{1,2} )
B ( cdot{-1,2} )
( mathbf{c} cdot{-1,1,0} )
D. ( left{1, frac{1}{2}, 0right} )
12
565 The value of ( cos left(tan ^{-1} tan 4right) ) is
A ( frac{1}{sqrt{17}} )
B. ( -frac{1}{sqrt{17}} )
( c cdot cos 4 )
D. – cos 4
12
566 39. The solution of the inequality log12 sin x > log1/2 cos x
a. XEO,
b. x 1 1 1
XE
c. xe0T2)
d. None of these
12
567 61. The sum of roots of the equation
1
tan-1
– + tan
1+2x
– + ton-
1
= tan-1

1+ 4x

a. 2
c. 4
b. 3
d. none of these
12
568 ( fleft(frac{1}{3}right)+cos ^{-1} x=frac{pi}{2}, ) then find ( x ) 12
569 If ( 2 sinh ^{-1}left(frac{a}{sqrt{1-a^{2}}}right)=log left(frac{1+X}{1-X}right) )
then ( X= )
( A )
B.
c. ( sqrt{1-a^{2}} )
D. ( frac{1}{sqrt{1-a^{2}}} )
12
570 If ( boldsymbol{alpha} leq 2 sin ^{-1} boldsymbol{x}+cos ^{-1} boldsymbol{x} leq boldsymbol{beta}, ) then
A ( cdot alpha=frac{pi}{2}, beta=frac{pi}{2} )
B. ( quad alpha=frac{pi}{2}, beta=frac{3 pi}{2} )
c. ( alpha=0, beta=pi )
12
571 89. The number of real solutions of the equation V1 + cos2x
= V2 sin? (sin x), -ASxSt, is
a. 0
c. 2
b. 1
d. infinite
12
572 If ( sin ^{-1} x=frac{pi}{5}, ) for some ( x in(-1,1) )
then find the value of ( cos ^{-1} x )
A. ( -frac{3 pi}{10} )
в. ( frac{3 pi}{10} )
c. ( -frac{7 pi}{10} )
D. ( frac{7 pi}{10} )
12
573 10. The number of real solutions of the equation
tan! Vx2 – 3x + 2 + cos 14x – x² – 3 = is
a. one
b. two
c. zero
d. infinite
12
574 Find the principal value of:
( sec ^{-1}left(2 sin frac{3 pi}{4}right) )
12
575 58. If x, y, z are natural numbers such that cot ‘x + cotly
= cot-‘z then the number of ordered triplets (x, y, z) that
satisfy the equation is
a. O
b. 1
c. 2
d. Infinite solutions
12
576 Evaluate the following:
i. ( sin left(cot ^{-1} xright) )
ii. ( sin left(frac{pi}{2}-sin ^{-1}left(-frac{sqrt{3}}{2}right)right) )
( ^{mathbf{A}} cdot(mathrm{i}) frac{1}{sqrt{x^{2}+1}},(mathrm{ii}) frac{1}{2} )
B. ( quad(i) frac{1}{sqrt{x^{2}-1}},left(text { ii) } frac{-sqrt{3}}{2}right. )
( c cdot(i) frac{-1}{sqrt{x^{2}+1}},left(text { ii) } frac{1}{2}right. )
D. ( left(text { i) } frac{-1}{sqrt{x^{2}-1}} text { , (ii) } frac{+sqrt{3}}{2}right. )
12
577 Assertion
If ( alpha, beta ) are the roots of the equation
( 18left(tan ^{-1} xright)^{2}-9 pi tan ^{-1} x+pi^{2}=0 )
( operatorname{then} boldsymbol{alpha}+boldsymbol{beta}=frac{boldsymbol{4}}{sqrt{mathbf{3}}} )
Reason
( sec ^{2}left(cos ^{-1}left(frac{1}{4}right)right)+ )
( operatorname{cosec}^{2}left(sin ^{-1}left(frac{1}{5}right)right)=41 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
578 The value of
( sin ^{-1}left[cos left[sin ^{-1}left(frac{1}{2} tan frac{pi}{3}right)right]right] ) is
( mathbf{A} cdot 2 tan ^{-1}(2-sqrt{3}) )
B ( cdot 2 tan ^{-1}(sqrt{2}-1) )
( mathbf{c} cdot 2 tan ^{-1}(sqrt{2}+1) )
D.
12
579 68. The value of tan-14
… equals
+ tan-1
19 + tan 14
20 + tan-1 4+
39
a. tan-‘1 + tan-
tan-11
b. tan1 + cott3
c. cot ‘1 + cot-1 + cott
d. cot ! 1+tan 3
12
580 ( cos left[2 cos ^{-1} frac{1}{5}+sin ^{-1} frac{1}{5}right]= ) 12
581 in? – 10n +21.6)
10. If cot-1
E N, then n can be
a. 3
c. 4
b. 2
d. 8
12
582 ( cos ^{-1}left(cos left(frac{5 pi}{4}right)right) ) is given by
( mathbf{A} cdot 5 pi / 4 )
B . ( 3 pi / 4 )
( mathbf{c} .-pi / 4 )
D. None of these
12
583 If ( sum_{i=1}^{2 n} sin ^{-1} x_{i}=n pi, ) then ( sum_{i=1}^{2 n} x_{i} ) is equal
to
A ( . n )
B. ( 2 n )
c. ( frac{n(n+1)}{2} )
D. None of these
12
584 The principal value of ( tan ^{-1}left(cot frac{3 pi}{4}right) )
is :
A. ( -frac{3 pi}{4} )
в. ( frac{3 pi}{4} )
( c cdot-frac{pi}{4} )
D.
12
585 Prove that ( : cos ^{-1}left(frac{3}{5}right)+cos ^{-1}left(frac{4}{5}right)= )
( frac{pi}{2} )
12
586 ( cot ^{-1} frac{x y+1}{x-y}+cot ^{1} frac{y z+1}{y-z}+ )
( cot ^{-1} frac{x z+1}{z-x} )
( A )
B. –
( c cdot 0 )
D. none of these
12
587 If ( sin ^{-1}(1-x)-2 sin ^{-1} x=pi / 2, ) then
( x ) is equals?
( mathbf{A} cdot{0,-1 / 2} )
B cdot ( {1 / 2,0} )
( c cdot{0} )
D. (-1,0)
12
588 If ( alpha, beta ) are the roots of the equation
( left(tan ^{-1}(x / 5)right)^{2}+(sqrt{3}- )
1) ( tan ^{-1}(x / 5)-sqrt{3}=0,|alpha|>|beta| ) then
This question has multiple correct options
A ( cdot alpha+beta=-5 pi / 12 )
в. ( |alpha-beta|=35 pi / 12 )
( mathbf{c} cdot alpha beta=-25 pi^{2} / 12 )
D. ( 3 alpha+4 beta=0 )
12
589 Find the value of :
( sec ^{2}left(tan ^{-1} 2right)+csc ^{2}left(cot ^{-1} 3right) )
A . 11
B. 15
c. 17
D. 21
12
590 ( int frac{x^{2}}{sqrt{1-x^{6}}} d x=frac{1}{k} sin ^{-1}left(x^{k}right)+C . ) what 12
591 ( tan left(cot ^{-1} xright) ) is equal to:
A ( cdot frac{pi}{2}-x )
B. ( cot left(tan ^{-1} xright) )
( c cdot tan x )
D.
12
592 ( sin ^{-1}(117 / 125) ) is equal to
This question has multiple correct options
A ( .2 alpha )
в. ( 3 alpha )
c. ( pi / 2-2 alpha )
D. ( pi-3 alpha )
12
593 Illustration 5.42 If sin-‘x = 7d/5, for some x € (-1, 1), then
find the value of cos-‘x.
12
594 The equation ( 2 cos ^{-1} x+sin ^{-1} x=frac{11 pi}{6} )
has
A. No solution
B. One solution
c. Two solutions
D. Three solutions
12
595 Illustration 5.30 Prove that
cosec(tan’ (cos(cot’ (sec(sin-‘ a))))) = 13 – a?,
where a € [0, 1].
12
596 Solve the equation:cos ( ^{-1}left(frac{x^{2}-1}{x^{2}+1}right)+ )
( sin ^{-1}left(frac{2 x}{x^{2}+1}right)+tan ^{-1}left(frac{2 x}{x^{2}-1}right)= )
3
A ( cdot frac{1}{sqrt{3}} )
B. ( cot 20^{circ} )
( c cdot frac{-1}{sqrt{3}} )
D. ( tan 20 )
12
597 Prove:
( 3 cos ^{-1} x=cos ^{-1}left(4 x^{3}-3 xright), x inleft[frac{1}{2}, 1right] )
12
598 The value of ( x ) satisfying ( tan left(sec ^{-1} xright)=sin left(cos ^{-1} frac{1}{sqrt{5}}right) ) is
( ^{A}+frac{3}{sqrt{5}} )
( ^{mathrm{B}} pm frac{5}{sqrt{3}} )
c. ( =frac{sqrt{2}}{3} )
D. ( pm frac{3}{5} )
12
599 If ( boldsymbol{A}=tan ^{-1}left(frac{boldsymbol{x} sqrt{mathbf{3}}}{mathbf{2 K}-boldsymbol{x}}right) ) and ( boldsymbol{B}= )
( tan ^{-1}left(frac{2 x-K}{K sqrt{3}}right), ) then the value of
( A-B ) is
( A )
B. 45
( c .60 )
D. 30
12
600 Find the principal value of the following
( tan ^{-1}left(tan frac{3 pi}{4}right) )
12
601 ( operatorname{Let} tan ^{-1} y=tan ^{-1} x+ )
( tan ^{-1}left(frac{2 x}{1-x^{2}}right) ) where ( |x|<frac{1}{sqrt{3}} . ) Then
a value of y is
A ( cdot frac{3 x-x^{3}}{1-3 x^{2}} )
В. ( frac{3 x+x^{3}}{1-3 x^{2}} )
c. ( frac{3 x+x^{3}}{1+3 x^{2}} )
D. ( frac{3 x-x^{3}}{1+3 x^{2}} )
12
602 19. The range of values of p for which the
sin cos (cos(tan-‘x) = p has a solution is
a (te ta
1)
b. (0,1)
d. (-1,1)
d. (-1,1)
12
603 Find the principal value of:
( operatorname{cosec}^{-1}left(2 cos frac{2 pi}{3}right) )
12
604 If ( boldsymbol{x}=mathbf{2} cos ^{-1}left[frac{1}{2}right]+sin ^{-1}left[frac{mathbf{1}}{sqrt{mathbf{2}}}right]+ )
( tan ^{-1}(sqrt{3}) ) and ( y= )
( cos left[frac{1}{2} sin ^{-1}left[sin frac{x}{2}right]right] ) then which of the
following statements holds good?
A ( cdot y=cos frac{3 pi}{16} )
В. ( y=cos frac{5 pi}{16} )
c. ( x=4 cos ^{-1} y )
D. none of these
12
605 The principal value of ( sin ^{-1}left{cos left(sin ^{-1} frac{sqrt{3}}{2}right)right} )
A. ( frac{pi}{6} )
в. ( frac{pi}{3} )
( c cdot-frac{pi}{3} )
D. none of these
12
606 Find the value of ( sin ^{-1} x+sin ^{-1} frac{1}{x}+ )
( cos ^{-1} x+cos ^{-1} frac{1}{x} )
12
607 Assertion
( sin ^{-1}left[x-frac{x^{2}}{2}+frac{x^{3}}{4} dotsright]=pi / 2 )
( cos ^{-1}left[x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4} ldots .right] ) for ( 0<|x|< )
( sqrt{2} ) has a unique solution.
