Mathematics - Inverse Trigonometric Functions
Exam Duration: 45 Mins Total Questions : 30
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 \(\alpha ={ sin }^{ -1 }\{ sin({ cos }^{ -1 }x)\} \) and \(\beta ={ cos }^{ -1 }[sin({ cos }^{ -1 }x)]\), then the value tan \(\alpha\). tan \(\beta\) is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
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)
\(\left( \frac { 33 }{ 67 } \right) \)
If \(\alpha ,\beta (\alpha <\beta )\) are the roots of the equation 6x2 + 11x + 3 = 0, then which of the following is real ?
- (a)
\({ sin }^{ -1 }\alpha \)
- (b)
\({ cos }^{ -1 }\alpha \)
- (c)
\({ cosec }^{ -1 }\beta \)
- (d)
\({ cot }^{ -1 }\alpha \quad and\quad { cot }^{ -1 }\beta \)
\(2{ tan }^{ -1 }(-2)\) is equal to
- (a)
\({ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \)
- (b)
\(\pi +{ cos }^{ -1 }\frac { 3 }{ 5 } \)
- (c)
\(-\frac { \pi }{ 2 } +{ tan }^{ -1 }\left( -\frac { 3 }{ 4 } \right) \)
- (d)
\(-\pi +{ cot }^{ -1 }\left( -\frac { 3 }{ 4 } \right) \)
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)
\(\left( x+\frac { 1 }{ x } \right) +\left( y+\frac { 1 }{ y } \right) +\left( z+\frac { 1 }{ z } \right) \ge 6\)
If \({ cos }^{ -1 }x-{ cos }^{ -1 }\frac { y }{ 2 } =\alpha ,\)then 4x2 - 4xy cos \(\alpha\) + y2 is equal to
- (a)
\(-4{ sin }^{ 2 }\alpha \)
- (b)
\(4{ sin }^{ 2 }\alpha \)
- (c)
4
- (d)
\(2sin2\alpha \)
Find the principal values of tan-1(-1)
- (a)
\(\frac{2\pi}{3}\)
- (b)
\(\frac{\pi}{10}\)
- (c)
\(\frac{3\pi}{4}\)
- (d)
\(-\frac{\pi}{4}\)
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,\(\pm{1\over2}\)
4 tan-1\(\frac{1}{5}\)-tan-1\(1\over70\)+tan-1\(1\over99\) is equal to
- (a)
\(\pi/6\)
- (b)
\(\pi/4\)
- (c)
\(\pi/3\)
- (d)
\(\pi/2\)
If 2sin-1x=sin-1(2x\(\sqrt{1-x^2}\)), then x belongs to
- (a)
[-1,1]
- (b)
\([-\frac{1}{\sqrt{2}},1]\)
- (c)
\([-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}]\)
- (d)
None of these
The range of sin-1 x+cos-1x+tan-1x is
- (a)
[0,\(\pi\)]
- (b)
\(\left[ \frac { \pi }{ 4 } ,\frac { 3\pi }{ 4 } \right] \)
- (c)
[0,\(\pi\))
- (d)
\([0, \frac { \pi }{ 2 } ]\)
\({ tan }\left( \frac { \pi }{ 4 } +\frac { 1 }{ 2 } { cos }^{ -1 }x \right) +tan\left( \frac { \pi }{ 4 } -\frac { 1 }{ 2 } { cos }^{ -1 }x \right) =\)
- (a)
x
- (b)
\(\frac{1}{x}\)
- (c)
2x
- (d)
\(\frac{2}{x}\)
The value of cot-19+cosec-1\(\frac{\sqrt{41}}{4}\) is given by
- (a)
0
- (b)
\(\frac{\pi}{4}\)
- (c)
tan-12
- (d)
\(\frac{\pi}{2}\)
