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.

Inverse Trigonometric Functions Questions

List of inverse trigonometric functions Questions

Question NoQuestionsClass
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
229. 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
3Range 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
488. 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
5The 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
6If ( 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
7The 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
8Solve ( : boldsymbol{y}=sin ^{-1}(sec boldsymbol{x}) )12
9Prove that: ( sin ^{-1}left(frac{3}{5}right)+cos ^{-1}left(frac{12}{13}right)= )
( sin ^{-1}left(frac{56}{65}right) )
12
10The 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
11If 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
12Find the projection of the vector ( hat{mathbf{i}}-widehat{boldsymbol{j}} ) on the vector ( hat{mathbf{i}}+widehat{boldsymbol{j}} )12
13Solve:
( 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
14Solve
( cot ^{-1} cot left(frac{5 pi}{4}right) )
12
15The value of ( cos left(2 cos ^{-1} 0.8right) ) is
A . 0.48
B. 0.96
( c .0 .6 )
D. 0.28
12
1650. 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
17Show that
( tan ^{-1}left(frac{1}{2}right)+tan ^{-1}left(frac{1}{3}right)=frac{pi}{4} )
12
18Find 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
19The 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
20Prove that ( sin ^{-1}left(frac{8}{17}right)+sin ^{-1}left(frac{3}{5}right)= )
( cos ^{-1}left(frac{36}{85}right) )
12
21Inverse 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
22If ( 6 operatorname{Sin}^{-1}left(x^{2}-6 x+12right)=2 pi, ) then the
value of ( x, ) is
12
23Inverse 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
24If ( 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
25Find 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
26If ( 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
2786. 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
2813
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
29Find 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
31Find the value of ( boldsymbol{x} )
If ( , tan ^{-1}(x-1)+tan ^{-1} x+tan ^{-1}(x+ )
1) ( =tan ^{-1} 3 x )
12
32Illustration 5.67
Solve sin-‘ x + sin- (1 – x) = cos-‘x.
12
33For the principal value:
( cos ^{-1}left(frac{1}{2}right)-2 sin ^{-1}left(-frac{1}{2}right) )
12
34Find the principle value of ( tan ^{-1}(-sqrt{3}) )
( mathbf{A} cdot pi / 3 )
в. ( -pi / 3 )
c. ( pi / 6 )
D. ( -pi / 6 )
12
35Prove ( 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
37If ( -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
39If ( sin ^{-1} x=frac{pi}{5} ) for ( operatorname{somex} in[-1,1] ) then
find the value of ( cos ^{-1} x )
12
40The 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
41The 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
42If ( 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
4324. The value of sin-(cos(cos(cosx) + sin-‘(sin x))), where
Xe
is equal to
Bla
b. – 1
B
dond. -**
12
44The 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
45Show that: ( cos ^{-1} frac{4}{5}+cos ^{-1} frac{12}{13}= )
( cos ^{-1} frac{33}{65} )
12
46Illustration 5.22 Find the number of solutions of
2tan-‘tan x) = 6 – X.
12
47SE
4. Find the sum cosec. V10 + cosec- 50 + cosec – 7170
+ … + cosec Vln? +1) (x2 + 2n +2).
12
4813. 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
49Solve ( : sin ^{-1}left(frac{2 pi}{4}right) )12
50prove 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
52Find 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
54If ( 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
5541. If sin ‘ x | + |cos + x1 =
, then x e
a. R
c. [0, 1]
b. [-1, 1]
d. 0
12
56The 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
57Let ( 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
58Prove:
( 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
59If ( 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
603. Find the range of f(x) = cot- (2x – x?).12
6114. The value of lim cos (tan-‘(sin(tan-? x))) is equal to
(
xo
a. -1
d.
12
6214. If 0<a, <a2 <… <an, then prove that
17-
12
63Prove 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
64тл
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
65Find 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
67If ( 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
68Assertion
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
69Find the principle value of :
( tan ^{-1}left(frac{1}{sqrt{3}}right) )
12
70Find the principal value of:
( cot ^{-1}left(-frac{1}{sqrt{3}}right) )
12
71The 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
72Prove that ( frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right)=tan ^{-1} sqrt{x} )12
73Range 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
74If 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
75Find the value of ( tan ^{2}left(frac{1}{2} sin ^{-1} frac{2}{3}right) )12
76(JEE Adv. 2013)12
77Find 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
79Let ( 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
80The 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
81Зл
а. Тоr — Зл
12
82Find the range of f(x) = sin- x + tan-‘x+
Illustration 5.45
cos-‘x.
12
83Inverse 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
84If ( 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
86If ( 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
87f ( 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
89The 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
90Write the value of ( tan ^{-1}left(frac{1}{x}right) ) for ( x<0 ) in terms of ( cot ^{-1}(x) )12
91If ( 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
92If 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
932x
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
94Illustration 5.18 Evaluate the following:
i. sin-‘(sin 10)
ii. sin-‘(sin 5)
iii. cos(cos 10) iv. tan-‘(tan(-6))
12
9510
44. If tan-x + 2 cot-‘x = ***, then x is equal to
13 –
b. 3
d. 2
c. √3
12
96Solve :
( cos ^{-1}left(log _{2} xright)=0 )
12
97The 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
9860. The number of solutions of the equation tan-‘(1 + x) +
tan-‘(1 – x) =
a. 2
b. 3
c. 1
an
d. 0
12
9973. 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
100If ( 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
101Illustration 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
104The 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
105The 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
106If ( 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
107Find 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
108Prove 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
109Write the principal value of ( cos ^{-1}left(frac{1}{2}right)- ) ( 2 sin ^{-1}left(-frac{1}{2}right) )12
110If ( 4 sin ^{-1} x+cos ^{-1} x=pi, ) then find the
value of ( boldsymbol{x} )
12
111Match the following12
112Two 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
113The 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
114Which 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
11537. 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
116If ( 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
117If ( 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
118Find the value of ( x .left(tan ^{-1} xright)^{2}+ )
( left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} )
12
119Write the value of ( cot ^{-1}(-x) ) for all ( x epsilon R )
in terms of ( cot ^{-1} x )
12
120Evaluate ( 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
121If ( 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
122Solve for ( x ) :
( tan ^{-1} x=frac{1}{2} cot ^{-1} x )
12
123If 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
124If ( 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
125If ( 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
126The 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
127The 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
12820. 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
130Illustration 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
132Solve for ( boldsymbol{x} )
( 2 tan ^{-1}(cos x)=tan ^{-1}(2 operatorname{cosec} x) )
12
133The 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
134The 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
135Find 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
136Which 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
137Evaluate:
[
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
138The 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
139f ( x y+y z+z x=1 ) then find the value
of ( tan ^{-1} x+tan ^{-1} y+tan ^{-1} z )
12
140The 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
141If ( 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
142Solve: ( 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
143Simplify ( 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
144Find the principle value of ( cos ^{-1}left[cos left(frac{7 pi}{3}right)right] )12
145If ( 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
146If ( 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
147The 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
148Prove 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
150Solve: ( tan ^{-1} 2 x+tan ^{-1} 3 x=frac{pi}{4} )12
151Find the principal value of the following:
( operatorname{cosec}^{-1}(-sqrt{2}) )
12
152If ( 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
153The 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
154Which 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
155The range of ( a r c sin x+a r c cos x+ )
