Mathematics - Inverse Trigonometric Functions
Exam Duration: 45 Mins Total Questions : 30
The solution of \({ sin }^{ -1 }x\le { cos }^{ -1 }x\) is
- (a)
\(\left( -1,\frac { 1 }{ \sqrt { 2 } } \right) \)
- (b)
\(\left[ -1,\frac { 1 }{ \sqrt { 2 } } \right] \)
- (c)
\(\left[ 1,\frac { 1 }{ \sqrt { 2 } } \right] \)
- (d)
\(\left( 1,\frac { 1 }{ \sqrt { 2 } } \right) \)
The solution of tan \(\left( { sin }^{ -1 }\frac { 3 }{ 5 } +{ cot }^{ -1 }\frac { 3 }{ 2 } \right) \) is
- (a)
\(\frac { 17 }{ 6 } \)
- (b)
\(\frac { 17 }{ 7 } \)
- (c)
\(\frac { 17 }{ 5 } \)
- (d)
\(\frac { 17 }{ 4 } \)
The greatest and least values of (sin-1 x)2 + (cos-1 x)2 are respectively
- (a)
\(\frac { { 5\pi }^{ 2 } }{ 4 } and\frac { { \pi }^{ 2 } }{ 8 } \)
- (b)
\(\frac { \pi }{ 2 } and\frac { -\pi }{ 2 } \)
- (c)
\(\frac { { \pi }^{ 2 } }{ 4 } and\frac { { -\pi }^{ 2 } }{ 4 } \)
- (d)
\(\frac { { \pi }^{ 2 } }{ 4 } and\quad 0\)
In a \(\triangle\)ABC, if \(A={ sin }^{ -1 }\frac { 1 }{ \sqrt { 5 } } ,B={ cos }^{ -1 }\left( \frac { 3 }{ \sqrt { 10 } } \right) ,\) then
- (a)
\(\triangle ABC\quad \)is a right angled triangle
- (b)
\(C=\frac { 3\pi }{ 4 } ,A+B=\frac { \pi }{ 2 } \)
- (c)
\((1+tanA).(1+tanB)=2\)
- (d)
\(C=\frac { \pi }{ 2 } \)
The value of \({ tan }^{ -1 }\sqrt { \frac { a(a+b+c) }{ bc } } +{ tan }^{ -1 }\sqrt { \frac { b(a+b+c) }{ ca } } +{ tan }^{ -1 }\sqrt { \frac { c(a+b+c) }{ ab } } \) is
- (a)
\(-\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(-\pi \)
- (d)
0
Find the principal values of \(sec^{-1}(\frac{-2}{\sqrt{3}})\)
- (a)
\(\frac{\pi}{6}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{5\pi}{6}\)
- (d)
\(\frac{2\pi}{3}\)
Find the principal values of \(sin^{-1}(\frac{1}{\sqrt{2}})\)
- (a)
\(\frac{\pi}{4}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{\pi}{6}\)
- (d)
\(\frac{\pi}{2}\)
Find the principal values of \(tan^{-1}1+cos^{-1}(\frac{-1}{2})+sin^{-1}(\frac{-1}{2})\)
- (a)
\(\frac{2\pi}{3}\)
- (b)
\(\frac{3\pi}{4}\)
- (c)
\(\frac{\pi}{2}\)
- (d)
6π
Domain of cos-1 [x] is
- (a)
[-1,2]
- (b)
[-1,2)
- (c)
(-1,2]
- (d)
None of these
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)
\(\pi/2\)
3 tan-1a, a∈\((-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})\) is equal to
- (a)
\(tan^{-1}(\frac{3a+a^3}{1+3a^2})\)
- (b)
\(tan^{-1}(\frac{3a-a^3}{1+3a^2})\)
- (c)
\(tan^{-1}(\frac{3a+a^3}{1-3a^2})\)
- (d)
\(tan^{-1}(\frac{3a-a^3}{1-3a^2})\)
If x∈\((-\frac{\pi}{2},\frac{\pi}{2})\), then the value of \(tan^{-1}(\frac{tan x}{4})+tan^{-1}(\frac{3sin2x}{5+3cos2x})\) is
- (a)
\(\frac{x}{2}\)
- (b)
2x
- (c)
3x
- (d)
x
\(tan^{-1}(\frac{x}{y})-tan^{-1}(\frac{x-y}{x+y})\) is equal to (where x>y>0)
- (a)
\(-\frac{\pi}{4}\)
- (b)
\(\frac{\pi}{4}\)
- (c)
\(\frac{3\pi}{4}\)
- (d)
None of these
4tan-1\(1\over5\)-tan-1\(1\over239\) is equal to