Reason
( tan ^{-1} sqrt{x(x+1)}+ )
( sin ^{-1} sqrt{x^{2}+x+1}=pi / 2 ) has no
solution for ( -sqrt{2}<x<0 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
608 (2x²+4)
Illustration 5.19
Solve sin-| sin
< 1 – 3
(1+x2
12
609 If ( tan ^{-1} frac{a+x}{a}+tan ^{-1} frac{a-x}{a}=frac{pi}{6} )
then ( x^{2}=? )
A ( cdot 2 sqrt{3} a )
an
B. ( sqrt{3} a )
( c cdot 2 sqrt{3} a^{2} )
( ^{2} )
D. None of these
12
610 If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then
( cos ^{-1} x+cos ^{-1} y= )
A ( cdot frac{pi}{6} )
в.
( c cdot frac{pi}{3} )
D.
12
611 ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} Rightarrow x= )
( A cdot-1 )
B.
( c cdot c )
D. ( pi sqrt{frac{5}{8}} )
12
612 Evaluate :
( int x^{2} tan ^{-1} frac{x}{2} d x )
12
613 Illustration 5.25
Find the value of sin
-cos
12
614
а – х
а
55. If tan-4° + tan-1″
а
a. 2/за
с. 2/за”
b. За
d. none of these
12
615 Find the value of ( x )
if ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3} )
12
616 represents the graph of the function ( f(x)=lim _{n rightarrow infty} frac{2}{pi} tan ^{-1}(n x) ? )
( A )
B.
( c )
( D )
12
617 The domain of function ( f(x)=sin ^{-1} 5 x )
is
A ( cdotleft(-frac{1}{5}, frac{1}{5}right) )
B. ( left[-frac{1}{5}, frac{1}{5}right] )
c. ( R )
D. ( left(0, frac{1}{5}right) )
12
618 ( cos ^{-1}left(cos left(frac{-17 pi}{5}right)right) ) is equal to
A. ( -frac{17 pi}{5} )
в. ( frac{3 pi}{5} )
( c cdot frac{2 pi}{5} )
D. none of these
12
619 Express the following in the simplest
form ( tan ^{-1}left(frac{cos x}{1+sin x}right), frac{-pi}{2}<x<frac{pi}{2} )
12
620 1.
COS
Find the value of : cos(2cos-tx + sin- x) at x = , where
Oscos-?xst and -1/2 <sin x S /2.
(1981 – 2 Marks)
12
621 Find the value of ( cos left(sec ^{-1} x+right. )
( left.operatorname{cosec}^{-1} xright),|x| geq 1 )
12
622 If ( A=tan ^{-1} frac{1}{7} ) and ( B=tan ^{-1} frac{1}{3} ) then
This question has multiple correct options
A ( cdot cos 2 A=frac{24}{25} )
B. ( cos 2 B=frac{4}{5} )
c. ( cos 2 A=sin 4 B )
D. ( tan 2 B=frac{3}{4} )
12
623 The greatest and least value of
( left(sin ^{-1} xright)^{2}+left(cos ^{-1} xright)^{2} ) are respectively
A ( cdot frac{pi^{2}}{4} a n d 0 )
B . ( frac{pi}{2} a n d-frac{pi}{2} )
C. ( frac{5 pi^{2}}{4} ) and ( frac{pi^{2}}{8} )
D. ( frac{pi^{2}}{4} ) and ( frac{-pi}{4} )
12
624 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=pi, ) prove
that ( x sqrt{1-x^{2}}+y sqrt{1-y^{2}}+ )
( z sqrt{1-z^{2}}=2 x y z )
12
625 ( sec ^{2}left(tan ^{-1} 2right)+operatorname{cosec}^{2}left(cot ^{-1} 3right) ) 12
626 Illustration 5.60
If x > y > z> 0, then find the value of
cot-1 *y + 1
– + cot-1 YZ +1
– + cor-1 ZX + 1
x – y
Y – Z
z
– x
12
627 2. 2 tan- ‘(- 2) is equal to
a.
– COS
b. – Te + cos2
12
628 Tllustration 2.8
Find the value of cos+ (-1/2).
12
629 53. The exhaustive set of values of a for which a-cot- 3x =
2tan-‘3x + cos ‘x v3 + sin ‘x 73 may have solution, is
TTT
(1
370
a.

c. 172
[20 40
[ 31 71
I
2
3
1
6
6
12
630 Differentiate ( tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right) ) 12
631 Find the principal value of:
( sin ^{-1}left(frac{sqrt{mathbf{3}}-1}{2 sqrt{2}}right) )
12
632 Solve the equation ( sin ^{-1}(3 x)=-frac{1}{3} pi )
giving the solution in an exact form.
12
633 Solve ( int frac{sin ^{-1} sqrt{x}-cos ^{-1} sqrt{x}}{sin ^{-1} sqrt{x}+cos ^{-1} sqrt{x}} d x ) 12
634 Show that ( sin ^{-1}(2 x sqrt{1-x^{2}})=2 sin ^{-1} x ) 12
635 Evaluate: ( tan ^{-1} sqrt{3}-cot ^{-1}(-sqrt{3}) )
( mathbf{A} cdot mathbf{0} )
B. ( 2 sqrt{3} )
( c cdot-frac{pi}{2} )
D.
12
636 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) and
( boldsymbol{f}(mathbf{2})=mathbf{2}, boldsymbol{f}(boldsymbol{a}+boldsymbol{b})= )
( boldsymbol{f}(boldsymbol{a}) boldsymbol{f}(boldsymbol{b}), forall boldsymbol{a}, boldsymbol{b} boldsymbol{epsilon} boldsymbol{R}, ) then
( boldsymbol{x}^{f(mathbf{2})}, boldsymbol{y}^{f(4)}, boldsymbol{z}^{f(boldsymbol{6})} ) are in
This question has multiple correct options
A. A.P
в. G.
c. н.
D. None
12
637 Prove that:
( tan ^{-1}left(frac{6 x-8 x^{3}}{1-12 x^{2}}right) )
( tan ^{-1}left(frac{4 x}{1-4 x^{2}}right)=tan ^{-1} 2 x ;|2 x|< )
( frac{1}{sqrt{3}} )
12
638 ( sin left[2 cos ^{-1} cot left(2 tan ^{-1} frac{1}{2}right)right] ) is equal
to
A ( cdot frac{3 sqrt{7}}{8} )
B. ( frac{5 sqrt{7}}{8} )
( c cdot frac{5 sqrt{7}}{2} )
D. ( frac{3 sqrt{7}}{2} )
12
639 Assertion: The value of ( frac{tan ^{-1} frac{4}{3}}{tan ^{-1} frac{1}{2}} ) is equal
to
2
Reason: ( forall boldsymbol{x} in[mathbf{0}, mathbf{1}], tan ^{-1}left(frac{mathbf{2} boldsymbol{x}}{mathbf{1 – x}^{2}}right)= )
( 2 tan ^{-1} x )
A. Both A and R are true and R is the correct explanation of
B. Both A and R are true but R is not the correct explanation of A
c. ( R ) is true but ( A ) is false
D. A is true but R is false
12
640 ( sec h^{-1}left(frac{1}{5}right)= )
( mathbf{A} cdot log (sqrt{24}+5) )
( mathbf{B} cdot log 5+sqrt{27} )
( mathbf{C} cdot log 26+sqrt{5} )
D. ( log 27+sqrt{5} )
12
641 ( cos ^{-1}left(cos left(2 cot ^{-1}(sqrt{2}-1)right)right) ) is equal to
A ( cdot sqrt{2}-1 )
в.
( c cdot frac{3 pi}{4} )
D. none of these
12
642 Write the principal value of ( tan ^{-1}(1)+ ) ( cos ^{-1}left(-frac{1}{2}right) ) 12
643 Solve:
( sin ^{-1}left(frac{2 x}{1+x^{2}}right) )
12
644 52. The product of all values of x satisfying the equation
sin-cos(2x+ + 10|x|+ 4 = cotcor- (2-18ſx))+
0$ x2 + 5|x +3 ) coco (9|x|JJ*2
is
a. 9
b. -9
c.
-3
12
645 ( cos ^{-1}(sqrt{frac{boldsymbol{a}-boldsymbol{x}}{boldsymbol{a}-boldsymbol{b}}})=sin ^{-1}(sqrt{frac{boldsymbol{x}-boldsymbol{b}}{boldsymbol{a}-boldsymbol{b}}}) )
possible if
A ( . a>x>b ) or ( a<xb ) and ( x ) takes any value
D. ( a<b ) and ( x ) takes any value
12
646 Illustration 5.41 Prove that 2 tan-‘(cosec tan ‘x- tan cotx)
= tan- x (x 0).
12
647 The value of ( x ) for which
( sin left(cot ^{-1}(1+x)right)=cos left(tan ^{-1} xright) ) is
( A cdot frac{1}{2} )
B.
( c )
D. ( -frac{1}{2} )
12
648 Prove that:
( cos ^{-1}(x)+cos ^{-1}left{frac{x}{2}+frac{sqrt{3-3 x^{2}}}{2}right}=frac{pi}{3} )
12
649 If ( 2 sin ^{-1} x=sin ^{-1}(2 x sqrt{1-x^{2}}), ) then
( boldsymbol{x} in )
A ( .[-1,1] )
B. ( left[-frac{1}{sqrt{2}}, 1right] )
( c cdotleft[-frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right] )
D. none of these
12
650 f ( sin ^{-1}left(frac{2 a}{1+a^{2}}right)+sin ^{-1}left(frac{2 b}{1+a^{2}}right)= )
( 2 tan ^{-1} x, ) then ( x ) is equal to
A ( frac{a-b}{1+a b} )
в. ( frac{b}{1+a b} )
c. ( frac{b}{1-a b} )
D. ( frac{a+b}{1-a b} )
12
651 ( operatorname{Let} f(x)=cos left(tan ^{-1} 2 xright)- )
( sin left{tan ^{-1}left(frac{1}{2 x+1}right)right} ) and ( a= )
( cos left(tan ^{-1}left(sin left(cot ^{-1} 2 xright)right)right) ) and ( b= )
cos ( left(frac{pi}{2}+cos ^{-1} 2 xright) cdot ) If ( f(x)=0, ) then
( b= )
A. ( frac{1}{sqrt{2}} )
B. ( -frac{sqrt{3}}{2} )
c. ( frac{sqrt{3}}{2} )
D. ( -frac{1}{sqrt{2}} )
12
652 Topic-wise SULULUI
Ras
ve
3.
For any positive integer n, define f.,:(0,00)
For any positi
f, (*) – Ez- tan” (+68+)(x+;-)) for all
x 0,00).
Here, the inverse trigonometric function tan” xassumes
values in
Then, which of the following statement(s) is (are) TRUE?