( arctan x ) is
12
156The 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
158The 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
15942. If (sin x)2 – (cos- ‘x)2 = an? then find the range of a.
dc. 1-1, 1]
d. -1,
12
1601. The principal value of sin “sin 25) is (1986- 2 Marks)
(a) 20 (6) 24 ( 47 (a) none
12
161Write the following in the simplest form:
( tan ^{-1}left{frac{sqrt{1+x^{2}}-1}{x}right}, x neq 0 )
12
16283. 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
16433. 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
165If 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
16612. 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
167Write the principal value of ( cos ^{-1}left(frac{1}{2}right)-2 sin ^{-1}left(-frac{1}{2}right) )12
168Evaluate ( sin left(frac{pi}{6}+cos ^{-1} frac{1}{4}right) )12
169If ( 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
170If ( 3 tan ^{-1} x+cot ^{-1} x=pi, ) then ( x )
equals:
( mathbf{A} cdot mathbf{0} )
B.
c. -1
D.
12
171The 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
172The 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
173The 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
174Simplify ( 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
17515. 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
177If ( 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
17938. 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
181If ( 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
182If ( 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
1845. If cot-1
,ne N, then the maximum value of n is
a. 6
c. 5
b. 7
d. none of these
12
185If ( 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
186Prove that ( 3 sin ^{-1}=sin ^{-1}(3 x-1) )
( left.4 x^{3}right), x epsilonleft[frac{-1}{2}, frac{1}{2}right] )
12
187The 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
188The 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
189Find the principal value of the following
( cos ^{-1}left(cos frac{7 pi}{6}right) )
12
190Find the value of: ( sin left(2 tan ^{-1} frac{1}{4}right)+ )
( cos left(tan ^{-1} 2 sqrt{2}right) )
12
191Assertion
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
192for 0<x< 12, then x equals
(a) 12 (b) I (c) -1/2
(20015)
(d) 1
12
193Solve 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
194Illustration 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
195If ( 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
196The value of
( sin ^{-1}left[cot left[sin ^{-1}(sqrt{frac{2-sqrt{3}}{4}})+cos ^{-1}right.right. )
is
12
197Inverse 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
198Solve ( : cos ^{-1} sqrt{frac{1+cos x}{2}} )12
199The value of p for which system has a solution is
A . 1
B. 2
c. 0
D. –
12
200Find ( 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
201Solve ( 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
202Evaluate:
( 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
204Which 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
205Write the value of ( 2 sin ^{-1} frac{1}{2}+ )
( cos ^{1}left(-frac{1}{2}right) )
12
206Find 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
208If ( 4 sin ^{-1} x+cos ^{-1} x=pi, ) then ( x ) is
equal to:
A ( cdot frac{1}{2} )
B. 2
c. 1
D.
12
2093. Which of the following is/are the value of
COS
cos(cos” (cos(-1947)]
(211
d. -cos
C. COS
12
210Solve
( 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
212Evaluate the following:
( cos ^{-1}(cos 12) )
12
213Write the principal value of :
( left[cos ^{-1} frac{sqrt{3}}{2}+cos ^{-1}left(-frac{1}{2}right)right] )
12
214Illustration 5.7 Find the range of f(x) = 13 tan ‘x- cos’O
– cos(-1).
12
215f ( 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
216If ( 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
217The 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
218If ( 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
219Illustration 5.33
Prove that
12
220If ( tan ^{-1} 4=4 tan ^{-1} x, ) then ( x^{5}-7 x^{3}+ )
( 5 x^{2}+2 x ) is equal to
12
221If ( cos ^{-1} x+cos ^{-1} y=frac{pi}{2} ) then prove that
( cos ^{-1} x=sin ^{-1} y )
12
222Solution 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
223The 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
224Illustration 5.64
If (x – 1) (x² + 1) > 0, then find the value
tan
12
225Prove 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
226Evaluate the following:
( cos ^{-1}(cos 4) )
12
227The 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
229Find the principal value of ( cos ^{-1}left(cos frac{7 pi}{6}right) )12
230Write the principal value of ( sin ^{-1}left(-frac{1}{2}right) ) ( ? )12
231Solve ( 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
232The 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
233The 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
234The 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
235For ( 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
236Evaluate ( 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
237If ( 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
238The 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
239Prove 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
240Prove the following:
( cos ^{-1}left(frac{12}{13}right)+sin ^{-1}left(frac{3}{5}right)= )
( sin ^{-1}left(frac{56}{65}right) )
12
241f ( 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
242The 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
243From 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
244The 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
245Show that:
( sin ^{-1}left(frac{12}{13}right)+cos ^{-1}left(frac{4}{5}right)+ )
( tan ^{-1}left(frac{63}{16}right)=pi )
12
24617. 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
247i) 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
249If ( 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
250The 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
251If ( 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
252Find
( int tan ^{-1} frac{x}{sqrt{a^{2}-x^{2}}} d x ;|x|langle a )
12
253If ( 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
2542.