- (a)
π
- (b)
\(\frac{\pi}{2}\)
- (c)
\(\frac{\pi}{3}\)
- (d)
\(\frac{\pi}{4}\)
Find the value of sec2 (tan-12)+cosec2(cot-13)
- (a)
12
- (b)
5
- (c)
15
- (d)
9
The equation sin-1x-cos-1x-cos-1\( \left( \frac { \sqrt { 3 } }{ 2 } \right) \) has
- (a)
unique solution
- (b)
no solution
- (c)
infinitely many solutions
- (d)
none of these
Solve for x:sin-12x+sin-13x=\(\frac { \pi }{ 3 } \)
- (a)
\(\sqrt { \frac { 76 }{ 3 } } \)
- (b)
\(\sqrt { \frac { 3 }{ 76 } } \)
- (c)
\(\frac { 3 }{ \sqrt { 76 } } \)
- (d)
\(\frac { \sqrt { 3 } }{ 76 } \)
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
\(sin\{2cos^{-1}(\frac{-3}{5})\}\) is equal to
- (a)
\(\frac{6}{25}\)
- (b)
\(\frac{24}{25}\)
- (c)
\(\frac{4}{5}\)
- (d)
\(-\frac{24}{25}\)
Solve the following equation sin-1(1-x)-2sin-1x=\(\frac{\pi}{2}\)
- (a)
0
- (b)
1/2
- (c)
0,1/2
- (d)
-1/2
tan-12x+tan-13x=\(\frac{\pi}{4}\)
- (a)
1
- (b)
\(1\over6\)
- (c)
\(1\over3\)
- (d)
\(-{1\over6}\)
sin[cot-1{cos(tan-1x)}]=
- (a)
\(\sqrt{\frac{x^2+1}{x^2+2}}\)
- (b)
\(\sqrt{\frac{x^2-1}{x^2-2}}\)
- (c)
\(\sqrt{\frac{x-1}{x-2}}\)
- (d)
\(\sqrt{\frac{x+1}{x+2}}\)
\(2sin^{-1}\sqrt{\frac{1-x}{2}}=\)
- (a)
cos-1x
- (b)
\(2cos^{-1}\sqrt{\frac{1+x}{2}}\)
- (c)
Both (a) and (b)
- (d)
None of these
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
Which of the following is the principal value branch of cosec -1x?
- (a)
\((\frac{-\pi}{2},\frac{\pi}{2})\)
- (b)
[0,π]-\(\{\frac{\pi}{2}\}\)
- (c)
\([\frac{-\pi}{2},\frac{\pi}{2}]\)
- (d)
\([\frac{-\pi}{2},\frac{\pi}{2}]\)-[0]
If cos-1∝+cos-1β+cos-1γ=3π, then ∝(β+γ)+β(⋎+∝)+γ(∝+β) equals
- (a)
0
- (b)
1
- (c)
6
- (d)
12
Match the following
Column I | Column II |
---|---|
(i) \(tan^{-1}(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^2}}{\sqrt{1+x^2}-{\sqrt{1-x^2}}})=\) | (p) \(tan^{-1}\frac{4}{3}-x\) |
(ii) \(cos^{-1}(\frac{3}{5}cosx+\frac{4}{5}sinx),\)where \(x∈(-\frac{3\pi}{4},\frac{\pi}{4})\)is equal to | (q) \(\sqrt{\frac{1+x^2}{2+x^2}}\) |
(iii) \(cot^{-1}(\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}}),\) where \(x∈(0,\frac{\pi}{4})\) is equal to | (r) \(\frac{\pi}{4}+\frac{1}{2}cos^{-1}x^2\) |
(iv) cos(tan-1(sin(cot-1x)))= | (s) \(\frac{x}{2}\) |
- (a)
(i) ⟶ (s), (ii) ⟶ (q), (iii) ⟶ (r), (iv) ⟶ (p)
- (b)
(i) ⟶ (r), (ii) ⟶ (s), (iii) ⟶ (r), (iv) ⟶ (p)
- (c)
(i) ⟶ (r), (ii) ⟶ (p), (iii) ⟶ (s), (iv) ⟶ (q)
- (d)
(i) ⟶ (p), (ii) ⟶ (r), (iii) ⟶ (q), (iv) ⟶ (s)
Statement I: \(sin^{-1}\frac{12}{13}+cos^{-1}\frac{4}{5}+tan^{-1}\frac{63}{16}=\frac{\pi}{2}\)
Statement II: \(tan^{-1}x+tan^{-1}y=\pi+tan^{-1}(\frac{x+y}{1-xy})\) , if 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:
\(sec^{-1}(\frac{3}{2})+sin^{-1}(\frac{2}{3})-2tan^{-1}3-tan^{-1}(\frac{1}{3})\) 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.