(JEE Adv. 2018)
(a)
(b)
:-1 tan? (,0)=55
1 (1+f;(0)) sec (5,0)=10
©
For any fixed positive integer n, lim tan (, (x)=
*+00
(d) For any fixed positive integer n, lim
sec (f(x))=1
Com
12
653 Find the value of ( sin ^{-1} x+sin ^{-1} frac{1}{x}+ )
( cos ^{-1} x+cos ^{-1} frac{1}{x} )
A. ( -pi )
в. ( +pi )
( c .-2 pi )
D. ( +2 pi )
12
654 If ( a sin ^{-1} x-b cos ^{-1} x=c, ) then the
value of ( a sin ^{-1} x+b c o s^{-1} x ) (whenever
exists) is equal to
A . 0
B. ( frac{pi a b+c(b-a)}{a+b} )
( c cdot frac{pi}{2} )
D. ( frac{pi a b+c(a-b)}{a+b} )
12
655 7. sec?(tan-? 2) + cosec?(cot-3) is equal to
a. 5
b. 13
190 c. 1575] 100 d. 6 ostalo
12
656 Find general solution of the following equations:
( sin theta=frac{1}{2} ? )
12
657 If ( left[cot ^{-1} xright]+left[cos ^{-1} xright]=0, ) where [
denotes the greatest integer function, then the complete set of values of ( x ) is
A ( .(cos 1,1] )
B. ( (cos 1,-cos 1) )
c. ( (cot 1,1] )
D. none of these
12
658 Find the value of ( tan left[frac{1}{2} cos ^{-1} frac{sqrt{5}}{3}right] ) 12
659 Illustration 5.73
Solve sin ‘ x – cos’ x = sin-‘(3x – 2).
COS
12
660 The principal value of
( cos ^{-1}left[frac{1}{sqrt{2}}left(cos left(frac{9 pi}{10}right)-sin left(frac{9 pi}{10}right)right)right] ) is
( mathbf{A} cdot frac{3 pi}{20} )
B. ( frac{7 pi}{200} )
( mathbf{C} cdot frac{7 pi}{10} )
D. none of these
12
661 The value of ( : tan ^{-1} frac{1}{5}+tan ^{-1} frac{1}{7}+ )
( tan ^{-1} frac{1}{3}+tan ^{-1} frac{1}{8}-frac{pi}{4} )
12
662 Write the simplest form of :
( tan ^{-1} frac{sqrt{1+x^{2}}-1}{x}, x neq 0 )
12
663 32. There exists a positive real number x satisfying
cos(tan-‘x) = x. Then the value of cos!
Tu labb. Tebe
c. 21
12
664 If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots inftyright)=frac{pi}{2} )
and ( 0<x<sqrt{2}, ) then ( x= )
A ( cdot frac{1}{2} )
в.
( c cdot-frac{1}{2} )
D. ( frac{sqrt{3}}{2} )
12
665 Find ( n ) if ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+ )
( sin ^{-1}left(frac{16}{65}right)=frac{n pi}{2} )
12
666 Illustration 5.40 Find the value of sin-‘(sin 5) +
cos(cos 10) + tan-‘{tan(-6)} + cot-‘{cot(-10)}.
12
667 TT
377
30. If sin” : [-1, 1]
and cos!: (-1, 1] → [0, 1]
he two bijective functions, respectively inverses of
bijective functions sin:
→ [-1, 1] and cos : [0, 1]
+ [-1, 1], then sin!x + cos ix is
b.
TT
31
o sito
d. not a constant
12
668 If ( a_{1}, a_{2}, a_{3} dots dots dots a_{n} ) are in A.P.with
common difference ( d, ) then
( tan left[tan ^{-1}left(frac{d}{1+a_{1} a_{2}}right)+tan ^{-1}left(frac{d}{1+a_{2} a_{3}}right)+right. )
( dots dots dots dots dots+ )
( left.tan ^{-1}left(frac{d}{1+a_{n-1} a_{n}}right)right]=? )
A ( cdot frac{(n-1) d}{a_{1}+a_{n}} )
B. ( frac{(n-1) d}{1+a_{1} a_{n}} )
c. ( frac{n d}{1+a_{1} a_{n}} )
D. ( frac{a_{n}-a_{1}}{a_{n}+a_{1}} )
12
669 18. The number of integral values of k for which the equation
sin-‘x + tan-?x=2k + 1 has a solution is
a. 110 0 b . 2 acting
c. 3
d. 4
12
670 The principal solution of the equation ( cot x=-sqrt{3} ) is
A ( cdot frac{pi}{6} )
в. ( frac{pi}{3} )
c. ( frac{5 pi}{6} )
D. ( -frac{5 pi}{6} )
12
671 If ( cos ^{-1} x-cos ^{-1} frac{y}{2}=alpha ) where -1
( 1 leq x leq 1,-2 leq y leq 2, x leq frac{y}{2} ) then for
all ( 4 x^{2}-4 x y cos alpha+y^{2} ) is equal to
A ( cdot 4 sin ^{2} alpha-2 x^{2} y^{2} )
B. ( 4 cos ^{2} alpha+2 x^{2} y^{2} )
( mathbf{c} cdot 4 sin ^{2} alpha )
( D cdot 2 sin ^{2} alpha )
12
672 Assertion
The equation ( 2left(sin ^{-1} xright)^{2}- )
( 5left(sin ^{-1} x+2right)=0 )
Reason
( sin ^{-1}(sin x)=x ) if ( x epsilon[-1.57,1.57] )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
12
673 State True or False ( sin ^{-1} 2+cos ^{-1} 2=frac{pi}{2} )
A. True
B. False
12
674 If ( tan ^{-1} x+tan ^{-1} y=frac{pi}{4}, x y<1, ) then
write the value of ( boldsymbol{x}+boldsymbol{y}+boldsymbol{x} boldsymbol{y} )
12
675 Prove that ( tan ^{-1}left(frac{sin x}{1+cos x}right),-pi<x<pi ) 12
676 If ( sin ^{-1}left(frac{2 p}{1+p^{2}}right)-cos ^{-1}left(frac{1-q^{2}}{1+q^{2}}right)= )
( tan ^{-1}left(frac{2 x}{1+x^{2}}right), ) then the value of ( x ) is equal to
A ( cdot frac{p+q}{1+p q} )
в. ( frac{p-q}{1-p q} )
c. ( frac{p-q}{p q-1} )
D. ( frac{p-q}{p q+1} )
12
677 Equations ( 2 sin ^{-1} x+3 sin ^{-1} y=frac{5 pi}{2} )
and ( y=k x-5 ) hold simultaneously
when k equals
( A cdot 2 )
B. 4
( c cdot 6 )
D. no such k exists
12
678 Find the principal value of ( cos ^{-1}left(-frac{1}{2}right) ) 12
679 Find the principal value of the following:
( tan ^{-1}(-1) )
12
680 illustration 5:26. Find the value of sin(3cot” (22)
Illustration 5.24
Find the value of sin
CO2
12
681 Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
begin{aligned}
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \
x y1
end{aligned}
]
Solve
(a) ( cos left(2 sin ^{-1} xright)=1 / 9 )
(b) ( cos ^{-1}(3 / 5)-sin ^{-1}(4 / 5)=cos ^{-1} x )
(c) If ( sin left(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then
prove that ( x ) is equal to ( 1 / 5 )
12
682 Solve:tan ( ^{-1} mathbf{2} boldsymbol{x}+tan ^{-1} mathbf{3} boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}} ) 12
683 If ( alpha ) is a real number for which ( f(x)= )
( log _{e} cos ^{-1} x ) is defined, then a possible
value of ( [boldsymbol{alpha}] ) (where [] denotes the
greatest function) is This question has multiple correct options
A . 0
B.
( c cdot-1 )
D. – –
12
684 Assertion
The value of the determinant
( begin{array}{|ccc|}tan ^{-1} x & cot ^{-1} x & pi / 2 \ sin ^{-1}(4 / 5) & sin ^{-1}(3 / 5) & sin ^{-1} 1 \ cos ^{-1}(3 / 5) & cos ^{-1}(4 / 5) & 1end{array} )
equal to zero for all values of ( x )
Reason
( 2 cos ^{-1} x=cos ^{-1}left(2 x^{2}-1right) ) if ( -1 leq )
( x leq 1 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
685 Illustration 5.31 If x < 0, then prove that
cos-"x = 1 – sin- 1 – x?.
12
686 Prove that ( sin ^{-1} frac{3}{5}-sin ^{-1} frac{8}{17}= )
( cos ^{-1}left(frac{84}{85}right) )
12
687 Express ( tan ^{-1} x+tan ^{-1} frac{2 x}{1-x^{2}} ) in the
terms of ( tan ^{-1} frac{3 x-x^{3}}{1-3 x^{2}} )
12
688 n
Illustration 5.35
Prove that cos- |
= 2 tan-
x
(1+z2n
0<x<
.
12
689 If two angle of a triangle are ( sin ^{-1}left(frac{1}{sqrt{5}}right) ) and ( sin ^{-1}left(frac{1}{sqrt{10}}right), ) then third angle is
( A cdot frac{pi}{4} )
B.
( c cdot frac{3 pi}{4} )
D. ( frac{2 pi}{3} )
12
690 6. If tan ‘y=4 tan ‘x (14|<tan), find y as an algebraic
function of x, and, hence, prove that tan 7/8 is a root of
the equation x4 – 6×2 + 1 = 0.
12
691 If ( (x-1)left(x^{2}+1right)>0, ) then find the
value of ( sin left(frac{1}{2} tan ^{-1} frac{2 x}{1-x^{2}}-tan ^{-1} xright) )
A . ( -1 / 2 )
B. –
( c cdot 1 / 2 )
D.
12
692 16. If [cot-‘ x] + [cos-1 x] = 0, where [.] denotes the greatest
integer function, then the complete set of values of x is
a. (cos 1, 1]
b. (cos 1, cos 1)
c. (cot 1, 1]
d. none of these
12
693 The range of the function, ( boldsymbol{f}(boldsymbol{x})= )
( left(1+sec ^{-1} xright)left(1+cos ^{-1} xright) ) is
( A cdot(-infty, infty) )
В ( cdot(-infty, 0] cup[4, infty) )
( ^{c} cdotleft{0,(1+pi)^{2}right} )
D. ( left[1,(1+pi)^{2}right] )
12
694 If ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8}, ) then
( boldsymbol{x}= )
A . -1
B.
( c cdot 0 )
D.
12
695 Find the value of sin-‘x+sin! – + cos x
Illustration 5.51
+ cos-1-
12
696 Find the value of ( sin ^{-1}left(2 cos ^{2} x-1right)+ )
( cos ^{-1}left(1-2 sin ^{2} xright) )
A ( cdot frac{pi}{2} )
в.
c.
D.
12
697 23. The value of the expression
sin-
sin 221
cos” (cos Spa) + tan” (tan 5x) + sin-” (cos 2)is
a. 1772 -2 6. – 2
d. none of these
12
698 The number of solutions of
( sin ^{-1}left(1+b+b^{2}+cdots inftyright)+ )
( cos ^{-1}left(a-frac{a^{2}}{3}+frac{a^{2}}{9} cdots inftyright)=frac{pi}{2} ) is
( A )
B. 2
( c .3 )
( D )
12
699 ( sin ^{-1}|sin x|=sqrt{sin ^{-1}|sin x|} ) then ( x= )
This question has multiple correct options
A ( . n pi-1 )
в. ( n pi )
c. ( n pi+1 )
D. ( n frac{pi}{2}+1 )
12
700 Evaluate the following:
( sin ^{-1}left(frac{2 pi}{4}right) )
ii. ( cos ^{-1}left(cos frac{7 pi}{6}right) )
iii. ( tan ^{-1}left(tan frac{2 pi}{3}right) )
iv. ( cos left(cos ^{-1}left(frac{sqrt{3}}{2}right)+frac{pi}{6}right) )
A ( cdot ) i. ( -frac{pi}{3} ) ii. ( -frac{5 pi}{6} ) iii. ( frac{pi}{3} ) iv. 1
B . i. ( frac{2 pi}{3} ) ii. ( frac{pi}{6} ) iii. ( -frac{2 pi}{3} ) iv. 1
C ( cdot ) i. ( -frac{2 pi}{3} ) ii. ( -frac{pi}{6} ) iii. ( frac{2 pi}{3} ) iv. -1
D cdot i. ( frac{pi}{3} ) ii. ( frac{5 pi}{6} ) iii. ( -frac{pi}{3} ) iv. -1
12
701 34. If sin + x + sin ‘y
, then
in 1 + x² + y
5 is equal to
x² – x² y + y²
b. 2
a.