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
255Solve:
( sin ^{-1}(cos x) )
12
256Write the principal values of the
following: ( sin ^{-1}left(-frac{1}{2}right)+cos ^{-1}left(-frac{1}{2}right) )
12
257If ( 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
2586x

Illustration 5.77 If cos-1
find the values of x.
1+ 9×2 –
5 + 2 tan-‘3x, then
12
259If ( 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
260Find the value of ( sin ^{-1}left(cos frac{33 pi}{5}right) )12
261Write ( tan ^{-1}left[frac{sqrt{1+x^{2}}-1}{x}right], x neq 0 )
the simplest form.
12
2621+r2
45. The number of solutions of the equation cos
– cos x = + sin ‘x is
b. 1
c. 2
d. 3
a. 0
12
263Show that ( sin ^{-1} frac{12}{13}+cos ^{-1} frac{4}{5}+ )
( cot ^{-1} frac{63}{16}=frac{pi}{2} )
12
264Assertion
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
265The 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
266Find ( y=left(sin ^{-1} xright)^{x^{2}}, ) then ( y^{prime}(0)=? )12
267If 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
269Find 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
271Calculate ( (192-214) )
( sin ^{-1}+2 tan ^{-1}(-sqrt{3}) )
12
272If 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
273Find the value of ( cos ^{-1}left(cos frac{2 pi}{3}right)+ )
( sin ^{-1}left(sin frac{2 pi}{3}right) )
12
274Find ( sin ^{-1}left(frac{sqrt{mathbf{3}}+1}{2 sqrt{2}}right)= )12
275Let ( 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
276Write the value of ( 2 sin ^{-1} frac{1}{2}+ )
( cos ^{1}left(-frac{1}{2}right) )
12
277Solve 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
278Illustration 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
279Solve the equation ( tan ^{-1}left[frac{1-x}{1+x}right]= )
( frac{1}{2} tan ^{-1} x,(x>0) )
12
28015. 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
281Find 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
282If 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
283Solve:
( cos ^{-1}left(frac{x-x^{-1}}{x+x^{-1}}right) )
12
284If ( 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
285Find 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
287The 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
28862. 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
289Prove 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
291Find the principal value of:
( cos ^{-1}left(sin frac{4 pi}{3}right) )
12
29264. 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
293If ( 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
2945. 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
295Illustration 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
296If ( sin ^{-1} x=frac{pi}{5}, ) for some ( x in(-1,1) )
then find the value of ( cos ^{-1} x )
12
2977. 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
2982.
Find all the solution of 4 cos xsin x – 2 sinx = 3 sin x
(1983 – 2 Marks
12
299If ( 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
30090. 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
30116. 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
30265. If x2 + y2 + z2 = r2, then tan-“|
+ tan
(ar)
+ tan-” ) is equal to
To
a.
b.
c. O
d. none of these
12
303Find the value of ( x ) if
( sin left{sin ^{-1} frac{1}{5}+cos ^{-1} xright}=1 )
12
304Find the value of ( sin ^{-1}left[sin left(-frac{17 pi}{8}right)right] )12
305The 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
308Evaluate the following:
( tan ^{-1}(tan 2) )
12
309Value 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
311The 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
312Consider 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
313If ( 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
314Evaluate :
( 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
315The 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
316Find the value of ( cos ^{-1}left(cos frac{13 pi}{6}right) )12
317Solve:
( tan ^{-1}left(frac{2 x}{1-x^{2}}right) )
12
318If ( 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
319Evaluate: ( 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
320tan-‘ 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
321Find 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
323Solve the equation for ( x ) ( sin ^{-1} x+sin ^{-1}(1-x)=cos ^{-1} x, x neq 0 )12
324For the principal value:
( tan ^{-1}left{2 sin left(4 cos ^{-1} frac{sqrt{3}}{2}right)right} )
12
325If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then find the
value of ( cos ^{-1} x+cos ^{-1} y )
12
326If ( 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
328Find 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
329Solve the following:
( tan left(frac{1}{2} sin ^{-1} frac{3}{4}right) )
12
33020. 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
33157. 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
332Find the value of ( cos ^{-1}left(cos frac{5 pi}{3}right) )12
3333.
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
334The 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
336Show that ( 2 tan ^{-1} frac{3}{5}=tan ^{-1} frac{15}{8} )12
337Assertion
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
338Assertion
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
339The 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
340If ( 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
341Assertion ( (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
342If ( 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
343Assertion
( 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
344Evaluate the following:
( cos ^{-1}(cos 5) )
12
345Find the principal values of ( sin ^{-1}left(-frac{1}{sqrt{2}}right) )12
346Illustration 5.29
Simplify sin cot’ tan cos’x, x > 0.
12
347f ( 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
348The 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
349Find 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
3511. Solve 2 cos + s = sinº (23 W1–?)
1. Solve
= sin
12
352Find the principal value:
( tan ^{-1}left(-frac{1}{sqrt{3}}right) )
12
353Illustration 5.9 If sin-‘(x2 – 4x + 5) + cos='(02- 2y + 2) =
then find the value of x and y.
12
354Evaluate the following:
( sin ^{-1}(sin 2) )
12
35535. 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
356The 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
357llustration 5.20
Solve cos'(cos x) > sin
(sin x),x € [0,21].
12
358Let ( 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
359ff ( 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
360The 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
361Illustration 5.54
Prove that cos– + cos!
13
-cos-1 33
5
12
362Consider 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
363The 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
364Let ( 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
36580. 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
36612. 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
3671 + x
cos- x
Prove that cos-
Illustration 5.32
-1<x< 1.
12
368Solve:( tan ^{-1}left(frac{6 x}{1-8 x^{2}}right) )12
369Write the function in the simplest form:
( tan ^{-1} frac{1}{sqrt{x^{2}-1}},|x|>1 )
12
370The 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
371Solve the equation for ( x: sin ^{-1} frac{5}{x}+ ) ( sin ^{-1} frac{12}{x}=frac{pi}{2}, x neq 0 )12
372Find the principal value of:
( sec ^{-1}(2) )
12
373The 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 ) equals12
375The 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
376The value of ( sin left(tan ^{-1} x+cot ^{-1} xright) ) is12
377The 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
378Illustration 5.49 If sec !x= cosecly, then find the value of
cos –
– + cos-
12
379Value 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
380The 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
381Illustration 5.11 If cos’2+ cos’u + cos-‘ y=31, then find
the value of λμ + μγ+ γλ.
