JEE Main Mathematics - Inverse Circular Functions
Exam Duration: 60 Mins Total Questions : 30
The greatest of tan1, tan-1 1, sin-11, sin1, cos 1, is
- (a)
sin 1
- (b)
tan 1
- (c)
tan-1 1
- (d)
none of these
If the mapping f(x)=ax+b,a>0 maps [-1,1] onto [0, 2], then cot [cot-1 7 + cot-1 8 + cot-1 18] is equal to
- (a)
f( - 1)
- (b)
f(0)
- (c)
f(1)
- (d)
f(2)
The sum of the infinite series \(sin^{-1}\left(1\over \sqrt2\right)+sin^{-1}\left(\sqrt2-1\over \sqrt6\right)+sin^{-1}\left(\sqrt3-\sqrt2\over \sqrt{12}\right)+.......+........+sin^{-1}\left(\sqrt n-\sqrt{(n-1)}\over \sqrt\{n(n+1)\}\right)+.........\)is
- (a)
\(\pi\over 8\)
- (b)
\(\pi\over 4\)
- (c)
\(\pi\over 2\)
- (d)
\(\pi\)
If [sin-1 cos-1 sin-1 tan-1 x] = 1, where [.] denotes the greatest integer function, then x belongs to the interval
- (a)
[tan sin cos 1, tan sin cos sin 1]
- (b)
(tan sin cos 1, tan sin cos sin 1)
- (c)
[- 1, 1]
- (d)
[sin cos tan 1, sin cos sin tan 1]
If \(cos^{-1}\left(x\over 3\right)+cos^{-1}\left(y\over 2\right)={\theta\over 2}\)then the value of \(4x^2-12xy\ cos\left(\theta\over 2\right)+9y^2\)is equal to
- (a)
18(1 + cos θ)
- (b)
18(1- cos θ)
- (c)
36 (1 + cos θ)
- (d)
36 (1 - cos θ)
If \(sin^{-1}x+sin^{-1}y={2\pi\over 3}\)then cos-1 x + cos-1 y is equal to
- (a)
\(2\pi\over 3\)
- (b)
\(\pi\over 3\)
- (c)
\(\pi\over 6\)
- (d)
\(\pi\)
\(x+{1\over x}=2\) the principal value of sin-1 x is
- (a)
\(\pi\over 4\)
- (b)
\(\pi\over 2\)
- (c)
\(\pi\)
- (d)
\(3\pi\over 2\)
Asolution of the equation tan-1(1 + x) + tan-1(1- x) = \(\pi\over 2\) is
- (a)
x=1
- (b)
x=-1
- (c)
x=0
- (d)
x=ㅠ
sin -1 x> cos-1 x holds for
- (a)
all values of x
- (b)
\(x\epsilon \left( 0,\frac { 1 }{ \sqrt { 2 } } \right) \)
- (c)
\(x\epsilon \left( \frac { 1 }{ \sqrt { 2 } } ,1 \right) \)
- (d)
x = 0.75
If 6 sin-1 (x2- 6x + 8.5) = \(\pi\), then
- (a)
x = 1
- (b)
x = 2
- (c)
x = 3
- (d)
x = 4
If \(\sum_{i=i}^{2n}sin^{-1}x_i=n\pi\ then\ \sum_{i=1}^{2n}\)is equal to
- (a)
n
- (b)
2n
- (c)
\(n(n+1)\over 2\)
- (d)
none of these
The inequality sin-1 (sin 5) > X2 - 4x holds, if
- (a)
\(x=2-\sqrt{(9-2\pi)}\)
- (b)
\(x=2+\sqrt{(9-2\pi)}\)
- (c)
\(x\epsilon (2-\sqrt{(9-2\pi)}, 2+\sqrt{(9-2\pi)}\)
- (d)
\(x>2+\sqrt{(9-2\pi)}\)
\(\theta =\tan ^{ -1 }{ \left( 2\quad { tan }^{ 2 }\theta \right) } -{ tan }^{ -1 }\left\{ \left( \frac { 1 }{ 3 } \right) tan\quad \theta \right\} ,\)if
- (a)
\(tan\ \theta =-2\)
- (b)
\(tan\ \theta =0\)
- (c)
\(tan\ \theta =1\)
- (d)
\(tan\ \theta =2\)
If cosec-1x = sin-1\(\left( \frac { 1 }{ x } \right) \), then x may be
- (a)
1
- (b)
\(-\frac { 1 }{ 2 } \)
- (c)
\(\frac { 3 }{ 2 } \)