1
d. none of these
12
702 If ( boldsymbol{alpha}, boldsymbol{beta}(boldsymbol{alpha}<boldsymbol{beta}) ) are the roots of the
equation ( 6 x^{2}+11 x+3=0 ) then which
of the following are real? This question has multiple correct options
( A cdot cos ^{-1} alpha )
B. ( sin ^{-1} beta )
( mathbf{c} cdot operatorname{cosec}^{-1} alpha )
D. Both ( cot ^{-1} alpha ) and ( cot ^{-1} beta )
12
703 ff ( y=tan ^{-1}left(frac{x sin alpha}{1-x cos alpha}right) . ) Find ( cot y ) 12
704 If
( a, b, c ) are distinct non-zero real numbers having the same sign, then prove that ( cot ^{-1}left(frac{a b+1}{a-b}right)+cot ^{-1}left(frac{b c+1}{b-c}right)+ )
( cot ^{-1}left(frac{c a+1}{c-a}right)=pi quad o r )
12
705 Write the simplest form of ( tan ^{-1}[sqrt{frac{1-cos x}{1+cos x}}] ) 12
706 Find the value of the expression ( sec ^{-1}left(frac{x+1}{x-1}right)+sin ^{-1}left(frac{x-1}{x+1}right) ) 12
707 If ( boldsymbol{A}=frac{1}{1} cot ^{-1}left(frac{1}{1}right)+frac{1}{2} cot ^{-1}left(frac{1}{2}right)+ )
( frac{1}{3} cot ^{-1}left(frac{1}{3}right) ) and ( B=1 cot ^{-1} 1+ )
( 2 cot ^{-1} 2+3 cot ^{-1} 3 operatorname{then}|B-A| ) is
equal to ( frac{a pi}{b}+frac{c}{d} ) cot ( ^{-1} 3 ) where
( a, b, c, d in N ) and are in their lowest
form then ( a+b+c+d ) equal to
12
708 Illustration 5.26
Prove that
/1 + sin x + 71-sin x
Cor-1
XE
1 + sin x –
sin x
12
709 The value of ( sin ^{-1}left(x^{2}-4 x+6right)+ )
( cos ^{-1}left(x^{2}-4 x+6right) ) for all ( x epsilon R ) is
A ( cdot frac{pi}{2} )
в. ( pi )
( c cdot 0 )
D. none of these
12
710 ( operatorname{Let} sin ^{-1}left(frac{1-x^{2}}{1+x^{2}}right), ) then ( frac{d y}{d x} ) is
A ( cdot frac{2}{1+x^{2}} )
в. ( frac{1}{2left(1+x^{2}right)} )
c. ( frac{-2}{1+x^{2}} )
D. ( frac{2}{2-x^{2}} )
12
711 If ( frac{3 pi}{2} leq x leq frac{5 pi}{2}, ) then ( sin ^{-1}(sin x) ) is
equal to-
( A )
B. ( -x )
c. ( 2 pi-x )
D. ( x-2 pi )
12
712 ( sin left(2 tan ^{-1} sqrt{frac{1-x}{1+x}}right) ) 12
713 13. Range of f(x) = sin-4x + tan– x + sec-?x is
d. none of these
12
714 Assertion
Let ( boldsymbol{f}:[mathbf{8} boldsymbol{pi}, mathbf{9} boldsymbol{pi}] rightarrow[-mathbf{1}, mathbf{1}], boldsymbol{f}(boldsymbol{x})=boldsymbol{operatorname { c o s } boldsymbol { x }} )
then
Statement-1: ( boldsymbol{f}^{-1}(-1)=mathbf{9} boldsymbol{pi} ) because
Reason
Statement-2 : ( boldsymbol{f}^{-1}(boldsymbol{x})=mathbf{1 0} boldsymbol{pi}- )
( cos ^{-1} x forall x infty[-1,1] )
A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-
B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT correct explanation for Statement-
c. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true
12
715 If ( cos left(2 sin ^{-1} xright)=frac{1}{9}, ) the value of ( x )
which satify equation is ( pm frac{a}{b} . ) Find the value of ( a+b )
A . 2
B. 3
( c cdot 4 )
D.
12
716 Assertion
The area bounded by the curve ( y= ) ( sin ^{-1} x & ) the line ( x=0 &|y|=frac{pi}{2} ) is ( sqrt{2} )
square units.
Reason
The domain & principal value branch of ( y=sin ^{-1} x operatorname{are}[-1,1] &left[frac{-pi}{2}, frac{pi}{2}right] )
respectively
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
717 ( 2 tan ^{-1}left(frac{1}{3}right)+tan ^{-1}left(frac{1}{7}right) ) is equal to
A. ( frac{pi}{6} )
в. ( frac{pi}{4} )
( c cdot frac{pi}{3} )
( D cdot frac{pi}{2} )
12
718 Assertion ( f_{i=1}^{2 n} sin ^{-1} x_{i}=n pi forall n epsilon N ) then ( sum_{i=1}^{2 n} x_{i}= )
( sum_{i=1}^{2 n} x_{i}^{2}=sum_{i=1}^{2 n} x_{i}^{n}=2 n )
Reason ( -frac{pi}{2} leq sin ^{-1} x leq frac{pi}{2} forall x epsilon[-1,1] )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
12
719 Find the principal value of ( tan ^{-1} sqrt{3}- )
( sec ^{-1}(-2) )
12
720 The value of ( sin ^{-1} x+cos ^{-1} x, forall x in )
[-1,1] is
A ( cdot frac{pi}{2} )
в. ( frac{-5 pi}{3} )
c. ( frac{-3 pi}{2} )
D.
12
721 Prove: ( tan ^{-1}left(frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}right)= )
( frac{pi}{4}-frac{1}{2} cos ^{-1} x,-frac{1}{sqrt{2}} leq x leq 1 )
[Hint: ( p u t x=cos 2 theta] )
12
722 Solve:
( cos ^{-1}(cos x)=pi+x, ) then ( x ) belongs
to
В. ( (pi, 2 pi) )
D. None of these
12
723 Let ( boldsymbol{f}(boldsymbol{x})=operatorname{cosec}^{-1}left[1+sin ^{2} boldsymbol{x}right], ) where
[.] denotes the greatest integer function
Then ( f(x) ) equals;
( ^{A} cdotleft{frac{pi}{2}right} )
В ( cdotleft{frac{pi}{2}, operatorname{cosec}^{-1} 2right} )
c. ( left{operatorname{cosec}^{-1} 2right} )
D. none of these
12
724 6. If x <0, then tan-'x is equal to
a. – 1 + cot-1 ]
b.
sin-1
1+x²
a. – cos dit
d. – cosec V1 + x²
12
725 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} )
then ( frac{sum_{k=1}^{2}left(x^{100 k}+y^{106 k}right)}{sum x^{207} cdot y^{207}} ) is
A ( cdot frac{1}{3} )
B. ( frac{4}{3} )
( c cdot frac{2}{3} )
D. None of these
12
726 If the number ( 93215 x 2 ) is completely divisible by ( 11, ) then ( x ) is equal to
( A cdot 2 )
B. 3
( c cdot 1 )
D. 4
12
727 Find the value of ( x ) which satisfy equation ( : sin ^{-1} x+sin ^{-1} 2 x=frac{pi}{3} )
A ( cdot x=frac{1}{2} sqrt{frac{3}{7}} )
B・ ( x=frac{1}{3} sqrt{frac{4}{7}} )
c. ( x=frac{1}{3} sqrt{frac{3}{7}} )
D. ( x=frac{1}{2} sqrt{frac{4}{7}} )
12
728 If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) and
( boldsymbol{f}(1)=2, f(x+y)=f(x) f(y) ) for all
( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} . ) Then ( boldsymbol{x}^{f(1)}+boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(3)}- )
( frac{boldsymbol{x}+boldsymbol{y}+boldsymbol{z}}{boldsymbol{x}^{f(1)}+boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(3)}} ) is equal to
( A cdot 0 )
B. 1
( c cdot 2 )
D. 3
12
729 22. The value of sin ‘(sin 12) + cos'(cos 12) is equal to
a. zero
O b. 24 – 21
c. 41-24
d. none of these
12
730 Find the possible value of ( cos x, ) if ( cot x )
( +operatorname{cosec} x=5 )
12
731 The sum of all the solution(s) of the
equation ( sin ^{-1} 2 x=cos ^{-1} x ) is
12
732 2x
18. If 2 tan-‘x+ sin-1-
et sin
is independent of x, then
1+x
o
n
a. x >1
c. 0<x< 1
b. x<-1
d. – 1<x<0
20
12
733 Find the principal value of ( tan left(cos ^{-1} frac{1}{2}right) ) 12
734 The trigonometric equation ( sin ^{-1} x= )
( 2 sin ^{-1} 2 a ) has a real solution, if
A ( cdot|a|>frac{1}{sqrt{2}} )
в. ( frac{1}{2 sqrt{2}}<|a|frac{1}{2 sqrt{2}} )
D ( cdot|a| leq frac{1}{2 sqrt{2}} )
12
735 f ( sin ^{-1}left(frac{x}{13}right)+operatorname{cosec}^{-1}left(frac{13}{12}right)=frac{pi}{2} )
then the value if ( x ) is
A . 5
B. 4
c. 12
D. 11
12
736 Evaluate the following:
( tan ^{-1}(tan 12) )
12
737 1. cos’ (cos(2 cot'( 12 – 1))) is equal to
a. √2-1
d. none of these
12
738 ( sin ^{-1}left(cos left(sin ^{-1} xright)right)+ )
( cos ^{-1}left(sin left(cos ^{-1} xright)right) ) is equal to
( A cdot frac{pi}{2} )
B.
( c cdot frac{3 pi}{4} )
D.
12
739 Theorem: For any ( boldsymbol{x} in boldsymbol{R} quad sinh ^{-1} boldsymbol{x}= )
( log _{e}(x+sqrt{x^{2}+1}) )
12
740 Find the principal value of:
( tan ^{-1}left(2 cos frac{2 pi}{3}right) )
12
741 пп
Illustration 5.12 If sin-‘x, + sin-x2 + … + sin ‘x, S-
ne N, n = 2m + 1, m > 1, then find the value of
x1 + x3 + x +…(m+1) terms
xż + x + x +…m terms
12
742 If ( 4 cos ^{-1} x+sin ^{-1} x=pi, ) then the
value of ( x ) is.
