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Inverse Trigonometric Functions — previous year questions · Alror

Mathematics

Inverse Trigonometric Functions

Appeared 66 times across 14 years of JEE Main papers — about 4.7 questions a year. Every one of them is below, with a full worked solution.

Questions per year

1’02

1’03

1’05

1’07

1’08

1’13

1’15

2’17

8’19

3’20

17’21

17’22

6’23

6’24

66 questions in the bank

Practise this topic →

01Easy66×since 2002Q4071

Considering only the principal values of inverse functions, the set A = { x

\ge

0: tan^

-

1(2x) + tan^

-

1(3x) =

{\pi \over 4}

}

02Easy66×since 2002Q4064

The domain of the function

f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right)

is :

03Easy66×since 2002Q4105

Let M and m respectively be the maximum and minimum values of the function f(x) = tan^

-

1 (sin x + cos x) in

\left[ {0,{\pi \over 2}} \right]

, then the value of tan(M

-

m) is equal to :

04Easy66×since 2002Q4058

The trigonometric equation

{\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a

has a solution for :

05Hard66×since 2002Q4059

The domain of the function f(x) =

{\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)

is (–

\infty

, -a]

\cup

[a,

\infty

). Then a is equal to :

06Medium66×since 2002Q4060

The number of solutions of the equation

{\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}

, for x

\in

[

-

1, 1], and [x] denotes the greatest integer less than or equal to x, is :

07Easy66×since 2002Q4061

The number of real roots of the equation

{\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4}

is :

08Medium66×since 2002Q4062

If the domain of the function

f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}

is the interval (

\alpha

\beta

], then

\alpha

\beta

is equal to :

09Easy66×since 2002Q4063

The domain of the function

{{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)

is :

10Medium66×since 2002Q4065

The domain of the function

{\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)

is :

11Medium66×since 2002Q4066

The domain of the function

f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}

is :

12Medium66×since 2002Q4067

The domain of the function

f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)

, where [t] is the greatest integer function, is :

13Easy66×since 2002Q4068

The domain of the function

f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)

is :

14Hard66×since 2002Q4069

If the domain of the function

f(x)=\log _{e}\left(4 x^{2}+11 x+6\right)+\sin ^{-1}(4 x+3)+\cos ^{-1}\left(\frac{10 x+6}{3}\right)

(\alpha, \beta]

, then

36|\alpha+\beta|

is equal to :

15Medium66×since 2002Q4070

If the domain of the function

\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)

(\alpha, \beta]

, then

3 \alpha+10 \beta

is equal to:

16Medium66×since 2002Q4072

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy

{\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x

is equal to :

17Easy66×since 2002Q4073

{\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))

is equal to : (The inverse trigonometric functions take the principal values)

18Easy66×since 2002Q4074

{\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over 4}} \right)

is equal to :

19Medium66×since 2002Q4075

If the inverse trigonometric functions take principal values then

{\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right) + {2 \over 5}\sin \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right)} \right)

is equal to :

20Hard66×since 2002Q4076

Let

\alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right)

and

\beta = \cos \left( {{{\sin }^{ - 1}}\left( {{4 \over 5}} \right) + {{\sec }^{ - 1}}\left( {{5 \over 3}} \right)} \right)

where the inverse trigonometric functions take principal values. Then, the equation whose roots are

\alpha

and

\beta

is :

Showing the first 20 of 66. Practise this topic to work through the rest.

Mathematics

Inverse Trigonometric Functions

Appeared 66 times across 14 years of JEE Main papers — about 4.7 questions a year. Every one of them is below, with a full worked solution.

Questions per year

1’02

1’03

1’05

1’07

1’08

1’13

1’15

2’17

8’19

3’20

17’21

17’22

6’23

6’24

66 questions in the bank

Practise this topic →

01Easy66×since 2002Q4071

Considering only the principal values of inverse functions, the set A = { x

\ge

0: tan^

-

1(2x) + tan^

-

1(3x) =

{\pi \over 4}

}

02Easy66×since 2002Q4064

The domain of the function

f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right)

is :

03Easy66×since 2002Q4105

Let M and m respectively be the maximum and minimum values of the function f(x) = tan^

-

1 (sin x + cos x) in

\left[ {0,{\pi \over 2}} \right]

, then the value of tan(M

-

m) is equal to :

04Easy66×since 2002Q4058

The trigonometric equation

{\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a

has a solution for :

05Hard66×since 2002Q4059

The domain of the function f(x) =

{\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)

is (–

\infty

, -a]

\cup

[a,

\infty

). Then a is equal to :

06Medium66×since 2002Q4060

The number of solutions of the equation

{\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}

, for x

\in

[

-

1, 1], and [x] denotes the greatest integer less than or equal to x, is :

07Easy66×since 2002Q4061

The number of real roots of the equation

{\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4}

is :

08Medium66×since 2002Q4062

If the domain of the function

f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}

is the interval (

\alpha

\beta

], then

\alpha

\beta

is equal to :

09Easy66×since 2002Q4063

The domain of the function

{{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)

is :

10Medium66×since 2002Q4065

The domain of the function

{\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)

is :

11Medium66×since 2002Q4066

The domain of the function

f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}

is :

12Medium66×since 2002Q4067

The domain of the function

f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)

, where [t] is the greatest integer function, is :

13Easy66×since 2002Q4068

The domain of the function

f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)

is :

14Hard66×since 2002Q4069

If the domain of the function

f(x)=\log _{e}\left(4 x^{2}+11 x+6\right)+\sin ^{-1}(4 x+3)+\cos ^{-1}\left(\frac{10 x+6}{3}\right)

(\alpha, \beta]

, then

36|\alpha+\beta|

is equal to :

15Medium66×since 2002Q4070

If the domain of the function

\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)

(\alpha, \beta]

, then

3 \alpha+10 \beta

is equal to:

16Medium66×since 2002Q4072

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy

{\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x

is equal to :

17Easy66×since 2002Q4073

{\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))

is equal to : (The inverse trigonometric functions take the principal values)

18Easy66×since 2002Q4074

{\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over 4}} \right)

is equal to :

19Medium66×since 2002Q4075

If the inverse trigonometric functions take principal values then

{\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right) + {2 \over 5}\sin \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right)} \right)

is equal to :

20Hard66×since 2002Q4076

Let

\alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right)

and

\beta = \cos \left( {{{\sin }^{ - 1}}\left( {{4 \over 5}} \right) + {{\sec }^{ - 1}}\left( {{5 \over 3}} \right)} \right)

where the inverse trigonometric functions take principal values. Then, the equation whose roots are

\alpha

and

\beta

is :

Showing the first 20 of 66. Practise this topic to work through the rest.