12
382If ( 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
383Illustration 5.69
Ifx e
then show that
cos” (3 (1 + cos2x) + Vísinºx – 48 cos?r) sin x)
= x – cos-‘(7 cos x)
12
384Write the following in simplest form:
( tan ^{-1}left(frac{sqrt{left(1+x^{2}right)}-1}{x}right) )
12
385Find the principal value of:
( sin ^{-1}left(cos frac{3 pi}{4}right) )
12
386The 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
387Find the real solution of the equation ( tan ^{-1} sqrt{x(x+1)}+sin ^{-1} sqrt{x^{2}+x+1}= )12
388Itsin-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
38936. 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
390If cos(2 sin-‘x) = -, then find the values
Illustration 5.23
of x.
12
391Solve ( : tan ^{-1}left(frac{x-1}{x-2}right)+ )
( tan ^{-1}left(frac{x+1}{x+2}right)=frac{pi}{4} )
12
392For 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
3939. 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
39421. Complete solution set of tan”(sin’x) > 1 is
– (-a) 6) (Ta’ tal-c03
c. (-1,1) – {0}
d. None of these
12
395The 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
396If ( 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
397If ( 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
398Solve:
( 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
399The 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
400Number 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
401f ( tan ^{-1}left(frac{x-1}{x-2}right)+tan ^{-1}left(frac{x+1}{x+2}right)= )
4
then ( x ) is
12
402The 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
404Simplify ( tan ^{-1} sqrt{2}-cot ^{-1}(1 / sqrt{2}) )12
405If ( tan left(2 tan ^{-1}left(frac{1}{5}right)-frac{pi}{4}right)=-frac{lambda}{17}, ) then ( lambda ) is
equal to
12
406Illustration 5.65
Find the value of cot- – + sin
13
12
407The 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
408The 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
409The 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
410Illustration 5.62 Find the value of
tan
(l+rtph
r=0
12
411Find the principal value:
( tan ^{-1}left(frac{1}{sqrt{3}}right) )
12
412Illustration 5.1 Find the principal value of the following:
(i) cosec-‘(2) (ii) tan-‘ (-13)
(ii) cos( – – –
12
413Find tan-
in terms of sin,
Illustration 5.28
where x e (0, a).
ſa² – x
12
414Illustration 5.47
Solve sin- –
+ sin
Nia
12
415Illustration 5.44
Solve
12
416Illustration 5.39
Find the range of y = (cot- x)(cot ‘(-x)).
12
417illustration 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
418For the principal value:
( sin ^{-1}left(-frac{1}{2}right)+2 cos ^{-1}left(-frac{sqrt{3}}{2}right) )
12
419The 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
420Prove that ( tan ^{-1} frac{63}{16}=sin ^{-1} frac{5}{13}+ )
( cos ^{-1} frac{3}{5} )
12
421Write the following into simple test form;
(1) ( sin left{2 tan ^{-1} sqrt{frac{1-n}{1+n}}right} )
12
422If ( 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
423Illustration 5.4
Solve for x if (cot- x)2 – 3 (cot-‘x) +2>0.
12
424Show 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
42631. 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
427Illustration 5.56
Find the value of tan-1
tan 2A
+
I
tan (cot A) + tan-‘(cot? A), for 0 <A<*.
12
428The 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
429The 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
430Illustration 5.27 Solve sin-‘(1 – x) – 2 sin-‘x =12
4318. 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
432The 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
433Find 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
43578. 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
4372. 2 tan- ‘(- 2) is equal to
a. –
b. – 1+ cos-1
5
c.
– –
2
+ tan-1
d. – T + cot-
Cot-1
12
438Evaluate:
( sum_{r=1}^{infty} tan ^{-1}left(frac{2}{1+(2 r+1)(2 r-1)}right) )
12
439Evaluate ( tan left[frac{1}{2} cos ^{-1}left(frac{3}{sqrt{11}}right)right] )12
440If ( 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
441The 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
443Find the principle value of: ( cos ^{-1}left(-frac{1}{2}right) )12
44410. 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
44510
10
Illustration 5.52
Find the value of
tan ºr
r=1s=1
12
446The 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
447If ( 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
4486. 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
449If ( 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
450Prove that:
( tan ^{-1} frac{1}{4}+tan ^{-1} frac{2}{9}=frac{1}{2} cos ^{-1} frac{3}{5} )
12
451Given 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
4525. 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
453Solve :
( 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
454Find 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
455If ( theta=sin ^{-1} x+cos ^{-1} x-tan ^{-1} x, 1 leq )
( x<infty, ) the smallest interval in which ( theta )
lies is
12
456If ( 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
4581 COS
The value of tam cor” ()un (1) –
n (1) Bosch (a) none
(1983-1 Mark)
12
459Illustration 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
460If ( 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
461Write the simplest form of ( tan ^{-1}left(frac{cos x-sin x}{cos x+sin x}right), 0<x<frac{pi}{2} )12
462Find 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
463There 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
464Match the column12
465If ( 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
466Find the principal value of:
( tan ^{-1}(-1)+cos ^{-1}left(-frac{1}{sqrt{2}}right) )
12
467Solve ( sin ^{-1}left{frac{sin x+cos x}{sqrt{2}}right},-frac{3 pi}{4}< )
( x<frac{pi}{4} )
12
468Find the principal value of the following:
( sec ^{-1}(-2) )
12
4695.
The value of cot( cosec
t an
12
470A 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
47156. = tan’ (2 tan’) – tan-
– tan 0 then tan
=
a. 2
c. 2/3
b. -1
d. 2
12
472Show that ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is
constant for ( x geq 1 . ) Also find that
constant.
12
473If ( 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
474Number 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
475If ( 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
476Illustration 5.14 Find x satisfying (tan-‘x] + [cot–x] = 2,
where [:] represents the greatest integer function.
12
477There 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
478The 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
479Write 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
480Prove: ( 2 sin ^{-1} frac{3}{5}=tan ^{-1} frac{24}{7} )12
481If sin sin-
Illustration 5.43
value of x.