A ( cdot frac{1}{2} )
в. ( frac{1}{sqrt{2}} )
( c cdot frac{sqrt{3}}{2} )
D. ( frac{2}{sqrt{3}} )
12
743 1. If a, Ba<B) are the roots of the equation 6×2 + 11x + 3
= 0, then which of the following are real?
a. cosa
b. sin'B
c. cosec-'a
d. Both cota and cot B
12
744 Let ( x_{1}, x_{2}, x_{3}, x_{4} ) be four non zero
numbers satisfying the equation ( tan ^{-1} frac{a}{x}+tan ^{-1} frac{b}{x}+tan ^{-1} frac{c}{x}+ )
( tan ^{-1} frac{d}{x}=frac{pi}{2} )
This question has multiple correct options
( ^{mathbf{A}} cdot sum_{i=1}^{4} x_{i}=a+b+c=d )
( ^{mathrm{B}} cdot sum_{i=1}^{4} frac{1}{x_{1}}=0 )
( mathbf{c} cdot Pi_{i=1}^{4} x_{i}=a b c d )
D. ( left(x_{1}+x_{2}+x_{3}right)left(x_{2}+x_{3}+x_{4}right)left(x_{3}+x_{4}+x_{1}right)left(x_{4}+x_{1}+right. )
( left.x_{2}right)=a b c d )
12
745 The number of solutions of the
equation ( 1+x^{2}+2 x sin left(cos ^{-1} yright)=0 )
is
( A )
B. 2
( c cdot 3 )
( D cdot 4 )
12
746 What is ( sin ^{-1} sin frac{3 pi}{5} ) equal to ( ? )
( mathbf{A} cdot frac{3 pi}{5} )
B. ( frac{2 pi}{5} )
( c cdot frac{pi}{5} )
D. None of the above
12
747 Evaluate: ( cos left[2 tan ^{-1}left[frac{1}{7}right]right] )
( A cdot sin left(4 cot ^{-1} 3right) )
B. ( sin left(3 cot ^{-1} 4right) )
( c cdot cos left(3 cot ^{-1} 4right) )
D. ( cos left(4 cot ^{-1} 4right) )
12
748 Solve for ( x ; cos ^{-1} sqrt{x}>sin ^{-1} sqrt{x} ) 12
749 Solve ( : 2 tan ^{-1}(-3)= )
This question has multiple correct options
( mathbf{A} cdot-cos ^{-1}(-4 / 5) )
( mathbf{B} cdot-pi+cos ^{-1}(4 / 5) )
C ( cdot-frac{pi}{2}+tan ^{-1}(-4 / 3) )
D. ( cot ^{-1}(4 / 3) )
12
750 The value of ( cos ^{-1}left(cos frac{7 pi}{6}right)= )
( A cdot frac{7 pi}{6} )
в. ( frac{5 pi}{6} )
( c cdot frac{pi}{3} )
D.
12
751 Solve ( : int frac{tan ^{-1} x}{1+x^{2}} d x ) 12
752 Write in simplest form ( sin ^{-1}left[frac{sqrt{1+x}+sqrt{1-x}}{2}right] ) 12
753 Evaluate the following:
( sin ^{-1}(sin 5) )
12
754 Which one of the following statement is meaningless?
( ^{mathbf{A}} cdot cos ^{-1}left(ln left(frac{2 e+4}{3}right)right) )
B. ( operatorname{cosec}^{-1}left(frac{pi}{3}right) )
c. ( cot ^{-1}left(frac{pi}{2}right) )
D ( cdot sec ^{-1}(pi) )
12
755 where x < 1, then x is equal to
ماده
b. –
c
ده
d.
12
756 63. If cos ‘x + cos’y + cos’z = , then
a. x2 + y2 + z2 + xyz = 0 b. x² + y2 + z2 + 2xyz = 0)
c. x2 + y2 + 2 + xyz = 1 d. x2 + y2 + 2 + 2xyz = 1
12
757 ( cot ^{-1}(2+sqrt{3})= )
( A cdot frac{pi}{12} )
B. ( frac{pi}{15} )
( c cdot frac{pi}{5} )
( D cdot frac{3 pi}{10 pi} )
12
758 The domain of ( mathbf{f}(mathbf{x})= ) ( cot ^{-1}left(frac{mathbf{x}}{sqrt{mathbf{x}^{2}-left[mathbf{x}^{2}right]}}right) ) is
( ([.] ) denotes the greatest integer function)
A. ( (0, infty) )
)
в. ( mathrm{R}-{0 )
c. ( R-{x: x in Z} )
D. ( (-infty, 0) )
12
759 Solve: ( 3 tan ^{-1} x+cot ^{-1} x=pi ) 12
760 The value of ( sin ^{-1}(sin 2) ) is?
A ( .2+n pi )
B. ( 2-pi )
( c cdot-2+pi )
D. ( 2-2 n pi )
12
761 49. If cos? Vp + cos’ V1- p + cos’ V1-9 =
37, then
the value of q is
a. 1
-la
12
762 ( boldsymbol{alpha}=sin ^{-1}left(cos left(sin ^{-1} xright)right) ) and ( beta= )
( cos ^{-1}left(sin left(cos ^{-1} xright)right) ) then:
( A cdot tan alpha=cot beta )
B. ( tan alpha=-cot beta )
( mathbf{c} cdot tan alpha=tan beta )
D. ( tan alpha=-tan beta )
12
763 Solve ( y=tan ^{-1}left(frac{cos x}{1-sin x}right) ) 12
764 Find the value of ( sin left[frac{1}{2} cot ^{-1}left(frac{-3}{4}right)right] ) 12
765 Solve ( tan x<2 ) 12
766 Find the principal value of the following
( tan ^{-1}left(tan frac{7 pi}{6}right) )
12
767 Find the principal value of:
( cot ^{-1}(sqrt{3}) )
12
768 If ( boldsymbol{f}:left(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) rightarrow(-infty, infty) ) is defined
by ( f(x)=tan x, ) then ( f^{-1}(sqrt{3})= )
12
769 How do you simplify
( sin x+cot x cdot cos x )
12
770 Integrate the function ( tan ^{-1}(sqrt{frac{1-sin x}{1+sin x}}) ) w.r.t d ( x ) 12
771 If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots inftyright)=frac{pi}{2} ) for
( mathbf{0}<|boldsymbol{x}|<sqrt{mathbf{2}}, ) then ( boldsymbol{x} ) equal
( A cdot frac{1}{2} )
B.
( c cdot frac{-1}{2} )
D. -1
12
772 If ( tan ^{-1}(2 x)+tan ^{-1}(3 x)=pi / 4 ) then
( x=? )
12
773 92. If 227/sin-‘x – 2(a+ 2) 24/sin- x + 8a < 0 for at least one
real x, then
a. Isa<2
b. a<2
20
c. a € R-{2}
d. ae (o 1 u 12,0)
d.
a e
12
774 If ( tan ^{-1}(cot theta)=2 theta ) then ( theta= )
A ( cdot frac{pi}{3} )
в. ( frac{pi}{4} )
c.
D. None of the above
12
775 The number of solutions of the equation
( mathbf{2}left(boldsymbol{operatorname { S i n }}^{-1} boldsymbol{x}right)^{2}-mathbf{5} operatorname{Sin}^{-1} boldsymbol{x}+mathbf{2}=mathbf{0} ) is
A .
в.
( c cdot 2 )
( D )
12
776 Find the domain of the following
function:
( boldsymbol{f}(boldsymbol{x})=operatorname{cosec}^{-1}left[mathbf{1}+sin ^{2} boldsymbol{x}right], ) where
denotes the greatest integer function
12
777 ( tan left[2 tan ^{-1}left(frac{sqrt{1+x^{2}}-1}{x}right)right]= )
( A cdot x )
B. ( 2 x )
c. ( x / 2 )
D. ( 3 x )
12
778 Calculate. ( arctan 1+arccos left(-frac{1}{2}right)+ )
( arcsin left(-frac{1}{2}right)=? )
12
779 to our
Σ sin-
is equal to
P=
Irort1
c. tan”(Tn)
d. tan” (Jn+1)
12
780 If value of ( x ) which satisfy equation
( cos ^{-1} x<2 ) is ( x epsilon(a, b] )
Find the value of ( a+b )
A. ( -1-cos 2 )
B. ( 1-cos 2 )
c. ( -1+cos 2 )
D. ( 1+cos 2 )
12
781 Solve:
( sin ^{-1} x+sin ^{-1} sqrt{1-x^{2}} )
12
782 If range of the function ( boldsymbol{f}(boldsymbol{x})= ) ( sin ^{-1} x+2 tan ^{-1} x+x^{2}+4 x+1 ) is
( [p, q], ) then the value of ( (p+q) ) is
12
783 Solve the equation ( 2 tan ^{-1}(cos x)=tan ^{-1}(2 csc x) ) 12
784 Find the principal value of the following
( cos ^{-1} frac{1}{2}+2 sin ^{-1} frac{1}{2} )
12
785 The value of ( tan left[cos ^{-1} frac{4}{5}+tan ^{-1} frac{2}{3}right] ) is
A ( cdot frac{6}{17} )
в. ( frac{7}{16} )
c. ( frac{17}{6} )
D. none of these
12
786 Solve: ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} ) 12
787 ( f cot left(cos ^{-1} frac{3}{5}+sin ^{-1} xright)=0, ) find the
value of ( boldsymbol{x} )
12
788 ( tan ^{-1}left(tan frac{2 pi}{3}right)= )
A ( cdot frac{pi}{3} )
B ( cdot frac{2 pi}{3} )
c. ( -frac{pi}{3} )
D. ( -frac{2 pi}{3} )
12
789 toppr
Q Type your question.
