Mathematics - Inverse Trigonometric Functions 1
Exam Duration: 45 Mins Total Questions : 30
If \({ sin }^{ -1 }\frac { 14 }{ \left| x \right| } +{ sin }^{ -1 }\frac { 2\sqrt { 15 } }{ \left| x \right| } =\frac { \pi }{ 2 } \) , then the value of x is
- (a)
\(\pm 10\)
- (b)
\(\pm 16\)
- (c)
\(\pm 4\)
- (d)
\(\pm 11\)
If sin \(\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\), then the value of x is
- (a)
\(\frac { 1 }{ 4 } \)
- (b)
\(\frac { 1 }{ 3 } \)
- (c)
\(\frac { 1 }{ 5 } \)
- (d)
\(\frac { 1 }{ 2 } \)
Observe the following columns
ColumnI | ColumnII | ||
A. | Rnage of f(x) = sin-1 x + cos-1 x + cot-1 x is | p | \(\left[ \left( 0,\frac { \pi }{ 2 } \right) \cup \left( \frac { \pi }{ 2 } ,\pi \right) \right] \) |
B. | Range of f(x) = cot-1 x + tan-1 x + cosec-1x is | q | \(\left[ \frac { \pi }{ 2 } ,\frac { 3\pi }{ 2 } \right] \) |
C. | Range of f(x) = cot-1 x + tan-1 x + cos-1 x is | r | \(\{ 0,\pi \} \) |
D. | Range of f(x) = sec-1 x + cosec-1 x + sin-1 x is | s | \(\left[ \frac { 3\pi }{ 4 } ,\frac { 5\pi }{ 4 } \right] \) |
- (a)
s p q r
- (b)
r s p q
- (c)
q r s p
- (d)
None of these
If \({ 3tan }^{ -1 }x+{ cot }^{ -1 }x=\pi \), then x is equal to
- (a)
0
- (b)
1
- (c)
-1
- (d)
\(\frac { 1 }{ 2 } \)
The value of \(tan\left\{ { cos }^{ -1 }\left( -\frac { 2 }{ 7 } \right) -\frac { \pi }{ 2 } \right\} \) is
- (a)
\(\frac { 2 }{ 3\sqrt { 5 } } \)
- (b)
\(\frac { 2 }{ 3 } \)
- (c)
\(\frac { 1 }{ \sqrt { 5 } } \)
- (d)
\(\frac { 4 }{ 5 } \)
The value of \(tan\left\{ \frac { 1 }{ 2 } { sin }^{ 2 }\left( \frac { 2x }{ { 1+x }^{ 2 } } \right) +\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { { 1-y }^{ 2 } }{ { 1+y }^{ 2 } } \right) \right\} \) is
- (a)
\(\frac { x-y }{ 1-xy } \)
- (b)
\(\frac { x-y }{ 1+xy } \)
- (c)
\(\frac { x+y }{ 1+xy } \)
- (d)
\(\frac { x+y }{ 1-xy } \)
If 2 tan-1(cos x) = tan-1 (2 cosec x), then the value of x is
- (a)
\(\frac { 2\pi }{ 3 } \)
- (b)
\(\frac { \pi }{ 4 } \)
- (c)
\(\frac { 3\pi }{ 4 } \)
- (d)
\(\frac { 3\pi }{ 2 } \)
The trognometric equation sin-1 x = 2 sin-1 a has a solution for
- (a)
\(\frac { 1 }{ 2 } <\left| a \right| <\frac { 1 }{ \sqrt { 2 } } \)
- (b)
all real values of a
- (c)
\(\left| a \right| \le \frac { 1 }{ \sqrt { 2 } } \)
- (d)
\(\left| a \right| \ge \frac { 1 }{ \sqrt { 2 } } \)
Find the principal values of tan-1(√3)
- (a)
\(\frac{\pi}{6}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{2\pi}{3}\)
- (d)
\(\frac{5\pi}{6}\)
Find the principal values of cot-1(-√3)
- (a)
\(\frac{5\pi}{6}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{\pi}{2}\)
- (d)
\(\frac{\pi}{4}\)
\(cos^{-1}(\frac{\sqrt{3}}{2})=\)
- (a)
\(\frac{\pi}{2}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{\pi}{4}\)
- (d)
\(\frac{\pi}{6}\)
If cos-1x-cos-1\(\frac{y}{2}=\alpha\), then 4x2-4xycosα+y2 is equal to
- (a)
-4sin2α
- (b)
4sin2α
- (c)
4
- (d)
2sin2α
If \(sin(sin^{-1}\frac{1}{5}+cos^{-1}x)=1,\) then the value of x is
- (a)
-1
- (b)
\(\frac{2}{5}\)
- (c)
\(1\over3\)
- (d)
\(1\over5\)