???
+ cos-‘x = 1, then find the
12
482Prove 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
483If ( tan ^{-1}(1+x)+tan ^{-1}(1-x)=frac{pi}{2} )
then ( x=? )
A .
B. –
c.
D.
12
484A 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
4854:(**”)un jo sguvą 5112
486The 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
487Find the number of terms of the AP
( 121,117,113, dots,-3 ? )
A . 32
B. 30
c. 28
D . 26
12
488Illustration 5.72 If f(x) = sin-‘ x then prove that
lim f(3x – 4x?) = 1 – 3 lim sin-‘ x.
12
489If ( 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
490Find 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
491Consider 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
492If ( 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
493If ( 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
494Solve the equation: ( tan ^{-1} x+ ) ( 2 cot ^{-1} x=frac{2 pi}{3} )12
495What 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
497If we consider only the principle values of the inverse
trigonometric functions then the value of
tan cos
SUD
(1994)
12
498Find the principal value of:
( operatorname{cosec}^{-1}left(2 tan frac{11 pi}{6}right) )
12
499Illustration 5.57 Simplify
3 sin 2a
| 5 + 3 cos 2a
tan a

+ tan-1
1, where –
I < a < 1 2
12
500Which 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
501Inverse 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
502Evaluate ( sin ^{-1}(cos x) )12
503If 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
504If 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
506f ( 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
507Prove that ( frac{pi}{2}-sin ^{-1} x= )
( operatorname{cosec}^{-1} frac{1}{sqrt{1-x^{2}}} )
12
508ff ( y=left(tan ^{-1} xright)^{2} ) then show
( left(x^{2}+1right)^{2} y_{2}+2 xleft(x^{2}+1right) y_{1}=2 )
12
509Illustration 5.68
Solve
12
510If the value of ( tan ^{-1}left(tan frac{3 pi}{4}right) ) is ( -frac{pi}{k} )
then ( k ) is
12
511Simplify: ( sin ^{-1}(2 x sqrt{1-x^{2}})= )
( dots . .left(|x|<frac{1}{sqrt{2}}right) )
12
51247. The value of a for which ax? + sin (x2 – 2x + 2) +
cos (r? – 2x + 2) = 0 has a real solution is
12
51384. If f (x) = sin-
is equal to
SxS 1, then f(x)
c. sin x +*
c. sin-2x +
d. none of these
12
514If 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
515Solve:
( cot ^{-1} x+cot ^{-1} 2=frac{pi}{2} )
12
516If ( 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
519Prove 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
520The number of solutions of the
equation ( sin ^{-1}left(frac{1+x^{2}}{2 x}right)=frac{pi}{2} sec (x- )
1) is
12
521Number 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
523The 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
524Itxt
2. Find the domain for f(x) = sin
2x
12
525If ( 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
526Illustration 5.66 Solve sin-‘x + sin 2x =12
527Illustration 5.15 Find the number of solutions of the
equation cos(cos=’x) = cosec(cosec-‘x).
12
528Find 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
529The 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
530If ( 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
531Find the principal value of ( sin ^{-1}left(-frac{1}{2}right) )12
532Indicate 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
533Evaluate ( cos left[cos ^{-1}left(-frac{sqrt{3}}{2}right)+frac{pi}{6}right] )12
534Find 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
536n
Illustration 5.71 Find the value
-1/1+ /(k – 1)k(k+1)(k + 2)
lim
k(k+1)
COS
no
k=2
12
537Number 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
538The 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
539If ( 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
540Exhaustive 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
541Prove 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
543Show that ( tan ^{-1} 1 / 2+tan ^{-1} 2 / 11+ )
( tan ^{-1} 4 / 3=pi / 2 )
12
544Write the value of ( cos ^{-1}left(cos frac{5 pi}{4}right) )12
545Find the set of values of parameter a so that the equation
(sin-2x)3 + (cos’x)’ = at has a solution.
12
546If ( tan ^{-1} x=frac{pi}{10} ) for some ( x in R, ) then
find the value of ( cot ^{-1} x )
12
54712. Solve the equation tan ” * +1+tan **= tan” (-7).
x
-1
12
548T 1
-+-COS
4 2
X
+ tan
cos’x, x = 0, is equal
4
2
b. 2x
d. none of these
12
549If ( 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
55083. 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
551If 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
552Find 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
553The 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
554The value of ( sin ^{-1}(sin 10) ) is
A . 10
в. ( 10-3 pi )
( c .3 pi-10 )
D. none of these
12
555Find 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
556Illustration 5.36 If x € (-1, 0), then find the value of
cos-(2×2 – 1)-2 sin-‘x.
12
557Assertion
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
559Solve the following equations. ( sin ^{-1} frac{5}{x}+sin ^{-1} frac{x}{2}=frac{pi}{2} )12
5602x
8. The maximum value of f(x) = tan
osa. 18°
c. 22.5°
sus b. 36° ismert..
d. 15°
2
in se
12
56127. 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
562f ( 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
563f ( 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
564The 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
565The 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
56639. 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
56761. 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
569If ( 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
570If ( 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
57189. 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
572If ( 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
57310. 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
574Find the principal value of:
( sec ^{-1}left(2 sin frac{3 pi}{4}right) )
12
57558. 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
576Evaluate 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
577Assertion
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
578The 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
57968. 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
581in? – 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
583If ( 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
584The 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
585Prove 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
587If ( 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
588If ( 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
589Find 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 . ) what12
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
593Illustration 5.42 If sin-‘x = 7d/5, for some x € (-1, 1), then
find the value of cos-‘x.
12
594The 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
595Illustration 5.30 Prove that
cosec(tan’ (cos(cot’ (sec(sin-‘ a))))) = 13 – a?,
where a € [0, 1].