- (d)
\(-\frac { 3 }{ 2 } \)
\(2{ cot }^{ -1 }7+{ cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \) is equal to
- (a)
\({ cot }^{ -1 }\left( \frac { 44 }{ 117 } \right) \)
- (b)
\({ cosec }^{ -1 }\left( \frac { 125 }{ 117 } \right) \)
- (c)
\({ tan }^{ -1 }\left( \frac { 4 }{ 117 } \right) \)
- (d)
\({ cos }^{ -1 }\left( \frac { 44 }{ 125 } \right) \)
If the equation sin-1(x2+x+1)+cos-1(\(\lambda\) x+1) = \(\frac { \pi }{ 2 } \) has exactly two solutions, then \(\lambda \) cannot have the integral value
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
If a, b are positive quantities and, if \(a_1={a+b\over 2},b_1=\sqrt{a_1,b}\) \(a_2={a_1+b_1\over1},b_2\sqrt{a_1b_1}\) and so on, then
- (a)
\(a_\infty={sqrt{(b^2-a^2)}\over cos^{-1}\left(a\over b\right)}\)
- (b)
\(b_\infty={sqrt{(b^2-a^2)}\over cos^{-1}\left(a\over b\right)}\)
- (c)
\(b_\infty={sqrt{(a^2-b^2)}\over cos^{-1}\left(b\over a\right)}\)
- (d)
none of these
If cos-1x = tan-1 x, then
- (a)
\({ x }^{ 2 }=\left( \sqrt { 5 } -1 \right) /2\)
- (b)
\({ x }^{ 2 }=\left( \sqrt { 5 } +1 \right) /2\)
- (c)
\(sin\left( { cos }^{ -1 }x \right) =\left( \sqrt { 5 } -1 \right) /2\)
- (d)
\(\\ \\ tan\left( { cos }^{ -1 }x \right) =\left( \sqrt { 5 } -1 \right) /2\)
If \((tan^{-1}x)^2+(cot^{-1}x^2)={5\pi^2\over 8}\)then x equals
- (a)
0
- (b)
-1
- (c)
-2
- (d)
-3
If a1 a2, a3, ... , an is an AP with common difference d, then \(tan\left[tan^{-1}\left(d\over 1+a_1a_2\right)+tan^{-1}\left(d\over 1+a_2a_3\right)+...+tan^{-1}\left(d\over 1+a_{n-1}a_n\right)\right]\)
- (a)
\((n-1)d\over a_1+a_n\)
- (b)
\((n-1)d\over1+ a_1a_n\)
- (c)
\(nd\over1+ a_1a_n\)
- (d)
\(a_n-a_1\over a_n+a_1\)
Let f : A\(\rightarrow \)B be a function defined by y = f(x) such that f is both one-one (Injective) and onto (surjective)(ie, bijective), then there exists a unique function g: B\(\rightarrow \)A such that \(f\left( x \right) =y\Leftrightarrow g\left( y \right) =x,\ \forall x\epsilon A\ y\epsilon B\), then g is said to be inverse of f. Thus, g = f-1: B\(\rightarrow \)A = \(\left[ \left\{ f\left( x \right) ,x \right\} :\left\{ x,\ f(x) \right\} \epsilon { f }^{ -1 } \right] \).If no branch of an inverse trigonometric function is mentioned, then it means the principal value branch of that functon. On the basis of above information, answer the following question: If x>1, then the value of 2 tan-1x + sin-1 \(\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \) is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\pi\)
- (d)
\(\frac { 3\pi }{ 2 } \)
The principal \(cos^{-1}\left(cos{2\pi\over 3}\right)+sin^{-1}\left(sin{2\pi\over 3}\right)\)is
