IISER Mathematics - Inverse Circular Functions
Exam Duration: 45 Mins Total Questions : 30
The number of solutions of the equation,\(cos^{-1}+m\ cos^{-1}x={n\pi\over 2}\) where m> 0, n > 0, is
- (a)
0
- (b)
1
- (c)
3
- (d)
infinite
\(x+{1\over x}=2\) the principal value of sin-1 x is
- (a)
\(\pi\over 4\)
- (b)
\(\pi\over 2\)
- (c)
\(\pi\)
- (d)
\(3\pi\over 2\)
If \(\frac{1}{2}\) < | x |< 1, then which of the following are real?
- (a)
sin-1x
- (b)
tan-1x
- (c)
sec-1x
- (d)
cos-1x
Let f(x) = \({ e }^{ { cos }^{ -1s }in\left( x+\frac { \pi }{ 3 } \right) }\), then
- (a)
\(f\left( \frac { 8\pi }{ 9 } \right) ={ e }^{ \frac { 5\pi }{ 18 } }\)
- (b)
\(f\left( \frac { 8\pi }{ 9 } \right) ={ e }^{ \frac { 13\pi }{ 18 } }\)
- (c)
\(f\left( -\frac { 7\pi }{ 4 } \right) ={ e }^{ \frac { \pi }{ 12 } }\)
- (d)
\(f\left( -\frac { 7\pi }{ 4 } \right) ={ e }^{ \frac { 11\pi }{ 12 } }\)
The value of \(tan^{-1}(1)+cos^{-1}\left(-{1\over 2}\right)+sin^{-1}\left(-{1\over2}\right)\)is equal to
- (a)
\(\pi\over 4\)
- (b)
\(5\pi\over12\)
- (c)
\(3\pi\over 4\)
- (d)
\(13\pi\over 12\)
The value of tan2(sec-12) + cot2 (cosec-13) is
- (a)
13
- (b)
15
- (c)
11
- (d)
none of these
If \(\alpha \le \sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } +\tan ^{ -1 }{ x } \le \beta \), then
- (a)
\(\alpha =0\)
- (b)
\(\beta =\pi /2\)
- (c)
\(\alpha =0\pi /4\)
- (d)
\(\beta =\pi \)
The greatest and least values of (sin-1 x)3 + (cos-1 x)3 are
- (a)
\(\frac { \pi ^3 }{ 32 } \)
- (b)
\(-\frac { \pi ^3 }{ 8 } \)
- (c)
\(\frac {7 \pi ^3 }{ 8 } \)
- (d)
\(\frac { \pi }{ 2 } \)
\(\alpha ,\beta \ and\ \gamma \) are the angles given by \(\alpha =2{ tan }^{ -1 }\left( \sqrt { 2 } -1 \right) ,\ \beta =3{ sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) +{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \ and\ \gamma ={ cos }^{ -1 }\left( \frac { 1 }{ 3 } \right) \), then
- (a)
\(\alpha >\beta \)
- (b)
\(\beta >\gamma \)
- (c)
\(\gamma >\alpha \)
- (d)
none of these
Indicate the relation which is true
- (a)
tan |tan-1x| = |x|
- (b)
cot |cot-1x| = |x|
- (c)
tan-1 |tan x| = |x|
- (d)
sin |sin-1 x| = |x|
\({ cos }^{ -1 }\left( \sqrt { \frac { a-x }{ a-b } } \right) ={ sin }^{ -1 }\left( \sqrt { \frac { x-b }{ a-b } } \right) \) is possible, if
- (a)
a > x > b
- (b)
a < x < b
- (c)
a = x = b
- (d)
a>b and x, takes any value
The smallest and the largest values of \(tan^{-1}\left(1-x\over 1+x\right),0\le x\le1\)are
- (a)
0π
- (b)
\(0,{\pi\over 4}\)
- (c)
\(-,{\pi\over 4}{\pi\over 4}\)
- (d)
\({\pi\over 4},{\pi\over 2}\)
If - 1< x < 0, then sin-1 x equals
- (a)
\(\pi-cos^{-1}\{\sqrt{1-x^2}\}\)
- (b)
\(tan^{-1}\left\{x\over \sqrt{(1-x^2)}\right\}\)
- (c)
\(-cot^{-1}\left\{ \sqrt{(1-x^2)\over x}\right\}\)
- (d)
cosec-1 x
\(\theta =\tan ^{ -1 }{ \left( 2\quad { tan }^{ 2 }\theta \right) } -{ tan }^{ -1 }\left\{ \left( \frac { 1 }{ 3 } \right) tan\quad \theta \right\} ,\)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
