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12th Maths Monthly Test - 1(Inverse Trigonometric Functions)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1(Inverse Trigonometric Functions)-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The value of tan \(\left( { cos }^{ -1 }\cfrac { 3 }{ 5 } +{ tan }^{ -1 }\cfrac { 1 }{ 4 } \right) \) is ______

\(\cfrac { 19 }{ 8 } \)
\(\cfrac { 8 }{ 19 } \)
\(\cfrac { 19 }{ 12 } \)
\(\cfrac { 3 }{ 4 } \)

If the function f(x)sin^-1(x²-3), then x belongs to

[-1,1]
[\(\sqrt2\),2]
\(\\ \\ \\ \left[ -2,-\sqrt { 2 } \right] \cup \left[ \sqrt { 2 } ,2 \right] \)
\(\left[ -2,-\sqrt { 2 } \right] \cap \left[ \sqrt { 2 } ,2 \right] \)

\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 9 } \right) \)is equal to

\(\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)
\(\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)
\(\frac { 1 }{ 2 } {tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)
\({ tan}^{ -1 }\left( \frac { 1}{ 2 } \right) \)

If sin^-1 \(\frac{x}{5}+ cot^{-1}\frac{5}{4}=\frac{\pi}{2}\), then the value of x is

4
5
2
3

The value of sin^-1 (cos x),0\(\le x\le\pi\) is

\(\pi-x\)
\(x-\frac{\pi}{2}\)
\(\frac{\pi}{2}-x\)
\(\pi-x\)

If cot^-1 2 and cot^-1 3 are two angles of a triangle, then the third angle is

\(\frac{\pi}{4}\)
\(\frac{3\pi}{4}\)
\(\frac{\pi}{6}\)
\(\frac{\pi}{3}\)

The domain of the function defined by f(x)=sin⁻¹\(\sqrt{x-1} \) is

[1,2]
[-1,1]
[0,1]
[-1,0]

If cot⁻¹x=\(\frac{2\pi}{5}\) for some x\(\in\)R, the value of tan^-1 x is

\(-\frac{\pi}{10}\)
\(\frac{\pi}{5}\)
\(\frac{\pi}{10}\)
\(-\frac{\pi}{5}\)

The equation tan^-1x-cot^-1x=tan^-1\(\left( \frac { 1 }{ \sqrt { 3 } } \right) \)has

no solution
unique solution
two solutions
infinite number of solutions

The pricipal value of \({ sin }^{ -1 }\left( \cfrac { -1 }{ 2 } \right) \) is _________

\(\cfrac { \pi }{ 6 } \)
\(\cfrac { -\pi }{ 6 } \)
\(\cfrac { \pi }{ 3 } \)
\(\cfrac { -\pi }{ 3 } \)

\({ sin }^{ -1 }\left[ tan\frac { \pi }{ 4 } \right] -{ sin }^{ -1 }\left[ \sqrt { \frac { 3 }{ x } } \right] =\frac { \pi }{ 6 } \).Then x is a root of the equation
x²−x−6=0

x²−x−12=0

x²+x−12=0

x²+x−6=0
If x=\(\frac{1}{5}\), the valur of cos (cos^-1x+2sin^-1x) is
\(-\sqrt { \frac { 24 }{ 25 } } \)

\(\sqrt { \frac { 24 }{ 25 } } \)

\(\frac{1}{5}\)

-\(\frac{1}{5}\)

If sin^-1x = 2sin^-1α has a solution, then

| α | ≤ \(\frac{1}{\sqrt{2}}\)
| α | ≥ \(\frac{1}{\sqrt{2}}\)
| α | < \(\frac{1}{\sqrt{2}}\)
| α | > \(\frac{1}{\sqrt{2}}\)

\(sin\left\{ 2{ cos }^{ -1 }\left( \cfrac { -3 }{ 5 } \right) \right\} =\)

\(\cfrac { 6 }{ 15 } \)
\(\cfrac { 24 }{ 25 } \)
\(\cfrac { 4 }{ 5 } \)
\(\cfrac { -24 }{ 25 } \)

The domain of cos^-1(x² - 4) Is______

[3,5]
[-1. 1]
\(\left[ -\sqrt { 5 } ,-\sqrt { 3 } \right] \cup \left[ \sqrt { 3 } ,\sqrt { 5 } \right] \)
[0,1]