12
596Solve 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
597Prove:
( 3 cos ^{-1} x=cos ^{-1}left(4 x^{3}-3 xright), x inleft[frac{1}{2}, 1right] )
12
598The 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
599If ( 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
600Find 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
60219. 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
603Find the principal value of:
( operatorname{cosec}^{-1}left(2 cos frac{2 pi}{3}right) )
12
604If ( 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
605The 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
606Find the value of ( sin ^{-1} x+sin ^{-1} frac{1}{x}+ )
( cos ^{-1} x+cos ^{-1} frac{1}{x} )
12
607Assertion
( 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
609If ( 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
610If ( 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
612Evaluate :
( int x^{2} tan ^{-1} frac{x}{2} d x )
12
613Illustration 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
615Find the value of ( x )
if ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3} )
12
616represents the graph of the function ( f(x)=lim _{n rightarrow infty} frac{2}{pi} tan ^{-1}(n x) ? )
( A )
B.
( c )
( D )
12
617The 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
619Express the following in the simplest
form ( tan ^{-1}left(frac{cos x}{1+sin x}right), frac{-pi}{2}<x<frac{pi}{2} )
12
6201.
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
621Find the value of ( cos left(sec ^{-1} x+right. )
( left.operatorname{cosec}^{-1} xright),|x| geq 1 )
12
622If ( 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
623The 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
624If ( 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
626Illustration 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
6272. 2 tan- ‘(- 2) is equal to
a.
– COS
b. – Te + cos2
12
628Tllustration 2.8
Find the value of cos+ (-1/2).
12
62953. 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
630Differentiate ( tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right) )12
631Find the principal value of:
( sin ^{-1}left(frac{sqrt{mathbf{3}}-1}{2 sqrt{2}}right) )
12
632Solve the equation ( sin ^{-1}(3 x)=-frac{1}{3} pi )
giving the solution in an exact form.
12
633Solve ( int frac{sin ^{-1} sqrt{x}-cos ^{-1} sqrt{x}}{sin ^{-1} sqrt{x}+cos ^{-1} sqrt{x}} d x )12
634Show that ( sin ^{-1}(2 x sqrt{1-x^{2}})=2 sin ^{-1} x )12
635Evaluate: ( tan ^{-1} sqrt{3}-cot ^{-1}(-sqrt{3}) )
( mathbf{A} cdot mathbf{0} )
B. ( 2 sqrt{3} )
( c cdot-frac{pi}{2} )
D.
12
636If ( 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
637Prove 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
639Assertion: 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
642Write the principal value of ( tan ^{-1}(1)+ ) ( cos ^{-1}left(-frac{1}{2}right) )12
643Solve:
( sin ^{-1}left(frac{2 x}{1+x^{2}}right) )
12
64452. 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
646Illustration 5.41 Prove that 2 tan-‘(cosec tan ‘x- tan cotx)
= tan- x (x 0).
12
647The 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
648Prove that:
( cos ^{-1}(x)+cos ^{-1}left{frac{x}{2}+frac{sqrt{3-3 x^{2}}}{2}right}=frac{pi}{3} )
12
649If ( 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
650f ( 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
652Topic-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
653Find 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
654If ( 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
6557. sec?(tan-? 2) + cosec?(cot-3) is equal to
a. 5
b. 13
190 c. 1575] 100 d. 6 ostalo
12
656Find general solution of the following equations:
( sin theta=frac{1}{2} ? )
12
657If ( 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
658Find the value of ( tan left[frac{1}{2} cos ^{-1} frac{sqrt{5}}{3}right] )12
659Illustration 5.73
Solve sin ‘ x – cos’ x = sin-‘(3x – 2).
COS
12
660The 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
661The 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
662Write the simplest form of :
( tan ^{-1} frac{sqrt{1+x^{2}}-1}{x}, x neq 0 )
12
66332. There exists a positive real number x satisfying
cos(tan-‘x) = x. Then the value of cos!
Tu labb. Tebe
c. 21
12
664If ( 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
665Find ( n ) if ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+ )
( sin ^{-1}left(frac{16}{65}right)=frac{n pi}{2} )
12
666Illustration 5.40 Find the value of sin-‘(sin 5) +
cos(cos 10) + tan-‘{tan(-6)} + cot-‘{cot(-10)}.
12
667TT
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
668If ( 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
66918. 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
670The 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
671If ( 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
672Assertion
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
673State True or False ( sin ^{-1} 2+cos ^{-1} 2=frac{pi}{2} )
A. True
B. False
12
674If ( 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
675Prove that ( tan ^{-1}left(frac{sin x}{1+cos x}right),-pi<x<pi )12
676If ( 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
677Equations ( 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
678Find the principal value of ( cos ^{-1}left(-frac{1}{2}right) )12
679Find the principal value of the following:
( tan ^{-1}(-1) )
12
680illustration 5:26. Find the value of sin(3cot” (22)
Illustration 5.24
Find the value of sin
CO2
12
681Inverse 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
682Solve:tan ( ^{-1} mathbf{2} boldsymbol{x}+tan ^{-1} mathbf{3} boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}} )12
683If ( 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
684Assertion
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
685Illustration 5.31 If x < 0, then prove that
cos-"x = 1 – sin- 1 – x?.
12
686Prove that ( sin ^{-1} frac{3}{5}-sin ^{-1} frac{8}{17}= )
( cos ^{-1}left(frac{84}{85}right) )
12
687Express ( 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
688n
Illustration 5.35
Prove that cos- |
= 2 tan-
x
(1+z2n
0<x<
.
