Inverse Trigonometric Functions Test

Question 1
1 / -0

Let $$\displaystyle f:A\rightarrow B$$ be a function defined by $$\displaystyle y=f(x)$$ where f is a bijective function, means f is injective (one-one) as well as surjective (onto), then there exist a unique mapping $$\displaystyle g:B\rightarrow A$$ such that $$\displaystyle f(x)=y$$ if and only if $$\displaystyle g(y)=x\forall x \epsilon A,y \epsilon B $$ Then function g is said to be inverse of f and vice versa so we write $$\displaystyle g=f^{-1}:B\rightarrow A[\left \{ f(x),x \right \}:\left \{ x,f(x) \right \}\epsilon f^{-1}] $$when branch of an inverse function is not given (define) then we consider its principal value branch.

If $$\displaystyle -1<x<0$$,then $$\displaystyle \tan^{-1}x $$ equals?

A
$$\displaystyle \pi-\cos^{-1}(\sqrt{1-x^{2}})$$

B
$$\displaystyle \sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right )$$

C
$$\displaystyle -\cos^{-1}\left(\frac {\sqrt{1-x^{2}}}{x}\right)$$

D
$$\displaystyle \ cosec ^{-1}x$$

Question 2

1 / -0

Match the entries of Column - I and Column - II.

	Column - I		Column - II
a	If 4 $$sin^{-1} x + cos^{-1} x = \pi$$, then x equals	1	ab
b	If $$\angle C = 90^{0}$$, then the value of $$tan^{-1}$$ $$\dfrac{a}{b + c}$$ + $$tan^{-1}$$ $$\dfrac{b}{c +a}$$ is	2	$$\pi$$
c	$$tan^{-1}$$ 1 + $$tan^{-1}$$ 2 + $$tan^{-1}$$ 3 is	3	$$\pi$$/4
d	If $$sec^{-1}$$ $$\dfrac{x}{a}$$ - $$sec^{-1}$$ $$\dfrac{x}{b}$$ = $$sec^{-1}$$ b - $$sec^{-1}$$ a, then x equals	4	1/2

a-4, b-3, c-2, d-1

a-1, b-3, c-2, d-4

a-4, b-3, c-1, d-2

a-3, b-4, c-2, d-1

Question 3
1 / -0

The value of $$sin^{-1} x + cos^{-1} x (|x| \geq 1)$$ is

A
1

B
$$\pi$$

C
$$\pi / 2$$

D
$$ - \pi / 2$$

Question 4
1 / -0

Express in terms of an inverse function the angle formed at the intesection of the diagonals of a cube.

A
$$sin^{1} 2/3$$

B
$$cos^{1} 1/3$$

C
$$tan^{1} 1/3$$

D
$$sin^{1} 1/3$$

Question 5
1 / -0

The number of solutions of the equation $$sin^{-1} \displaystyle \left ( \frac{1 + x^2}{2x} \right ) = \frac{\pi}{2} (sec(x - 1))$$ is/are

A
0

B
1

C
2

D
3

Question 6
1 / -0

Let $$a, b, c$$ be a positive real numbers $$\theta = \tan^{-1} \sqrt{\dfrac{a(a + b +c)}{bc}} + \tan^{-1} \sqrt{\dfrac{b(a + b+ c)}{ca}} + \tan^{-1} \sqrt{\dfrac{c(a + b + c)}{ab}}$$, then $$\tan \theta$$

A
$$0$$

B
$$3 \pi$$

C
$$1$$

D
$$4 \pi$$

Question 7
1 / -0

Let $$a={ (\sin ^{ -1 }{ x) } }^{ \sin ^{ -1 }{ x } },\quad b={ \left( \sin ^{ -1 }{ x } \right) }^{ \cos ^{ -1 }{ x } },\quad c={ \left( \cos ^{ -1 }{ x } \right) }^{ \sin ^{ -1 }{ x } },\quad d={ \left( \cos ^{ -1 }{ x } \right) }^{ \cos ^{ -1 }{ x } }$$ and if $$x\in (0,1)$$then

A
$$a>b>d>c$$

B
$$b>a>d>c$$

C
$$d>c>a>b$$

D
none of these

Question 8
1 / -0

$$2{\tan ^{ - 1}}\left[ {\sqrt {\dfrac{{a - b}}{{a + b}}} \tan \dfrac{\theta }{2}} \right] = $$

A
$${\cos ^{ - 1}}\left( {\dfrac{{a\cos \theta + b}}{{a + b\cos \theta }}} \right)$$

B
$${\cos ^{ - 1}}\left( {\dfrac{{a + b\cos \theta }}{{a\cos \theta + b}}} \right)$$

C
$${\cos ^{ - 1}}\left( {\dfrac{{a\cos \theta }}{{a + b\cos \theta }}} \right)$$

D
$${\cos ^{ - 1}}\left( {\dfrac{{b\cos \theta }}{{a\cos \theta + b}}} \right)$$

Question 9
1 / -0

If $$\cos ^{ -1 }{ \cfrac { x }{ 2 } } +\cos ^{ -1 }{ \cfrac { y }{ 3 } } =\theta $$, then $$9{x}^{2}-12xy\cos{\theta}+4{y}^{2}$$ is equal to

A
$$36 \sin ^{ 2 }{ \theta } $$

B
$$36 \cos ^{ 2 }{ \theta } $$

C
$$36 \tan ^{ 2 }{ \theta } $$

D
None of these

Question 10
1 / -0

$$\sin^{-1} \sin{15}+\cos^{-1} \cos{20}+\tan^{-1}\tan{25}=$$ ?

A
$$1.04719754.$$

B
$$11.04719754.$$

C
$$111.04719754.$$

D
$$1111.04719754.$$

Submit Test

Inverse Trigono...

Question 1

$$\displaystyle \pi-\cos^{-1}(\sqrt{1-x^{2}})$$

$$\displaystyle \sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right )$$

$$\displaystyle -\cos^{-1}\left(\frac {\sqrt{1-x^{2}}}{x}\right)$$

$$\displaystyle \ cosec ^{-1}x$$

Question 2

a-4, b-3, c-2, d-1

a-1, b-3, c-2, d-4

a-4, b-3, c-1, d-2

a-3, b-4, c-2, d-1

Question 3

1

$$\pi$$

$$\pi / 2$$

$$ - \pi / 2$$

Question 4

$$sin^{1} 2/3$$

$$cos^{1} 1/3$$

$$tan^{1} 1/3$$

$$sin^{1} 1/3$$

Question 5

0

1

2

3

Question 6

$$0$$

$$3 \pi$$

$$1$$

$$4 \pi$$

Question 7

$$a>b>d>c$$

$$b>a>d>c$$

$$d>c>a>b$$

none of these

Question 8

$${\cos ^{ - 1}}\left( {\dfrac{{a\cos \theta + b}}{{a + b\cos \theta }}} \right)$$

$${\cos ^{ - 1}}\left( {\dfrac{{a + b\cos \theta }}{{a\cos \theta + b}}} \right)$$

$${\cos ^{ - 1}}\left( {\dfrac{{a\cos \theta }}{{a + b\cos \theta }}} \right)$$

$${\cos ^{ - 1}}\left( {\dfrac{{b\cos \theta }}{{a\cos \theta + b}}} \right)$$

Question 9

$$36 \sin ^{ 2 }{ \theta } $$

$$36 \cos ^{ 2 }{ \theta } $$

$$36 \tan ^{ 2 }{ \theta } $$

None of these

Question 10

$$1.04719754.$$

$$11.04719754.$$

$$111.04719754.$$

$$1111.04719754.$$

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