If |x|\(\le\)1, then 2tan^-1 x-sin^-1 \(\frac{2x}{1+x^2}\) is equal to

tan^-1x
sin^-1x
0
\(\pi\)

Part - B

Generic 2 - Answer All Question

Find the value of tan⁻¹\(\left(\sqrt{3} \right)\) − sec⁻¹( − 2)

Find all values of x such that
-6\(\pi\le x \le 6\pi\) and cos x -0

Find the value of the expression in terms of x, with the help of a reference triangle.
cos (tan^-1(3x-1))

Find the value of
\(cos\left[ \frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1 }{ 8 } \right) \right] \)

Prove that \(\frac{\pi}{2}\le sin^{-1}x+2 cos^{-1} x\le\frac{3\pi}{2}\).

For what value of x , the inequality\(\cfrac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi \) holds?

Find the value of
i) \(sin\left[ \frac { \pi }{ 3 } -{ sin }^{ 2 }\left( -\frac { 1 }{ 2 } \right) \right] \)

Find all the values of x such that
-8π ≤ x ≤ -8π and sin x=-1

Find the value of sin^-1\(\left(sin\frac{5\pi}{9}cos\frac{\pi}{9}+cos\frac{5\pi}{9}sin\frac{\pi}{9} \right)\)

Find the principal value of
sin^-1\(\left(sin\left(-\frac{π}{3}\right) \right)\)

For what value of x does sinx=sin^-1 x?

Find the principal value of cos^-1\((\frac{1}{2})\).

Find the principal value of cot^-1\(\left(\sqrt{3}\right)\)
If tan^-1 x+tan^-1y+tan^-1z=\(\pi\), show that x+y+z+=xyz

Find the value of tan(tan^-1(-0.2021)).
Find the principal value of
\({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

Find all values of x such that 6π ≤ x ≤ 6π and cos x -0

Part - C

Generic 3 - Answer All Question

Find the value of

\({ sin }^{ -1 }(-1)+{ cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ cot }^{ -1 }(2)\)

Find the value of tan^-1(−1 )+cos^-1\((\frac{1}{2})+sin^{-1}(-\frac{1}{2})\)
Solve \({ tan }^{ -1 }\left( \cfrac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \cfrac { 1-{ x }^{ 2 } }{ 2x } \right) =\cfrac { \pi }{ 3 } ,x>0\)

Find

i) tan^-1\( (-\sqrt3)\)

ii) tan^-1\((tan\frac{3\pi}{5})\)

iii) tan(tan^-1 −(2019))

Find the domain of
g(x)=sin^-1x+cos^-1x

If \(sin\left( { sin }^{ -1 }\cfrac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value of x.

Find the principal value of tan^-1\(\left(\sqrt{3}\right)\)

Find the domain of the following functions

\(\frac{1}{2}tan^{-1}(1-x^2)-\frac{\pi}{4}\)

Solve tan^-1 2x+tan^-1 3x=\(\frac{\pi}{4}\), if 6x²<1
Evaluate \(sin\left[ { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) +{ sec }^{ -1 }\left( \frac { 5 }{ 4 } \right) \right]\)

Evaluate \(sin\left[ { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) +{ sec }^{ -1 }\left( \frac { 5 }{ 4 } \right) \right] \)

Find the domain of
f(x)=sin^-1\(\left(\frac{| x | − 2}{3} \right)\)+cos^-1\(\left(\frac{1-| x | }{4} \right)\)

Part - D

Generic 5 - Answer All Question

Provethat \({ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\ \)

Solve \(cos\left( sin^{ -1 }\left( \frac { x }{ \sqrt { 1+{ x }^{ 2 } } } \right) \right) =sin\left\{ cot^{ -1 }\left( \frac { 3 }{ 4 } \right) \right\} \)

Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)
Simplify \({ sin }^{ -1 }\left( \cfrac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac{-\pi}{4}<x<\frac{\pi}{4}\)

Find the principal value of cos^-1\(\left( \frac { \sqrt { 3 } }{ 3 } \right)\)

If a₁, a₂, a₃ , ... an is an arithmetic progression with common difference d, prove that tan x = \({-b \pm \sqrt{b^2-4ac} \over 2a}\quad \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } }\)

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