12
689If 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
6906. 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
691If ( (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
69216. 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
693The 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
694If ( 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
695Find the value of sin-‘x+sin! – + cos x
Illustration 5.51
+ cos-1-
12
696Find 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
69723. 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
698The 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
700Evaluate 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
70134. 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
702If ( 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
703ff ( y=tan ^{-1}left(frac{x sin alpha}{1-x cos alpha}right) . ) Find ( cot y )12
704If
( 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
705Write the simplest form of ( tan ^{-1}[sqrt{frac{1-cos x}{1+cos x}}] )12
706Find the value of the expression ( sec ^{-1}left(frac{x+1}{x-1}right)+sin ^{-1}left(frac{x-1}{x+1}right) )12
707If ( 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
708Illustration 5.26
Prove that
/1 + sin x + 71-sin x
Cor-1
XE
1 + sin x –
sin x
12
709The 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
711If ( 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
71313. Range of f(x) = sin-4x + tan– x + sec-?x is
d. none of these
12
714Assertion
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
715If ( 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
716Assertion
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
718Assertion ( 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
719Find the principal value of ( tan ^{-1} sqrt{3}- )
( sec ^{-1}(-2) )
12
720The 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
721Prove: ( 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
722Solve:
( cos ^{-1}(cos x)=pi+x, ) then ( x ) belongs
to
В. ( (pi, 2 pi) )
D. None of these
12
723Let ( 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
7246. 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
725If ( 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
726If 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
727Find 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
728If ( 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
72922. 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
730Find the possible value of ( cos x, ) if ( cot x )
( +operatorname{cosec} x=5 )
12
731The sum of all the solution(s) of the
equation ( sin ^{-1} 2 x=cos ^{-1} x ) is
12
7322x
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
733Find the principal value of ( tan left(cos ^{-1} frac{1}{2}right) )12
734The 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
735f ( 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
736Evaluate the following:
( tan ^{-1}(tan 12) )
12
7371. 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
739Theorem: For any ( boldsymbol{x} in boldsymbol{R} quad sinh ^{-1} boldsymbol{x}= )
( log _{e}(x+sqrt{x^{2}+1}) )
12
740Find the principal value of:
( tan ^{-1}left(2 cos frac{2 pi}{3}right) )
12
741пп
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
742If ( 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
7431. 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
744Let ( 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
745The 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
746What 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
747Evaluate: ( 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
748Solve for ( x ; cos ^{-1} sqrt{x}>sin ^{-1} sqrt{x} )12
749Solve ( : 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
750The 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
751Solve ( : int frac{tan ^{-1} x}{1+x^{2}} d x )12
752Write in simplest form ( sin ^{-1}left[frac{sqrt{1+x}+sqrt{1-x}}{2}right] )12
753Evaluate the following:
( sin ^{-1}(sin 5) )
12
754Which 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
755where x < 1, then x is equal to
ماده
b. –
c
ده
d.
12
75663. 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
758The 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
759Solve: ( 3 tan ^{-1} x+cot ^{-1} x=pi )12
760The value of ( sin ^{-1}(sin 2) ) is?
A ( .2+n pi )
B. ( 2-pi )
( c cdot-2+pi )
D. ( 2-2 n pi )
12
76149. 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
763Solve ( y=tan ^{-1}left(frac{cos x}{1-sin x}right) )12
764Find the value of ( sin left[frac{1}{2} cot ^{-1}left(frac{-3}{4}right)right] )12
765Solve ( tan x<2 )12
766Find the principal value of the following
( tan ^{-1}left(tan frac{7 pi}{6}right) )
12
767Find the principal value of:
( cot ^{-1}(sqrt{3}) )
12
768If ( 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
769How do you simplify
( sin x+cot x cdot cos x )
12
770Integrate the function ( tan ^{-1}(sqrt{frac{1-sin x}{1+sin x}}) ) w.r.t d ( x )12
771If ( 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
772If ( tan ^{-1}(2 x)+tan ^{-1}(3 x)=pi / 4 ) then
( x=? )
12
77392. 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
774If ( tan ^{-1}(cot theta)=2 theta ) then ( theta= )
A ( cdot frac{pi}{3} )
в. ( frac{pi}{4} )
c.
D. None of the above
12
775The 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
776Find 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
778Calculate. ( arctan 1+arccos left(-frac{1}{2}right)+ )
( arcsin left(-frac{1}{2}right)=? )
12
779to our
Σ sin-
is equal to
P=
Irort1
c. tan”(Tn)
d. tan” (Jn+1)
12
780If 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
781Solve:
( sin ^{-1} x+sin ^{-1} sqrt{1-x^{2}} )
12
782If 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
783Solve the equation ( 2 tan ^{-1}(cos x)=tan ^{-1}(2 csc x) )12
784Find the principal value of the following
( cos ^{-1} frac{1}{2}+2 sin ^{-1} frac{1}{2} )
12
785The 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
786Solve: ( 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
789toppr
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
791Solve the equation ( tan ^{-1}left(frac{1-x}{1+x}right)= )
( frac{1}{2} tan ^{-1} x, x>0 )
12
792The 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
793Prove that ( tan ^{-1} frac{1}{2}+tan ^{-1} frac{2}{11}= )
( tan ^{-1} frac{3}{4} )
12
794find 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
796Show 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
797Evaluate ( cos ^{-1}(cos 3) )12
798Let ( 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
799Illustration 5.70 Which of the following angles is greater ?
COS
12
800Find the principal value of ( cot ^{-1}(sqrt{3}) )12
80111. 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
802If ( 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
803the 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
804f ( tan ^{-1} frac{x-3}{x-4}+tan ^{-1} frac{x+3}{x+4}=frac{3}{4}, ) then
find the value of ( x )
12
805Which of the following is/are the value of12
806The 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
807Find 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
809Find 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
811Find the value of ( cos left(sec ^{-1} x+right. )
( left.csc ^{-1} xright),|x| geq 1 )
12
812The 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
813Find 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
815For ( 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
816The 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
817The 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
818If ( 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
819ff ( y=sin left(cos ^{-1} xright) ) and ( x=99, ) then
( 1 / y^{2} ) is equal to
12
820Which of the following is/are a rational number?
1
b. cos

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

for x E
12
823What 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
825Find the principal value the following
expression:
( sin ^{-1}left(-frac{sqrt{3}}{2}right) )
12
826Solve 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
828Evaluate the following:
( sin ^{-1}(sin 10) )
12
829Illustration 5.46
(cosec-‘x)?
Find the minimum value of (sec- x)2 +
12
83075. If 3 tan-
– tan’ – = tan
, then x is equal to
a. 1
c. 3 Set
Sb. 2
ons d. 2
12
831Solve: ( 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
832The 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
833The 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
834If ( 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
8354
If sin
+ cosec
)
, then the values of x is
(a) 4
(c) 1
[20071
(b) 5
(d) 3
12
836Prove:
( 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
838Solve: ( sin ^{-1} frac{5}{x}+sin ^{-1} frac{12}{x}=frac{pi}{2} )12
839Differentiate ( cos ^{-1}left(4 x^{2}-3 xright) ; x epsilonleft(frac{1}{2}, 1right) )12
840Find 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
841If ( 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
8423.