( left.right|^{x in L}left|: sin left(log _{e} mid overline{x-1}right)right|^{operatorname{ls} a} )
(Here, the inverse trigonometric
function ( sin ^{-1} x ) assumes values in ( left.left[-frac{pi}{2}, frac{pi}{2}right]right) )
Let ( boldsymbol{f}: boldsymbol{E}_{1} rightarrow mathbb{R} ) be the function define by ( f(x)=log _{e}left(frac{x}{x-1}right) ) and ( g: E_{2} rightarrow mathbb{R} )
be the function defined by ( g(x)= ) ( sin ^{-1}left(log _{e}left(frac{x}{x-1}right)right) )
LIST-1
1. ( left(-infty, frac{1}{1-e}right] )
The range of ( f ) is ( quadleft[frac{e}{e-1}, inftyright) )
Q. The range of ( g )
contins
R. The domain of ( f )
contains
[
begin{array}{ll}
text { S. The domain of } g text { is } & 4 .(-infty, 0) cup(0, infty) \
& 5 .left(-infty, frac{e}{e-1}right] \
& text { 6. }(-infty, 0) cupleft(frac{1}{2}, frac{e}{e-1}right]
end{array}
]
The correct option is
A. ( P rightarrow 4 ; Q rightarrow 2 ; R rightarrow 1 ; S rightarrow 1 )
в. ( P rightarrow 3 ; Q rightarrow 3 ; R rightarrow 6 ; S rightarrow 5 )
c. ( P rightarrow 4 ; Q rightarrow 2 ; R rightarrow 1 ; S rightarrow 6 )
D. ( P rightarrow 4 ; Q rightarrow 3 ; R rightarrow 6 ; S rightarrow 5 )
12
790 ( operatorname{Let} 2 y=left(cot ^{-1}left(frac{sqrt{3} cos x+sin x}{cos x-sqrt{3} sin x}right)right)^{2} )
then ( frac{boldsymbol{a} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is equal to
A ( cdot x-frac{pi}{6} )
B. ( x+frac{pi}{6} )
c. ( 2 x-frac{pi}{6} )
D. ( 2 x-frac{pi}{3} )
12
791 Solve the equation ( tan ^{-1}left(frac{1-x}{1+x}right)= )
( frac{1}{2} tan ^{-1} x, x>0 )
12
792 The value of ( cos ^{-1}left(cos left(frac{4 pi}{3}right)right) ) is
A ( cdot 2 pi / 3 )
в. ( -2 pi / 3 )
c. ( 4 pi / 3 )
D. ( -4 pi / 3 )
12
793 Prove that ( tan ^{-1} frac{1}{2}+tan ^{-1} frac{2}{11}= )
( tan ^{-1} frac{3}{4} )
12
794 find the value of the following:
( (i) sin ^{-1}left(frac{-1}{2}right) )
( (i i) cos ^{-1}left(frac{sqrt{3}}{2}right) )
( (i i i) operatorname{cosec}^{-1}(2) )
( (i v) tan ^{-1}(-sqrt{3}) )
( (v) cos ^{-1}left(frac{-1}{2}right) )
( (v i) tan ^{-1}(-1) )
12
795 ( cos left(cot ^{-1}left(operatorname{cosec}left(cos ^{-1} aright)right)right)=dots )
(where ( 0<a<1) )
A ( cdot frac{1}{sqrt{2-a^{2}}} )
B. ( sqrt{3-a^{2}} )
c. ( sqrt{2-a^{2}} )
D. ( frac{1}{sqrt{2+a^{2}}} )
12
796 Show that ( cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)= )
( frac{x}{2} ) for ( x inleft(0, frac{pi}{2}right) )
12
797 Evaluate ( cos ^{-1}(cos 3) ) 12
798 Let ( boldsymbol{f}(boldsymbol{x})=sin ^{-1} boldsymbol{x}+cos ^{-1} boldsymbol{x}, ) then ( frac{boldsymbol{pi}}{2} ) is
equal to This question has multiple correct options
A ( cdot fleft(-frac{1}{2}right) )
B . ( fleft(k^{2}-2 k+3right), k in R )
c. ( fleft(frac{1}{1+k^{2}}right), k in R )
D. ( f(-2) )
12
799 Illustration 5.70 Which of the following angles is greater ?
COS
12
800 Find the principal value of ( cot ^{-1}(sqrt{3}) ) 12
801 11. If sin (x – 1) + cos(x – 3) + tan-
Tt, then the value of k is
= cos-‘k +
nie.
a. 1
b. – Ja
1
d. none of these 02.10
12
802 If ( 0<x<1, ) then ( tan ^{-1}left(frac{sqrt{1-x^{2}}}{1+x}right) ) is
equal to
( ^{mathbf{A} cdot} frac{1}{2} cos ^{-1} x )
B. ( cos ^{-1} frac{sqrt{1+x}}{2} )
c. ( sin ^{-1} sqrt{frac{1-x}{2}} )
D. ( frac{1}{2} sqrt{frac{1+x}{1-x}} )
12
803 the number of real solutions of the
equation ( tan ^{-1} sqrt{x^{2}-3 x+2}+ )
( cos ^{-1} sqrt{4 x-x^{2}-3}=pi ) is
A. one
B. two
c. zero
D. infinite
12
804 f ( tan ^{-1} frac{x-3}{x-4}+tan ^{-1} frac{x+3}{x+4}=frac{3}{4}, ) then
find the value of ( x )
12
805 Which of the following is/are the value of 12
806 The solutions set of inequality ( cos ^{-1} x<sin ^{-1} x ) is
A ( cdot[-1,1] )
B. ( left[frac{1}{sqrt{2}}, 1right] )
c. [0,1]
D. ( left(frac{1}{sqrt{2}}, 1right] )
12
807 Find the value of
[
tan left{frac{1}{2} sin ^{-1}left(frac{2 x}{1+x^{2}}right)+frac{1}{2} cos ^{-1}left(frac{1-}{1+}right.right.
]
if ( boldsymbol{x}>boldsymbol{y}>1 )
12
808 ( frac{x}{5} ) 12
809 Find the principal value of ( operatorname{cosec}^{-1}left(frac{2}{sqrt{3}}right) ) 12
810 (sin x)
Illustration 5.21 Find the area bounded by y=sin
and x-axis for x = [0, 1006].
12
811 Find the value of ( cos left(sec ^{-1} x+right. )
( left.csc ^{-1} xright),|x| geq 1 )
12
812 The largest interval lying in ( left(frac{-pi}{2}, frac{pi}{2}right) ) for which the function
( left[f(x)=4^{-x^{2}}+cos ^{-1}left(frac{x}{2}-1right)+log (c oright. )
is defined, is-
A . ( [0, pi] )
в. ( left(frac{-pi}{2}, frac{pi}{2}right) )
c. ( left[-frac{pi}{4}, frac{pi}{2}right) )
D. ( left[0, frac{pi}{2}right) )
12
813 Find the value of ( sin ^{-1}left(sin frac{3 pi}{5}right) ) 12
814 ( tan left(2 tan ^{-1}left(frac{sqrt{5}-1}{2}right)right)= )
( A )
B. 3
( c cdot 2 )
( D )
12
815 For ( boldsymbol{x} in(mathbf{0}, boldsymbol{pi} / mathbf{2}) )
( sin ^{-1}(cos x)=? )
A. ( pi-x )
B ( cdot frac{pi}{2}-x )
c. ( pi-frac{x}{2} )
D . ( pi-2 x )
12
816 The smallest and the largest values of ( tan ^{-1}left(frac{1-x}{1+x}right), 0 leq x leq 1 ) are
A ( .0, pi )
в. ( 0, frac{pi}{4} )
( c cdot-frac{pi}{4}, frac{pi}{4} )
D. ( frac{pi}{4}, frac{pi}{2} )
12
817 The trigonometric equation ( sin ^{-1} x= ) ( 2 sin ^{-1} 2 a ) has a real solution if
A ( cdot|a|>frac{1}{sqrt{2}} )
в. ( frac{1}{2 sqrt{2}}<|a|frac{1}{2 sqrt{2}} )
D ( cdot|a| leq frac{1}{2 sqrt{2}} )
12
818 If ( sin ^{-1}left(frac{1}{3}right)+sin ^{-1}left(frac{2}{3}right)=sin ^{-1} x, ) then
( x ) is equal to
( mathbf{A} cdot mathbf{0} )
B. ( frac{sqrt{5}+4 sqrt{2}}{9} )
c. ( frac{5 sqrt{2}-4 sqrt{5}}{9} )
D.
12
819 ff ( y=sin left(cos ^{-1} xright) ) and ( x=99, ) then
( 1 / y^{2} ) is equal to
12
820 Which of the following is/are a rational number?
1
b. cos

-sin
12
821 Calculating the principal value, find the value of ( sin left[2 sin ^{-1}left(frac{4}{5}right)right] ) 12
822 cos x
26.
tan
nx

for x E
12
823 What is the value of
( cos left{cos ^{-1} frac{4}{5}+cos ^{-1} frac{12}{13}right} ? )
A . ( 63 / 65 )
B. 33/65
c. 22/65
D. ( 11 / 65 )
12
824 ( f sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots dots dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots dots dots inftyright)=frac{pi}{2} )
and ( 0<x<sqrt{2} ) then ( x= )
( A cdot frac{1}{2} )
B.
( c cdot-frac{1}{2} )
D. – –
12
825 Find the principal value the following
expression:
( sin ^{-1}left(-frac{sqrt{3}}{2}right) )
12
826 Solve for ( x: )
( left(tan ^{-1} xright)^{2}+left(cos ^{-1} xright)^{2}=frac{5 pi^{2}}{8} )
12
827 ( frac{cos ^{-1}(41 / 49)}{sin ^{-1}(2 / 7)}= )
( A cdot 4 )
B. 3
( c cdot 2 )
( D )
12
828 Evaluate the following:
( sin ^{-1}(sin 10) )
12
829 Illustration 5.46
(cosec-‘x)?
Find the minimum value of (sec- x)2 +
12
830 75. If 3 tan-
– tan’ – = tan
, then x is equal to
a. 1
c. 3 Set
Sb. 2
ons d. 2
12
831 Solve: ( cot left(cos e c^{-1} frac{5}{3}+tan ^{-1} frac{2}{3}right) )
A ( cdot frac{6}{17} )
в. ( frac{3}{17} )
c. ( frac{4}{17} )
D. ( frac{5}{17} )
12
832 The value of ( cos ^{-1}left(cos frac{7 pi}{6}right) ) is equal to
A ( cdot frac{7 pi}{6} )
в. ( frac{5 pi}{6} )
( c cdot frac{pi}{3} )
D.
12
833 The value of ( sin left(2 sin ^{-1} 0.8right) ) is equal to.
( mathbf{A} cdot sin ^{-1} 1.2 )
B. ( sin ^{-1}(0.96) )
c. ( sin ^{-1}(0.48) )
D. sin ( 1.6^{circ} )
12
834 If ( cot ^{-1}left[(cos alpha)^{1 / 2}right]+ )
( left[tan ^{-1}(cos alpha)^{1 / 2}right]=x, ) then ( sin x )
equals
A .
B ( cdot cot ^{2}left(frac{alpha}{2}right) )
( mathbf{c} cdot tan alpha )
D. ( cot left(frac{alpha}{2}right) )
12
835 4
If sin
+ cosec
)
, then the values of x is
(a) 4
(c) 1
[20071
(b) 5
(d) 3
12
836 Prove:
( 2 tan ^{-1}left(sqrt{frac{a-b}{a+b}} tan frac{theta}{2}right)=cos ^{-1} )
( left(frac{a cos theta+b}{a+b cos theta}right) )
12
837 ( cos ^{-1}left{frac{1}{sqrt{2}}left(cos frac{9 pi}{10}-sin frac{9 pi}{10}right)right}= )
A ( cdot frac{23 pi}{20 pi} )
B. ( frac{7 pi}{10} )
( c cdot frac{7 pi}{20} )
D. ( frac{17 pi}{20 pi} )
12
838 Solve: ( sin ^{-1} frac{5}{x}+sin ^{-1} frac{12}{x}=frac{pi}{2} ) 12
839 Differentiate ( cos ^{-1}left(4 x^{2}-3 xright) ; x epsilonleft(frac{1}{2}, 1right) ) 12
840 Find the value of ( tan ^{-1}left(frac{1}{2} tan 2 Aright)+ )
( tan ^{-1}(cot A)+tan ^{-1}left(cot ^{3} Aright), ) for ( 0< )
( boldsymbol{A}<frac{boldsymbol{pi}}{boldsymbol{4}} )
( mathbf{A} cdot-pi / 2 )
в. ( +pi / 2 )
c. ( -pi )
D. ( +pi )
12
841 If ( cos ^{-1} x= )
( left{begin{array}{r}a pi-b cos ^{-1}left(2 x^{2}-1right), i f-1 leq x< \ c cos ^{-1}left(2 x^{2}-1right), text { if } 0 leq x leq 1end{array}right. )
Find the value of ( a+b+c )
A . 1
B. 2
( c .3 )
D. 4
12
842 3.