The value of \({ tan }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 3 } \right) +{ tan }^{ -1 }\left( \frac { 7 }{ 8 } \right) \)is
- (a)
\({ tan }^{ -1 }\left( \frac { 7 }{ 8 } \right) \)
- (b)
cot-1(15)
- (c)
tan-1(15)
- (d)
\({ tan }^{ -1 }\left( \frac { 25 }{ 24 } \right) \)
If p>q>0 and pr<-1<qr, then \({ tan }^{ -1 }\left( \frac { p-q }{ 1+pq } \right) +{ tan }^{ -1 }\left( \frac { q-r }{ 1+qr } += \right) +{ tan }^{ -1 }\left( \frac { r-p }{ 1+rp } \right) =\_ \_ \_ .\)
- (a)
\(\pi \)
- (b)
2\(\pi \)
- (c)
\(\frac { \pi }{ 2 } \)
- (d)
\(\frac { 3\pi }{ 2 } \)
If A=tan-1\(\left( \frac { x\sqrt { 3 } }{ 2k-x } \right) andB={ tan }^{ -1 }\left( \frac { 2x-k }{ k\sqrt { 3 } } \right) \) then the value of A-B is
- (a)
10o
- (b)
45o
- (c)
60o
- (d)
30o
If \({ tan }^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \frac { x-1 }{ x } \right) ={ tan }^{ -1 }(-7)\), then the value of x is
- (a)
0
- (b)
-2
- (c)
1
- (d)
2
If 4cos-1x+sin-1x=π, then the value of x is
- (a)
\(3\over2\)
- (b)
\(\frac{1}{\sqrt{2}}\)
- (c)
\(\frac{\sqrt{3}}{2}\)
- (d)
\(\frac{2}{\sqrt{3}}\)
Solve the following equation sin[2cos-1{cot(2tan-1x)}]=0
- (a)
士1,1-√2
- (b)
士1,-1士√2, 1士√2
- (c)
-1士√2, 1士√2
- (d)
1,1+√2
\(tan^{-1}\frac{x-1}{x+1}+tan^{-1}\frac{2x-1}{2x+1}=tan^{-1}\frac{23}{36}\)
- (a)
\(\frac{4}{3}\)
- (b)
\(-\frac{4}{3}\)
- (c)
\(\frac{1}{3}\)
- (d)
\(\frac{1}{3}\)
2tan-1(cosx)=tan-1(2cosecx)
- (a)
0
- (b)
π/3
- (c)
π/4
- (d)
π/2
If sin-1 x+sin-1y+sin-1z=π, then x4+y4+z4+4x2y2z2=
- (a)
x2+y2+y2z2
- (b)
2(x2y2+y2z2+z2x2)
- (c)
(x+y)2
- (d)
(x+y+z)2
The value of expression 2sec-1 2+sin-1\((\frac{1}{2})\) is
- (a)
\(\frac{\pi}{6}\)
- (b)
\(\frac{5\pi}{6}\)
- (c)
\(\frac{7\pi}{6}\)
- (d)
1
If tan-1x+tan-1y=\(\frac{4\pi}{5}\), then cot-1x+cot-1y equals
- (a)
\(\frac{\pi}{5}\)
- (b)
\(\frac{2\pi}{5}\)
- (c)
\(\frac{3\pi}{5}\)
- (d)
π
If |x|≤1, then 2tan-1x+sin-1\((\frac{2x}{1+x^2})\) is equal to
- (a)
4 tan-1x
- (b)
0
- (c)
π/2
- (d)
π
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 \(sin[tan^{-1}(-\sqrt{3})+cos^{-1}(-\frac{\sqrt{3}}{2})]\) is 1.
Statement II: tan-1x(-x)=-tan-1x and cos-1(-x)=cos-1x.
- (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:
\(sec^{-1}(\frac{3}{2})+sin^{-1}(\frac{2}{3})-2tan^{-1}3-tan^{-1}(\frac{1}{3})\) is equal to tan-13
Statement II: sin-1x+cos-1x=\(\frac{\pi}{2}\),
tan-1x+cot-1x=\(\frac{\pi}{2}\)
cosec-1 x=sin-1\((\frac{1}{x})\)
cot-1(x)=tan-1\((\frac{1}{x})\)
- (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.