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
843The 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
844If ( 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
845If ( 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
846The numerical value of tan ( left(2 tan ^{-1} frac{1}{5}-frac{pi}{4}right) ) is12
84740. 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
849Find 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
850If ( 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
851Find the principal value of the following:
( cot ^{-1}(-1) )
12
852The 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
853Assertion
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
854Find the principal value of:
( sec ^{-1}left(2 tan frac{3 pi}{4}right) )
12
855If ( 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
85772. If cot’ (Vcos a) – tan! (Vcos a ) = x, then sinx is
a. tang
b. cot a
c. tan
10
d. cot
810
12
85828. The trigonometric equation sin-‘x=2 sin ‘a has a solution
for
a. all real values
b.
V
12
8595.
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
860If ( 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
861Find the principal value:
( sin ^{-1}left(tan frac{5 pi}{4}right) )
12
862For the principal value:
( sin ^{-1}left(-frac{sqrt{mathbf{3}}}{mathbf{2}}right)+cos ^{-1}left(frac{sqrt{mathbf{3}}}{mathbf{2}}right) )
12
863Illustration 5.38
Find the minimum value of the function
Ax)=
16 cot-10–cot-‘x.
12
864Evaluate:
( tan ^{-1}left(frac{sqrt{1+cos x}-sqrt{1-cos x}}{sqrt{1+cos x}+sqrt{1-cos x}}right) )
12
865Prove that
( sin ^{-1} frac{3}{5}+sin ^{-1} frac{8}{17}=cos ^{-1} frac{36}{85} )
12
866a

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
867If ( y=cot ^{-1}(sqrt{cos x})- )
( tan ^{-1}(sqrt{cos x}) P . T sin y=tan ^{2} x / 2 )
12
868Write the value of
( tan ^{-1}left[2 sin left(2 cos ^{-1} frac{sqrt{3}}{2}right)right] )
12
869The 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
8708. 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
8712. The value of
sin
cot sin-1 12-13
los V 4
+cos-1 V12
+sec-12
4
Rim
d. none of these
12
872If ( I sin ^{-1} x-cos ^{-1} x=frac{pi}{6}, ) then solve for
( boldsymbol{x} ) ?
12
873Solve:
( 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
875If ( 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
876If ( 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
877ff ( 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
878The 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
879Solve:
( 2 tan ^{-1} frac{3}{4}-tan ^{-1} frac{17}{31} )
12
880Simplify ( 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
881Simplify: ( sin . cot ^{-1} cot x )12
882The 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
883Illustration 5.10 If sin- ‘(x2 + 2x + 2) + tan- ‘(x2 – 3x – K)
then find the values of k.
12
884Prove ( 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
885Illustration 5.16 Evaluate the following:
i. sin(sin at/4) ii. cos'(cos27/3)
iii. tan(tan 7/3)
12
886Prove that ( sin cot ^{-1} tan cos ^{-1} x= )
( sin operatorname{cosec}^{-1} cot tan ^{-1} x=x ) where
( x in(0,1] )
12
887Prove that :
( 3 cos ^{-1} x=cos ^{-1}left(4 x^{3}-3 xright) )
12
888Illustration 5.48 If a= sin-‘(cos(sin- x)) and ß=
cos-‘(sin(cos+ x)), then find tan a • tan B.
12
889The 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
890Find ( x ) if ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3} )12
8915. Find the number of positive integral solutions of the
3
equation tan’x + cos
= =sin-l
√ 1 – 2
TO
12
8924. The value of cos
-COS
is
Cool 7
one
c.
de
12
893The 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
894Evaluate: ( tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right) )12
895xo
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
896If 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
897The 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
898If ( 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
89948. If sin-1
48. If sin
+ sin(12) – , then x is equal to
+ sin-1
X
c. 13
12
900The 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
902Illustration 5.8
Find the value of x for which sec-‘x+sin-
12
903Evaluate:
( cos ^{-1}left(frac{2 x}{1+x^{2}}right) )
12
904If ( 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
905If ( 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
906Show 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
9076.
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
908If ( 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
909Illustration 5.74
IfA=2 tan-‘(2 V2 – 1) and B = 3 sin
+ sin-1

, then which is greater ?
12
910Find the value of ( cos ^{-1}left(frac{1}{2}right)+ )
( 2 sin ^{-1}left(frac{1}{2}right) )
12
91123. Domain of definition of the function
eal valued x, is
f(x)= /sin-‘(2x) + ” for real valued x, is (20035)
( ( ) [*
12
912Illustration 5.55 If two angles of a triangle are tan-‘(2) and
tan-‘(3), then find the third angle.
12
913Simplify: ( sin . cot ^{-1} tan cdot cos ^{-1} x )12
914If ( 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
916sin x ||
11. Solve the equation VI sin-‘| cos x | + | cos
sin- cosx – cos- | sin x |, ” <<<".
2
12
9171. Solve 2 cos-‘x = sin (2x v1 -x?).
1. Solv
12
918The 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
919Find the principal value of ( operatorname{cosec}^{-1}(-sqrt{2}) )12
920( sin ^{-1}left(frac{sqrt{1+x}+sqrt{1-x}}{2}right) )12
921The 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
924Prove that ( tan ^{-1} frac{sqrt{1+x^{2}}-1}{x}= )
( frac{1}{2} tan ^{-1} x, x neq 0 )
12
925If ( 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
926Consider ( 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
927If ( 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
9306. 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
932cot-(/cos a) – tan-(cosa) = x,then sinx=
(a) tana) (6) cot? )
(c) tan a (2) cot (2)
2002]
12
933A 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
934Solve ( tan ^{-1}left(frac{1-x}{1+x}right)= )
( frac{1}{2} tan ^{-1} x,(0<x<1) )
12
935Inverse 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
936Show 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
938Simplify:tan ( ^{-1}(1 / 2)+tan ^{-1}(1 / 3) )12
939If ( 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
940Find the principal value of ( operatorname{cosec}^{-1}(2) )12
941If ( 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
943If ( 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
944Find the value of ( cos left[frac{pi}{2}-sin ^{-1}left(frac{1}{3}right)right] )12
945The 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
946Find 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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