IISER Mathematics - Inverse Trigonometric Functions
Exam Duration: 45 Mins Total Questions : 30
Find the value of \(sin\left\{ \left( \frac { \pi }{ 2 } -{ sin }^{ -1 }\left( -\frac { \sqrt { 3 } }{ 2 } \right) \right) \right\} \)
- (a)
\(\frac { 1 }{ 3 } \)
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
\(-\frac { 1 }{ 2 } \)
- (d)
\(\frac { 2 }{ 3 } \)
If \({ tan }^{ -1 }x=\frac { \pi }{ 10 } \) for some \(x\epsilon R\), then the value of cot-1 x is
- (a)
\(\frac { \pi }{ 5 } \)
- (b)
\(\frac { 2\pi }{ 5 } \)
- (c)
\(\frac { 3\pi }{ 5 } \)
- (d)
\(\frac { 4\pi }{ 5 } \)
The number of solutions of the equation cos ( cos-1 x) = cosec (cosec-1 x) is
- (a)
2
- (b)
3
- (c)
4
- (d)
1
If the value of \({ cos }^{ -1 }\frac { 4 }{ 5 } +is\quad { cos }^{ -1 }\frac { 12 }{ 13 } { cos }^{ -1 }x\), then x is
- (a)
\(\left( \frac { 33 }{ 34 } \right) \)
- (b)
\(\left( \frac { 33 }{ 64 } \right) \)
- (c)
\(\left( \frac { 33 }{ 65 } \right) \)
- (d)
\(\left( \frac { 33 }{ 67 } \right) \)
Find the value of \({ 4tan }^{ -1 }\frac { 1 }{ 5 } -{ tan }^{ -1 }\frac { 1 }{ 70 } +{ tan }^{ -1 }\frac { 1 }{ 99 } \)
- (a)
\(\frac { \pi }{ 2 } \)
- (b)
\(\frac { 3\pi }{ 4 } \)
- (c)
\(\frac { \pi }{ 4 } \)
- (d)
\(\frac { 2\pi }{ 3 } \)
If \({ cosec }^{ -1 }x+{ cos }^{ -1 }y+{ sec }^{ -1 }z\ge { \alpha }^{ 2 }-\sqrt { 2\pi } \alpha +3\pi \), where \(\alpha\) is a real number, then
- (a)
x = 1, y = -1
- (b)
x = -1, z = -1
- (c)
x = 2, y = 1
- (d)
x = 1, y = -2
If \(f(x)={ e }^{ { cos }^{ -1 }sin\left( x+\frac { \pi }{ 3 } \right) }\), then
- (a)
\(f\left( -\frac { 7\pi }{ 4 } \right) ={ e }^{ \frac { \pi }{ 11 } }\)
- (b)
\(f\left( \frac { 8\pi }{ 9 } \right) ={ e }^{ \frac { 13\pi }{ 18 } }\)
- (c)
\(f\left( -\frac { 7\pi }{ 4 } \right) ={ e }^{ \frac { 3\pi }{ 12 } }\)
- (d)
\(f\left( -\frac { 7\pi }{ 4 } \right) ={ e }^{ \frac { 11\pi }{ 13 } }\)
If \({ cos }^{ -1 }x+{ cos }^{ -1 }y+{ cos }^{ -1 }z=\pi ,\) then
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }+2xyz=1\)
- (b)
\(({ sin }^{ -1 }x+{ sin }^{ -1 }y+{ sin }^{ -1 }z)={ cos }^{ -1 }x+{ cos }^{ -1 }y+{ cos }^{ -1 }z\)
- (c)
\(xy+yz+zx=x+y+z-1\)
- (d)
\(\left( x+\frac { 1 }{ x } \right) +\left( y+\frac { 1 }{ y } \right) +\left( z+\frac { 1 }{ z } \right) \ge 6\)
Find the principal values of \(cosec^{-1}(\frac{-2}{\sqrt{3}})\)
- (a)
\(-\frac{\pi}{3}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{\pi}{2}\)
- (d)
\(-\frac{\pi}{2}\)
\(cos^{-1}(\frac{1}{2})+2sin^{-1}(\frac{1}{2})\) is equal to
- (a)
\(\frac{\pi}{4}\)
- (b)
\(\frac{\pi}{6}\)
- (c)
\(\frac{\pi}{3}\)
- (d)
\(\frac{2\pi}{3}\)
\(cos^{-1}(\frac{1}{2})+2sin^{-1}(\frac{1}{2})+4tan^{-1}(\frac{1}{\sqrt{3}})\) is equal to
- (a)
\(\frac{\pi}{6}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{4\pi}{3}\)
- (d)
\(\frac{3\pi}{4}\)
If \(cot^{-1}(\sqrt{cos\alpha})-tan^{-1}(\sqrt{cos\alpha})=x,\) then sin x is equal to
- (a)
\(tan^2(\frac{\alpha}{2})\)
- (b)
\(cot^2(\frac{\alpha}{2})\)
- (c)
tanα
- (d)
\(cot(\frac{\alpha}{2})\)
If tan-1(x-1)+tan-1x+tan-1(x+1)=tan-13x, then the values of x are
- (a)
\(\pm\frac{1}{2}\)
- (b)
0,\(1\over2\)
- (c)
0,-\(1\over2\)
- (d)
0,\(\pm{1\over2}\)
\(cos^{-1}\{\frac{1}{2}x^2+\sqrt{1-x^2}\sqrt{1-\frac{x^2}{4}}\}=cos^{-1}\frac{x}{2}-cos^{-1}x\) holds for