The value of \(sin^{-1}\left\{(sin\pi/3){x\over \sqrt{(x^2+k^2-k\ x)}}\right\}-cos^{-1}\left\{cos{\pi\over 6}{x\over \sqrt{(x^2+k^2-kx)}}\right\}\)where \(\left( {k\over 2}< x<2k, k>0\right)\)
- (a)
\(tan^{-1}\left( 2x^2+xk-k^2\over x^2-2xk+k^2\right)\)
- (b)
\(tan^{-1}\left( x^2+2xk-k^2\over x^2-2xk+k^2\right)\)
- (c)
\(tan^{-1}\left( x^2+2xk-2k^2\over 2x^2-2xk+2k^2\right)\)
- (d)
none of these
The value \(\left\{\left(cos^{-1}\left(-{2\over 7}\right)-{\pi\over 2}\right)\right\}\)is
- (a)
\(2\over 3\sqrt5\)
- (b)
\(2\over 3\)
- (c)
\(1\over \sqrt5\)
- (d)
\(4\over \sqrt5\)
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 \(cot^{-1}\left(n\over \pi\right)>\left(\pi\over 6\right), n\epsilon N\) then the maximum value of n is
- (a)
1
- (b)
5
- (c)
9
- (d)
none of these
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 takes negative permissible value, then sin-1 x is equal to
- (a)
\({ cos }^{ -1 }\sqrt { { \left( 1-{ x }^{ 2 } \right) } } \)
- (b)
\({ cos }^{ -1 }\left( \frac { \sqrt { { 1-{ x }^{ 2 } } } }{ x } \right) \)
- (c)
\(\pi -{ cos }^{ -1 }\sqrt { \left( 1-{ x }^{ 2 } \right) } \)
- (d)
\(-\pi +{ cot }^{ -1 }\left( \frac { \sqrt { { 1-{ x }^{ 2 } } } }{ x } \right) \)
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 function. On the basis of above information, answer the following question: If \(\frac { 3\pi }{ 2 } \le x\le \frac { 5\pi }{ 2 } \) then sin-1(sin x) is equal to
- (a)
x
- (b)
-x
- (c)
\(2\pi -x\)
- (d)
\(x-2\pi \)
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
The solution of the inequality (cot-1 X)2 - 5 cot-1 X + 6> 0 is
- (a)
(cot 3, cot 2)
- (b)
(- ∞, cot 3) u (cot 2, ∞)
- (c)
(cot 2, ∞)
- (d)
none of the above
\(\sum _{ r=1 }^{ n }{ { tan }^{ -1 } } \left( \frac { { x }_{ r }-{ x }_{ r-1 } }{ 1+{ x }_{ r-1 }{ x }_{ r } } \right) =\sum _{ r=1 }^{ n }{ \left( { tan }^{ -1 }{ x }_{ r }-{ tan }^{ -1 }{ x }_{ r-1 } \right) } ={ tan }^{ -1 }{ x }_{ n }-{ tan }^{ -1 }{ x }_{ 0 },\forall n\epsilon N\)
On the basis of above information, answer the following questions:
The value of \({ cosec }^{ -1 }\sqrt { 5 } +{ cosec }^{ -1 }\sqrt { 65 } +{ cosec }^{ -1 }\sqrt { \left( 325 \right) } +.....\) to \(\infty \) is
- (a)
\(\pi \)
- (b)
\(\frac { 3\pi }{ 4 } \)
- (c)
\(\frac { \pi }{2 } \)
- (d)
\(\frac { \pi }{ 4 } \)
\(\sum _{ r=1 }^{ n }{ { tan }^{ -1 } } \left( \frac { { x }_{ r }-{ x }_{ r-1 } }{ 1+{ x }_{ r-1 }{ x }_{ r } } \right) =\sum _{ r=1 }^{ n }{ \left( { tan }^{ -1 }{ x }_{ r }-{ tan }^{ -1 }{ x }_{ r-1 } \right) } ={ tan }^{ -1 }{ x }_{ n }-{ tan }^{ -1 }{ x }_{ 0 },\forall n\epsilon N\)
On the basis of above information, answer the following questions:
The sum to infinite terms of the series \({ cot }^{ -1 }\left( { 2 }^{ 2 }+\frac { 1 }{ 2 } \right) +{ cot }^{ -1 }\left( { 2 }^{ 3 }+\frac { 1 }{ 2^{ 2 } } \right) +{ cot }^{ -1 }\left( { 2 }^{ 4 }+\frac { 1 }{ 2^{ 3 } } \right) +.....\)is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
cot-1 2
- (d)
-cot-1 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 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 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