- (a)
π
- (b)
π/2
- (c)
π/3
- (d)
4π/3
\(\sum _{ r=1 }^{ n }{ { tan }^{ -1 } } \left( \frac { { x }_{ r }-{ x }_{ r-1 } }{ 1+{ x }_{ r-1 }{ x }_{ r } } \right) =\sum _{ r=1 }^{ n }{ \left( { tan }^{ -1 }{ x }_{ r }-{ tan }^{ -1 }{ x }_{ r-1 } \right) } ={ tan }^{ -1 }{ x }_{ n }-{ tan }^{ -1 }{ x }_{ 0 },\forall n\epsilon N\)
On the basis of above information, answer the following questions:
The sum to infinite terms of the series \({ tan }^{ -1 }\left( \frac { 1 }{ 3 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 9 } \right) +....+{ tan }^{ -1 }\left( \frac { { 2 }^{ n-1 } }{ 1+{ 2 }^{ 2n-1 } } \right) +......\)is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\pi\)
- (d)
none of these
\(\sum _{ r=1 }^{ n }{ { tan }^{ -1 } } \left( \frac { { x }_{ r }-{ x }_{ r-1 } }{ 1+{ x }_{ r-1 }{ x }_{ r } } \right) =\sum _{ r=1 }^{ n }{ \left( { tan }^{ -1 }{ x }_{ r }-{ tan }^{ -1 }{ x }_{ r-1 } \right) } ={ tan }^{ -1 }{ x }_{ n }-{ tan }^{ -1 }{ x }_{ 0 },\forall n\epsilon N\)
On the basis of above information, answer the following questions:
The sum to infinite terms of the series \({ tan }^{ -1 }\left( \frac { 2 }{ 1-{ 1 }^{ 2 }+{ 1 }^{ 4 } } \right) +{ tan }^{ -1 }\left( \frac { 4 }{ 1-{ 2 }^{ 2 }+{ 2 }^{ 4 } } \right) +{ tan }^{ -1 }\left( \frac { 6 }{ 1-{ 3 }^{ 2 }+{ 3 }^{ 4 } } \right) +\) ......is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\frac { 3\pi }{ 4 } \)
- (d)
none of these
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The principal value of \(\sin ^{ -1 }{ \left( sin\ 5 \right) -\cos ^{ -1 }{ \left( cos\ 5 \right) } } \) is
- (a)
0
- (b)
\(2\pi -10\)
- (c)
\(-\pi \)
- (d)
\(3\pi -10\)
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The principal value of \({ tan }^{ -1 }\left( tan\left( -\frac { 3\pi }{ 4 } \right) \right) +{ cot }^{ -1 }cot\left( -\frac { 3\pi }{ 4 } \right) \) is
- (a)
\(\frac { \pi }{ 2 } \)
- (b)
\(\pi\)
- (c)
\(\frac { -3\pi }{ 2 } \)
- (d)
0
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The value of sin-1 [cos{cos-1(cos x)+sin-1(sin x)}], where \(x\epsilon \left( \frac { \pi }{ 2 } ,\pi \right) \) is
- (a)
\(\frac { \pi }{ 2 } \)
- (b)
\(-\pi\)
- (c)
\(\pi\)
- (d)
\(-\frac { \pi }{ 2 } \)
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The number of solutions of the equation \({ cos }^{ -1 }\left( \frac { { x }^{ 2 }-1 }{ { x }^{ 2 }+1 } \right) +{ sin }^{ -1 }\left( \frac { 2x }{ { x }^{ 2 }-1 } \right) +{ tan }^{ -1 }\left( \frac { 2x }{ { x }^{ 2 }-1 } \right) =\frac { 2\pi }{ 3 } \) is
- (a)
1
- (b)
2
- (c)
3
- (d)
infinite