- (a)
|x|≤1
- (b)
x∈R
- (c)
0≤x≤1
- (d)
-1≤x≤0
4tan-1\(1\over5\)-tan-1\(1\over239\) is equal to
- (a)
π
- (b)
\(\frac{\pi}{2}\)
- (c)
\(\frac{\pi}{3}\)
- (d)
\(\frac{\pi}{4}\)
The value of tan-1(1)+tan-1(0)+tan-1(2)+tan-1(3) is equal to
- (a)
π
- (b)
\(\frac{5\pi}{4}\)
- (c)
\(\frac{\pi}{2}\)
- (d)
None of these
\({ tan }\left( \frac { \pi }{ 4 } +\frac { 1 }{ 2 } { cos }^{ -1 }x \right) +tan\left( \frac { \pi }{ 4 } -\frac { 1 }{ 2 } { cos }^{ -1 }x \right) =\)
- (a)
x
- (b)
\(\frac{1}{x}\)
- (c)
2x
- (d)
\(\frac{2}{x}\)
\(sin^{-1}(\frac{2x}{1+x^2})=2tan^{-1}x\) for
- (a)
|x|≥1
- (b)
x≥0
- (c)
|x|≤1
- (d)
all x∈R
\(cot^{-1}(\frac{ab+1}{a-b})+cot^{-1}(\frac{bc+1}{b-c})+cot^{-1}(\frac{ca+1}{c-a})\) is equal to
- (a)
0
- (b)
\(\frac{\pi}{4}\)
- (c)
1
- (d)
5
Number of solutions of the equation \(tan^{-1}(\frac{1}{2x+1})+tan^{-1}(\frac{1}{4x+1})=tan^{-1}(\frac{2}{x^{2}})\) is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
\(cos(2tan^{-1}\frac{1}{7})-sin(4tan^{-1}\frac{1}{3})=\)
- (a)
1
- (b)
0
- (c)
\(1\over2\)
- (d)
\(-{1\over2}\)
The value of \({ tan }^{ -1 }\left( \frac { 3 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) \)is
- (a)
\(\pi \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\frac { 3\pi }{ 4 } \)
- (d)
\(\frac { \pi }{ 4 } \)
If sin-1 x-cos-1 x=\(\frac{\pi}{6}\), then x=
- (a)
\(\frac{1}{2}\)
- (b)
\(\frac{\sqrt3}{2}\)
- (c)
-\(\frac{1}{2}\)
- (d)
-\(\frac{\sqrt3}{2}\)
If tan-13+tan-1x=tan-18, then x=
- (a)
5
- (b)
\(\frac{1}{5}\)
- (c)
\(5\over14\)
- (d)
\(14\over5\)
tan-12x+tan-13x=\(\frac{\pi}{4}\)
- (a)
1
- (b)
\(1\over6\)
- (c)
\(1\over3\)
- (d)
\(-{1\over6}\)
\(2sin^{-1}\sqrt{\frac{1-x}{2}}=\)
- (a)
cos-1x
- (b)
\(2cos^{-1}\sqrt{\frac{1+x}{2}}\)
- (c)
Both (a) and (b)
- (d)
None of these
The value of cos-1\((cos\frac{3\pi}{2})\) is equal to
- (a)
\(\frac{\pi}{2}\)
- (b)
\(\frac{3\pi}{2}\)
- (c)
\(\frac{5\pi}{2}\)
- (d)
\(\frac{7\pi}{2}\)
If x=\(cosec[tan^{-1}\{cos(cot^{-1}(sec(sin^{-1}a)))\}]\) and y=\(sec[cot^{-1}\{sin(tan^{-1}(cosec(cos^{-1}a)))\}]\) where a ∈ [0,1]. Find the relationship between x and y in terms of a.
- (a)
x2=y2=a2
- (b)
x2=y2=3+a2
- (c)
x2=y2=3-a2
- (d)
none of these
Statement I:\(sin^{-1}\frac{8}{17}+sin^{-1}\frac{3}{5}=sin^{-1}\frac{77}{85}\)
Statement II: \(tan^{-1}x+tan^{-1}y=tan^{-1}(\frac{x+y}{1-xy}), xy<1\)
- (a)
If both statement I and statement II are true and statement II is the correct explanation of statement I
- (b)
If both statement I and statement II are true but statement II is not the correct explanation of statement I
- (c)
If statement I is true but statement II is false.
- (d)
If statement I is false and statement II is true.
Statement I: The value of \(tan^{-1}(\frac{3}{4})+tan^{-1}(\frac{1}{7})\) is \(\frac{\pi}{4}\)
Statement II: If x>0, y>0, then \(tan^{-1}(\frac{x}{y})+tan^{-1}(\frac{y-x}{y+x})=\frac{3\pi}{4}\)
- (a)
If both statement I and statement II are true and statement II is the correct explanation of statement I
- (b)
If both statement I and statement II are true but statement II is not the correct explanation of statement I
- (c)
If statement I is true but statement II is false.
- (d)
If statement I is false and statement II is true.