If cosx -cos-1y = a, then 4×2 – 4xy cos a + y is
equal to
[2005]
(a) 2 sin 2a (6) 4
(c) 4 sin? a (d) – 4 sin² a
12
843 The value of ( sin left(2 sin ^{-1}(0.8)right) ) is equal
to
( A cdot sin 1.2^{circ} )
B. ( sin 1.6^{circ} )
c. 0.48
D. 0.96
12
844 If ( boldsymbol{f}(boldsymbol{x})= )
( sin {[boldsymbol{x}+mathbf{5}]+{boldsymbol{x}-{boldsymbol{x}-{boldsymbol{x}}}}} ) for
( boldsymbol{x} inleft(mathbf{0}, frac{boldsymbol{pi}}{mathbf{4}}right) ) is invertible, where ( {.} )
and [.] represent fractional part and greatest integer functions respectively,
then ( boldsymbol{f}^{-1}(boldsymbol{x}) ) is
This question has multiple correct options
( mathbf{A} cdot sin ^{-1} x )
в. ( frac{pi}{2}-cos ^{-1} x )
( c cdot sin ^{-1}{x} )
D ( cdot cos ^{-1}{x} )
12
845 If ( 2 tan h^{-1} x=log y, ) then the value of ( y )
in the terms of ( x ) is
A ( .2 x )
в. ( frac{2 x}{1-x^{2}} )
c. ( x^{2} )
D. ( left(frac{1+x}{1-x}right) )
12
846 The numerical value of tan ( left(2 tan ^{-1} frac{1}{5}-frac{pi}{4}right) ) is 12
847 40. For 0 < 0 cos? (sin ) is true when
12
848 ( tan ^{-1}(1)+cos ^{-1}left(-frac{1}{2}right)+sin ^{-1}left(-frac{1}{2}right) ) 12
849 Find the value of ( x, ) if:
( tan ^{-1}left(frac{x-2}{x-1}right)+tan ^{-1}left(frac{x+2}{x+1}right)=frac{pi}{4} )
12
850 If ( tan ^{-1} frac{1-x}{1+x}=frac{1}{2} tan ^{-1} x, ) then ( x= )
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{3} )
c. ( frac{1}{sqrt{3}} )
D. None of these
12
851 Find the principal value of the following:
( cot ^{-1}(-1) )
12
852 The value of ( sin left(2 tan ^{-1}(1 / 3)right)+ )
( cos left(tan ^{-1} 2 sqrt{2}right) ) is
( A cdot 12 / 13 )
B. ( 13 / 14 )
c. ( 14 / 15 )
D. none of these
12
853 Assertion
If ( boldsymbol{x}<mathbf{0}, tan ^{-1} boldsymbol{x}+tan ^{-1} frac{mathbf{1}}{boldsymbol{x}}=frac{boldsymbol{pi}}{mathbf{2}} )
Reason ( tan ^{-1} x+cot ^{-1} x=frac{pi}{2} forall x in R )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
854 Find the principal value of:
( sec ^{-1}left(2 tan frac{3 pi}{4}right) )
12
855 If ( U=cot ^{-1} sqrt{cos 2 theta}-tan ^{-1} sqrt{cos 2 theta} )
then ( sin U ) is equal to
( A cdot sin ^{2} theta )
B. ( cos ^{2} theta )
( mathbf{c} cdot tan ^{2} theta )
( mathbf{D} cdot tan ^{2} 2 theta )
12
856 ( fleft(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then find
the value of ( x )
12
857 72. If cot’ (Vcos a) – tan! (Vcos a ) = x, then sinx is
a. tang
b. cot a
c. tan
10
d. cot
810
12
858 28. The trigonometric equation sin-‘x=2 sin ‘a has a solution
for
a. all real values
b.
V
12
859 5.
The value of x for which sin (cot -1 (1+x)) = cos (tan- x) is
(2004)
(a) 112 (b) 1 (C) 0 (d) -1/2
12
860 If ( tan ^{-1}left(frac{x}{sqrt{a^{p}-x^{q}}}right)= )
( sin ^{-1}left(frac{x}{a}right), a>0 . ) Find the value of
and ( q )
( mathbf{A} cdot p=1, q=1 )
( mathbf{B} cdot p=1, q=2 )
( mathbf{c} cdot p=2, q=1 )
D ( . p=2, q=2 )
12
861 Find the principal value:
( sin ^{-1}left(tan frac{5 pi}{4}right) )
12
862 For the principal value:
( sin ^{-1}left(-frac{sqrt{mathbf{3}}}{mathbf{2}}right)+cos ^{-1}left(frac{sqrt{mathbf{3}}}{mathbf{2}}right) )
12
863 Illustration 5.38
Find the minimum value of the function
Ax)=
16 cot-10–cot-‘x.
12
864 Evaluate:
( tan ^{-1}left(frac{sqrt{1+cos x}-sqrt{1-cos x}}{sqrt{1+cos x}+sqrt{1-cos x}}right) )
12
865 Prove that
( sin ^{-1} frac{3}{5}+sin ^{-1} frac{8}{17}=cos ^{-1} frac{36}{85} )
12
866 a

b
74. The value 2 ans (Ver mais equilito
74. The value 2 tan-1
tan
is equal to
Va+h
a. cos-1/ a cosO+b
b.
cos-1
(a+bcos e
a cos 0 + b
(a + bcos e)
c. cos-
a cos e
(a + bcos )
d. cos
(bcoso
a cos 0 +b )
12
867 If ( y=cot ^{-1}(sqrt{cos x})- )
( tan ^{-1}(sqrt{cos x}) P . T sin y=tan ^{2} x / 2 )
12
868 Write the value of
( tan ^{-1}left[2 sin left(2 cos ^{-1} frac{sqrt{3}}{2}right)right] )
12
869 The number of solution of the equation
( 1+x^{2}+2 x sin left(cos ^{-1} yright)=0 ) is :
( A )
B. 2
( c cdot 3 )
D. 4
12
870 8. If (sin-‘x + sin ‘w) (sin-y + sin- z) = re”, then
DE
23 WAN, N2, N₃, NEN)
a. has a maximum value of 2
b. has a minimum value of 0
c. 16 different D are possible
d. has a minimum value of -2
12
871 2. The value of
sin
cot sin-1 12-13
los V 4
+cos-1 V12
+sec-12
4
Rim
d. none of these
12
872 If ( I sin ^{-1} x-cos ^{-1} x=frac{pi}{6}, ) then solve for
( boldsymbol{x} ) ?
12
873 Solve:
( tan ^{-1}left(tan frac{7 pi}{6}right) )
12
874 ( f(x)=tan ^{-1}(sin x+cos x) ) is an
increasing function in This question has multiple correct options
A ( cdotleft(0, frac{pi}{4}right) )
В ( cdotleft(0, frac{pi}{2}right) )
c. ( left(frac{-pi}{4}, frac{pi}{4}right) )
D. None of these
12
875 If ( cot ^{-1} x+cot ^{-1} y+cot ^{-1} z=frac{pi}{2} ) then
( boldsymbol{x}+boldsymbol{y}+boldsymbol{z} ) equals
A ( . x y z )
в. ( x y+y z+z x )
( mathrm{c} cdot 2 x y z )
D. None of these
12
876 If ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is
independent of ( x ) then
A ( cdot x epsilon(1,+infty) )
В . ( x epsilon(-1,1) )
c. ( x epsilon(-infty,-1) )
D. none of these
12
877 ff ( y=2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) then
A. ( -pi / 2<y<pi / 2 )
в. ( -3 pi / 2<y<3 pi / 2 )
c. ( -pi<y<pi )
D. ( -pi / 4<y<pi / 4 )
12
878 The value of ( sin ^{-1} )
( left{tan left(cos ^{-1} sqrt{frac{2+sqrt{3}}{4}}+cos ^{-1} frac{sqrt{12}}{4}-right.right. )
is
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
( c cdot-frac{pi}{2} )
D.
12
879 Solve:
( 2 tan ^{-1} frac{3}{4}-tan ^{-1} frac{17}{31} )
12
880 Simplify ( cot ^{-1} frac{1}{sqrt{x^{2}-1}} ) for ( x<-1 )
( A cdot cos ^{-1} x )
( mathbf{B} cdot sec ^{-1} x )
( mathbf{c} cdot operatorname{cosec}^{-1} x )
D. ( tan ^{-1} x )
12
881 Simplify: ( sin . cot ^{-1} cot x ) 12
882 The value of ( cos left(tan ^{-1}left(frac{3}{4}right)right) ) is
( A cdot frac{4}{5} )
B.
( c cdot frac{3}{4} )
D.
12
883 Illustration 5.10 If sin- ‘(x2 + 2x + 2) + tan- ‘(x2 – 3x – K)
then find the values of k.
12
884 Prove ( 4 tan ^{-1}left(frac{1}{5}right)-tan ^{-1}left(frac{1}{70}right)+ )
( tan ^{-1}left(frac{1}{99}right)=frac{pi}{4} )
12
885 Illustration 5.16 Evaluate the following:
i. sin(sin at/4) ii. cos'(cos27/3)
iii. tan(tan 7/3)
12
886 Prove that ( sin cot ^{-1} tan cos ^{-1} x= )
( sin operatorname{cosec}^{-1} cot tan ^{-1} x=x ) where
( x in(0,1] )
12
887 Prove that :
( 3 cos ^{-1} x=cos ^{-1}left(4 x^{3}-3 xright) )
12
888 Illustration 5.48 If a= sin-‘(cos(sin- x)) and ß=
cos-‘(sin(cos+ x)), then find tan a • tan B.
12
889 The number of positive integral
solutions of the equation ( tan ^{-1} x+ ) ( cot ^{-1} y=tan ^{-1} 3 ) is :
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
890 Find ( x ) if ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3} ) 12
891 5. Find the number of positive integral solutions of the
3
equation tan’x + cos
= =sin-l
√ 1 – 2
TO
12
892 4. The value of cos
-COS
is
Cool 7
one
c.
de
12
893 The solution set of the equation ( sin ^{-1} sqrt{1-x}+cos ^{-1} x= )
( cot ^{-1}left(frac{sqrt{1-x^{2}}}{x}right)-sin ^{-1} x )
A ( cdot[-1,1]-{0} )
B . (0,1]( cup{-1} )
c. [-1,0)( cup{1} )
D. {1}
12
894 Evaluate: ( tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right) ) 12
895 xo
71. The value of tan-
-cot-1
cose
a. 20
c. 0/2
(1-xsin o)
x-sin e
b. 0
d. independent of e
12
896 If the minimum value of ( left(sec ^{-1} xright)^{2}+ )
( left(operatorname{cosec}^{-1} xright)^{2} ) is ( frac{pi^{a}}{b} . ) Find the value of
( a+b )
( mathbf{A} cdot mathbf{6} )
B. 8
c. 10
D. 12
12
897 The value of ( tan left[frac{1}{2} cos ^{-1}left(frac{2}{3}right)right] ) is
A ( cdot frac{1}{sqrt{5}} )
B. ( sqrt{frac{3}{10}} )
c. ( sqrt{frac{5}{2}} )
D. ( 1-sqrt{frac{5}{2}} )
12
898 If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots inftyright)=frac{pi}{2} )
and ( 0<x<sqrt{2} ) then ( x= )
( A cdot frac{1}{2} )
B.
( c cdot frac{-1}{2} )
D. –
12
899 48. If sin-1
48. If sin
+ sin(12) – , then x is equal to
+ sin-1
X
c. 13
12
900 The value of ( sec left[tan ^{-1} frac{b+a}{b-a}-tan ^{-1} frac{a}{b}right] ) is
( A )
B. ( sqrt{2} )
( c cdot 4 )
D.
12
901 (4
2.