If tan-12x+tan-13x=\(\frac{\pi}{4}\), then x is
- (a)
\(1\over6\)
- (b)
-1
- (c)
\((\frac{1}{6},-1)\)
- (d)
None of these
\({ tan }^{ -1 }\left[ cos\left( 2tan^{ -1 }\frac { 3 }{ 4 } \right) +sin\left( 2cot^{ -1 }\frac { 1 }{ 2 } \right) \right] \)is
- (a)
not defined
- (b)
\(\frac{\pi}{4}\)
- (c)
>\(\frac{\pi}{4}\)
- (d)
<\(\frac{\pi}{4}\)
The number of positive integral solutions of tan-1x+cos-1\(\left( \frac { y }{ \sqrt { 1+{ y }^{ 2 } } } \right) ={ sin }^{ -1 }\left( \frac { 3 }{ \sqrt { 10 } } \right) \)
- (a)
0
- (b)
1
- (c)
2
- (d)
>2
\(2{ tan }^{ -1 }\left( \frac { 1 }{ 3 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\)
- (a)
\(\frac{\pi}{4}\)
- (b)
\(\frac{\pi}{2}\)
- (c)
\(\frac{\pi}{3}\)
- (d)
\({\pi}\)
If tan-1\((\frac{a}{x})+tan^{-1}(\frac{b}{x})=\frac{\pi}{2}\), then x is equal to
- (a)
\(\sqrt{ab}\)
- (b)
\(\sqrt{2ab}\)
- (c)
2ab
- (d)
ab
The number of the real solutions of \(tan^{-1}\sqrt{x(x+1)}+sin^{-1}\sqrt{x^2+x+1}=\frac{\pi}{2}\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
∞
If \(tan^{-1}(\frac{x-1}{x+2})+tan^{-1}(\frac{x+1}{x+2})=\frac{\pi}{4}\) then x is equal to
- (a)
\(\frac{1}{\sqrt{2}}\)
- (b)
\(-\frac{1}{\sqrt{2}}\)
- (c)
\(\pm\sqrt{\frac{5}{2}}\)
- (d)
\(\pm\frac{1}{2}\)
If θ=tan-1a, ф=tan-1b and ab=-1, then (θ-ф) is equal to
- (a)
0
- (b)
\(\frac{\pi}{4}\)
- (c)
\(\frac{\pi}{2}\)
- (d)
None of these
\(tan\left[ { cos }^{ -1 }\frac { 1 }{ \sqrt { 81 } } -{ sin }^{ -1 }\left( \frac { 5 }{ \sqrt { 26 } } \right) \right] \)is equal to
- (a)
\(\frac{2}{23}\)
- (b)
\(\frac{4}{31}\)
- (c)
\(\frac{29}{3}\)
- (d)
\(\frac{6}{13}\)
Solve for :\(\{ xcos({ cot }^{ -1 }x)+sin({ cot }^{ -1 }x)\} ^{ 2 }=\frac { 51 }{ 50 } \)
- (a)
\(\frac{1}{\sqrt2}\)
- (b)
\(\frac{1}{5\sqrt2}\)
- (c)
\({2\sqrt2}\)
- (d)
\(5{\sqrt2}\)
If 3 sin-1\(\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) -4{ cos }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) +tan^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) =\frac { \pi }{ 3 } \)the x is equal to
- (a)
\(\frac { 1 }{ \sqrt { 3 } } \)
- (b)
-\(\frac { 1 }{ \sqrt { 3 } } \)
- (c)
\({ \sqrt { 3 } } \)
- (d)
-\(\frac{ \sqrt { 3 } } {4 }\)
Solve the following equation \(sin^{-1}\frac{3x}{5}+sin^{-1}\frac{4x}{5}=sin^{-1}x\)
- (a)
0,1,-1
- (b)
0,-1
- (c)
0,1
- (d)
1,-1
\(tan^{-1}\frac{x-1}{x-2}+tan^{-1}\frac{x+1}{x+2}=\frac{\pi}{4}\)
- (a)
\(\frac{1}{\sqrt{2}}\)
- (b)
\(-\frac{1}{\sqrt{2}}\)
- (c)
\(\pm\frac{1}{\sqrt{2}}\)
- (d)
0
If 3tan-1x+cot-1x=π, then x equals
- (a)
0
- (b)
1
- (c)
-1
- (d)
\(1\over2\)
The value of \(sin^{-1}(cos(\frac{33\pi}{5}))\) is
- (a)
\(\frac{3\pi}{5}\)
- (b)
\(\frac{-7\pi}{5}\)
- (c)
\(\frac{\pi}{10}\)
- (d)
\(\frac{-\pi}{10}\)
Statement I: If x<0, tan-1x+tan-1\((\frac{1}{x})=\frac{\pi}{2}\)
Statement II: tan-1x+cot-1x=\(\frac{\pi}{2},∀x∈π\)
- (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.
cos[tan-1{sin(cot-1x)}] is equal to
- (a)
\(\sqrt{\frac{x^2+2}{x^2+3}}\)
- (b)
\(\sqrt{\frac{x^2+2}{x^2+1}}\)
- (c)
\(\sqrt{\frac{x^2+1}{x^2+2}}\)
- (d)
None of these