If a= 3sin and B = 3cos , where the inverse
trigonometric functions take only the principal values, then
the correct option(s) is (are)
(JEE Adv. 2015)
(a) cosß > 0
b) sinß0
d) cosa < 0
12
902 Illustration 5.8
Find the value of x for which sec-‘x+sin-
12
903 Evaluate:
( cos ^{-1}left(frac{2 x}{1+x^{2}}right) )
12
904 If ( tan A=-frac{1}{2} ) and ( tan B=-frac{1}{3} . ) (where
( A, B>0), ) then ( A+B ) can be
A ( cdot frac{pi}{4} )
в. ( frac{3 pi}{4} )
c. ( frac{5 pi}{4} )
D. ( frac{7 pi}{4} )
12
905 If ( boldsymbol{x}>1, ) then the value of ( 2 tan ^{-1} x+sin ^{-1}left(frac{2 x}{1+x^{2}}right) ) is
( ^{A} cdot frac{2 pi}{4} )
в.
( c )
D. ( frac{3 pi}{2} )
12
906 Show that ( sin ^{-1}(2 x sqrt{1-x^{2}})=2 sin ^{-1} x )
for ( frac{-1}{sqrt{2}} leq x leq frac{1}{sqrt{2}} )
12
907 6.
Ifx,y,z are in A.P. and tan-1x, tan-ly and tan- z are also in
A.P., then
[JEE M 2013
(a) x=y=z (b) 2x=3y=62
©) 6x=3y=22 (d) 6x=4y=3z
12
908 If ( 1<x<sqrt{2}, ) the number of solutions of
the equation ( tan ^{-1}(x-1)+tan ^{-1} x+ )
( tan ^{-1}(x+1)=tan ^{-1} 3 x ) is
A.
B.
( c cdot 2 )
D.
12
909 Illustration 5.74
IfA=2 tan-‘(2 V2 – 1) and B = 3 sin
+ sin-1

, then which is greater ?
12
910 Find the value of ( cos ^{-1}left(frac{1}{2}right)+ )
( 2 sin ^{-1}left(frac{1}{2}right) )
12
911 23. Domain of definition of the function
eal valued x, is
f(x)= /sin-‘(2x) + ” for real valued x, is (20035)
( ( ) [*
12
912 Illustration 5.55 If two angles of a triangle are tan-‘(2) and
tan-‘(3), then find the third angle.
12
913 Simplify: ( sin . cot ^{-1} tan cdot cos ^{-1} x ) 12
914 If ( cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=pi ) then,
prove that ( x^{2}+y^{2}+z^{2}+2 x y z=1 )
12
915 ( 2 tan ^{-1}left(frac{sqrt{a-b}}{a+b} tan frac{x}{2}right)= )
A ( cdot cos ^{-1}left(frac{b+a cos x}{a+b cos x}right) )
B. ( cos ^{-1}left(frac{b+a cos x}{a-b cos x}right) )
c. ( cos ^{-1}left(frac{b-a cos x}{a+b cos x}right) )
D. ( cos ^{-1}left(frac{b-a cos x}{a-b cos x}right) )
12
916 sin x ||
11. Solve the equation VI sin-‘| cos x | + | cos
sin- cosx – cos- | sin x |, ” <<<".
2
12
917 1. Solve 2 cos-‘x = sin (2x v1 -x?).
1. Solv
12
918 The value of ( cos left(frac{1}{2} cos ^{-1} frac{1}{8}right) ) is
A. ( frac{3}{4} )
в. ( -frac{3}{4} )
c. ( frac{1}{16} )
D.
12
919 Find the principal value of ( operatorname{cosec}^{-1}(-sqrt{2}) ) 12
920 ( sin ^{-1}left(frac{sqrt{1+x}+sqrt{1-x}}{2}right) ) 12
921 The principal value of ( sin ^{-1} x ) lies in the
interval
A ( cdotleft(-frac{pi}{2}, frac{pi}{2}right) )
B. ( left[-frac{pi}{2}, frac{pi}{2}right] )
c. ( left[0, frac{pi}{2}right] )
D. ( [0, pi] )
12
922 ( tan left(cos ^{-1} xright) ) is equal to
A ( cdot frac{x}{1+x^{2}} )
B. ( frac{sqrt{1+x^{2}}}{x} )
c. ( frac{sqrt{1-x^{2}}}{x} )
D. ( sqrt{1-2 x} )
12
923 ( boldsymbol{y}=boldsymbol{c o t}^{-1} frac{boldsymbol{2} boldsymbol{x}}{1-boldsymbol{x}^{2}}, boldsymbol{x} neq pm mathbf{1} ) 12
924 Prove that ( tan ^{-1} frac{sqrt{1+x^{2}}-1}{x}= )
( frac{1}{2} tan ^{-1} x, x neq 0 )
12
925 If ( boldsymbol{alpha}=boldsymbol{2} boldsymbol{s} boldsymbol{i} boldsymbol{n}^{-1}(boldsymbol{2} / boldsymbol{3}) ) and ( boldsymbol{beta}=boldsymbol{2} boldsymbol{t} boldsymbol{a} boldsymbol{n}^{-1} boldsymbol{9} )
then ( 80 operatorname{cosec}^{2} alpha+81 operatorname{cosec}^{2} beta ) is equal
to
12
926 Consider ( boldsymbol{x}=mathbf{4} tan ^{-1}left(frac{mathbf{1}}{mathbf{5}}right), boldsymbol{y}= )
( tan ^{-1}left(frac{1}{70}right) ) and ( z=tan ^{-1}left(frac{1}{99}right) )
What is ( x ) equal to?
A ( cdot tan ^{-1}left(frac{60}{119}right) )
в. ( tan ^{-1}left(frac{120}{119}right) )
( ^{mathbf{c}} cdot tan ^{-1}left(frac{90}{169}right) )
D. ( tan ^{-1}left(frac{170}{169}right) )
12
927 If ( boldsymbol{alpha} boldsymbol{epsilon}left(mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right), ) then the value of
( tan ^{-1}(cot alpha)-cot ^{-1}(tan alpha)+ )
( sin ^{-1}(sin alpha)-cos ^{-1}(cos alpha) ) is equal to
A ( .2 alpha )
B . ( pi+alpha )
( c cdot 0 )
D. ( pi-2 alpha )
12
928 ( cot ^{-1} 9+operatorname{cosec}^{-1} frac{sqrt{41}}{4}=? )
A ( cdot frac{pi}{6} )
в. ( frac{pi}{4} )
( c cdot frac{pi}{3} )
D. ( frac{3 pi}{4} )
12
929 ( sin ^{-1}left(3 x-2-x^{2}right)+cos ^{-1}left(x^{2}-4 x+right. )
3) ( =frac{pi}{4} ) can have a solution for ( x epsilon )
A . [1,2]
B ( cdotleft(frac{3+sqrt{5}}{2}, 2+sqrt{2}right) )
( left(frac{3-sqrt{5}}{2}, 2-sqrt{2}right) )
D ( cdotleft(2-sqrt{2}, frac{3-sqrt{5}}{2}right) cupleft(frac{3-sqrt{5}}{2}, 2+sqrt{2}right) cup{2} )
12
930 6. If cosec- (cosec x) and cosec(cosec- x) are equal
functions, then the maximum range of value of x is
TC
.-lul1
b
10,5
1919 L 2
L 2]
c. (-, -1] U[1,0) d. [-1, 0) U[0, 1)
a. T_T
T7 7
12
931 ( sin cot ^{-1} tan cos ^{-1} x ) is equal to
A . ( x )
B. ( sqrt{1-x^{2}} )
c. ( frac{1}{x} )
D. none of these
12
932 cot-(/cos a) – tan-(cosa) = x,then sinx=
(a) tana) (6) cot? )
(c) tan a (2) cot (2)
2002]
12
933 A function ( f(x)=sqrt{1-2 x}+x ) is
defined from ( D_{1} rightarrow D_{2} ) and is onto. If
the set ( D_{1} ) is its complete domain then
the set ( D_{2} ) is
A ( cdotleft(-infty, frac{1}{2}right] )
в. ( (-infty, 2) )
( mathbf{c} cdot(-infty, 1) )
D ( cdot(-infty, 1] )
12
934 Solve ( tan ^{-1}left(frac{1-x}{1+x}right)= )
( frac{1}{2} tan ^{-1} x,(0<x<1) )
12
935 Inverse circular functions,Principal
values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x )
[
tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y}
]
( boldsymbol{x} boldsymbol{y}1
]
(a) ( tan ^{-1} frac{1}{2}+tan ^{-1} frac{1}{3}=frac{pi}{4} )
( (b) tan ^{-1} frac{1}{2}+tan ^{-1} frac{1}{5}+tan ^{-1} frac{1}{8}=frac{pi}{4} )
( (c) tan ^{-1} frac{3}{4}+tan ^{-1} frac{3}{5}-tan ^{-1} frac{8}{19}=frac{pi}{4} )
12
936 Show that
( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+sin ^{-1} frac{16}{65}=frac{pi}{2} )
12
937 ( cos ^{-1}left{frac{1}{2} x^{2}+sqrt{1-x^{2}} cdot sqrt{1-frac{x^{2}}{4}}right}= )
( cos ^{-1} frac{x}{2}-cos ^{-1} x ) holds for
A. ( |x| leq 1 )
В. ( x in R )
c. ( 0 leq x leq 1 )
D. ( -1 leq x leq 0 )
12
938 Simplify:tan ( ^{-1}(1 / 2)+tan ^{-1}(1 / 3) ) 12
939 If ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is
independent of ( x, ) then
This question has multiple correct options
( mathbf{A} cdot x>1 )
B. ( x<-1 )
c. ( 0<x<1 )
D. ( -1<x<0 )
12
940 Find the principal value of ( operatorname{cosec}^{-1}(2) ) 12
941 If ( frac{(x+1)^{2}}{x^{3}+x}=frac{A}{x}+frac{B x+C}{x^{2}+1}, ) then
( csc ^{-1}left(frac{1}{A}right)+cot ^{-1}left(frac{1}{B}right)+sec ^{-1} C= )
A ( cdot frac{5 pi}{6} )
B.
c.
D. ( frac{pi}{2} )
12
942 ( sin ^{-1} 0 ) is equal to:
( mathbf{A} cdot mathbf{0} )
в.
c. ( frac{pi}{2} )
D. ( frac{pi}{3} )
12
943 If ( f(x)=sin ^{-1} x+sec ^{-1} x ) is defined
then which of the following value/values is/are in its range?
A ( cdot frac{-pi}{2} )
в. ( frac{pi}{2} )
( c . pi )
D. ( frac{3 pi}{2} )
12
944 Find the value of ( cos left[frac{pi}{2}-sin ^{-1}left(frac{1}{3}right)right] ) 12
945 The value of ( sin left(frac{1}{4} sin ^{-1} frac{sqrt{63}}{8}right) ) is
A ( cdot frac{1}{2} )
B. ( frac{1}{3} )
( c cdot frac{1}{2 sqrt{2}} )
D.
12
946 Find the principal value of ( sec ^{-1}left(frac{2}{sqrt{3}}right) ) 12
947 ( operatorname{Let} cos ^{-1}left(4 x^{3}-3 xright)=a+b cos ^{-1} x )
( x inleft[-frac{1}{2}, frac{1}{2}right], ) then the principal value of ( sin ^{-1}left(sin frac{a}{b}right) ) is
( A cdot-frac{pi}{3} )
в.
( c cdot-frac{pi}{6} )
D. None of these
12

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