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NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

Edited By Ramraj Saini | Updated on Sep 13, 2023 08:44 PM IST | #CBSE Class 12th

Inverse Trigonometric Functions Class 12 Questions And Answers

Inverse Trigonometric Functions class 12 Maths solutions are provided here. These NCERT solutions are created by expert team at Careers360 considering the demand of class 12 students, CBSE latest syllabus and pattern 2023-23, and comprehensively covering step by step all the concepts. Therefore these are great resource for students to command the concepts and in-depth understanding of the concepts. These inverse trigonometric functions class 12 NCERT solutions is one of the most important chapters in the NCERT Class 12 Maths book. These class 12 Maths chapter 2 notes will be beneficial for board exams as well as for other entrance exams such as JEE.

This Story also Contains
  1. Inverse Trigonometric Functions Class 12 Questions And Answers
  2. Inverse Trigonometric Functions Class 12 Questions And Answers PDF Free Download
  3. Inverse Trigonometric Functions Class 12 NCERT Solutions - Important Formulae
  4. Class 12 Inverse Trigonometric Functions NCERT Solutions (Intext Questions and Exercise)
  5. Class 12 inverse trigonometric functions NCERT solutions - Topics
  6. Class 12 Maths Question Answer - Chapter Wise
  7. Class 12 Maths Question Answer - Subject Wise
  8. NCERT Books and NCERT Syllabus
NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions
NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

Concepts of maths chapter 2 class 12 are useful in other concepts like differential calculus and Integral calculus therefore class 12 inverse trigonometry becomes very important for students as it is base. Inverse trigonometric functions of NCERT class 12 chapter 2 also include exercises that are important as they help in practicing the problems and in depth understanding of class 12 chapter 2 inverse trigonometric functions of ncert.

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Inverse Trigonometric Functions Class 12 Questions And Answers PDF Free Download

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Inverse Trigonometric Functions Class 12 NCERT Solutions - Important Formulae

>> Basic Concepts: Inverse trigonometric functions map real numbers back to angles.

The inverse of the sine function: sin-1(x) or arcsin(x) is defined on [-1, 1].

Properties of Inverse Trigonometric Functions:

Function

Domain

Range

y = sin-1(x)

[-1, 1]

[-π/2, π/2]

y = cos-1(x)

[-1, 1]

[0, π]

y = cosec-1(x)

R - (-1, 1)

[-π/2, π/2] - {0}

y = sec-1(x)

R - (-1, 1)

[0, π] - {π/2, π/2}

y = tan-1(x)

R

(-π/2, π/2)

y = cot-1(x)

R

(0, π)

Self-Adjusting Trigonometric Property:

sin(sin-1(x)) = x

sin-1(sin(x)) = x

cos(cos-1(x)) = x

cos-1(cos(x)) = x

tan(tan-1(x)) = x

tan-1(tan(x)) = x

sec(sec-1(x)) = x

sec-1(sec(x)) = x

cosec-1(cosec(x)) = x

cosec(cosec-1(x)) = x

cot-1(cot(x)) = x

cot(cot-1(x)) = x

Reciprocal Relations:

sin-1(1/x) = cosec-1(x), x ≥ 1 or x ≤ -1

cos-1(1/x) = sec-1(x), x ≥ 1 or x ≤ -1

tan-1(1/x) = cot-1(x), x > 0

Even and Odd Functions:

sin-1(-x) = -sin-1(x), x ∈ [-1, 1]

tan-1(-x) = -tan-1(x), x ∈ R

cosec-1(-x) = -cosec-1(x), |x| ≥ 1

cos-1(-x) = π - cos-1(x), x ∈ [-1, 1]

sec-1(-x) = π - sec-1(x), |x| ≥ 1

cot-1(-x) = π - cot-1(x), x ∈ R

Complementary Relations:

sin-1(x) + cos-1(x) = π/2

tan-1(x) + cot-1(x) = π/2

cosec-1(x) + sec-1(x) = π/2

Sum and Difference Formulas:

tan-1(x) + tan-1(y) = tan-1((x+y)/(1-xy))

tan-1(x) - tan-1(y) = tan-1((x-y)/(1+xy))

sin-1(x) + sin-1(y) = sin-1[x√(1-y2)+y√(1-x2)]

sin-1(x) - sin-1(y) = sin-1[x√(1-y2)-y√(1-x2)]

cos-1(x) + cos-1(y) = cos-1[xy-√(1-x2)√(1-y2)]

cos-1(x) - cos-1(y) = cos-1[xy+√(1-x2)√(1-y2)]

cot-1(x) + cot-1(y) = cot-1((xy-1)/(x+y))

cot-1(x) - cot-1(y) = cot-1((xy+1)/(y-x))

Double Angle Formula:

2 tan-1(x) = sin-1(2x/(1+x2))

2 tan-1(x) = cos-1((1-x2)/(1+x2))

2 tan-1(x) = tan-1(2x/(1-x2))

2 sin-1(x) = sin-1(2x√(1+x2))

2 cos-1(x) = sin-1(2x√(1-x2))

Conversion Properties:

sin-1(x) = cos-1(√(1-x2)) = tan-1(x/√(1-x2)) = cot-1(√(1-x2)/x)

cos-1(x) = sin-1(√(1-x2)) = tan-1(√(1-x2)/x) = cot-1(x/√(1-x2))

tan-1(x) = sin-1(x/√(1-x2)) = cos-1(x/√(1+x2)) = sec-1(√(1+x2)) = cosec-1(√(1+x2)/x)

Free download Inverse Trigonometric Functions Class 12 NCERT solutions for CBSE Exam.

Class 12 Inverse Trigonometric Functions NCERT Solutions (Intext Questions and Exercise)

Inverse Trigonometric Functions Class 12 Solutions: Exercise 2.1

Question:1 Find the principal values of the following : \sin^{-1}\left ( \frac{-1}{2} \right )

Answer:

Let x = \sin^{-1}\left ( \frac{-1}{2} \right )

\implies \sin x = \frac{-1}{2}= -\sin(\frac{\pi}{6}) = \sin(-\frac{\pi}{6})
We know, principle value range of sin^{-1} is [-\frac{\pi}{2}, \frac{\pi}{2}]

\therefore The principal value of \sin^{-1}\left ( \frac{-1}{2} \right ) is -\frac{\pi}{6},

Question:2 Find the principal values of the following: \cos^{-1}\left(\frac{\sqrt3}{2} \right )

Answer:

So, let us assume that \cos^{-1}\left(\frac{\sqrt3}{2} \right ) = x then,

Taking inverse both sides we get;

cos\ x = (\frac{\sqrt{3}}{2}) , or cos (\frac{\pi}{6}) = (\frac{\sqrt{3}}{2})

and as we know that the principal values of cos^{-1} is from [0, \pi ],

Hence cos\ x = (\frac{\sqrt{3}}{2}) when x = \frac{\pi}{6} .

Therefore, the principal value for \cos^{-1}\left(\frac{\sqrt3}{2} \right ) is \frac{\pi}{6} .

Question:3 Find the principal values of the following: \textup{cosec}^{-1}(2)

Answer:

Let us assume that \textup{cosec}^{-1}(2) = x , then we have;

Cosec\ x = 2 , or

Cosec( \frac{\pi}{6}) = 2 .

And we know the range of principal values is [\frac{-\pi}{2},\frac{\pi}{2}] - \left \{ 0 \right \}.

Therefore the principal value of \textup{cosec}^{-1}(2) is \frac{\pi}{6} .

Question:4 Find the principal values of the following: \tan^{-1}(-\sqrt3)

Answer:

Let us assume that \tan^{-1}(-\sqrt3) = x , then we have;

\tan x = (-\sqrt 3) or

-\tan (\frac{\pi}{3}) = \tan \left ( \frac{-\pi}{3} \right ).

and as we know that the principal value of \tan^{-1} is \left ( \frac{-\pi}{2}, \frac{\pi}{2} \right ) .

Hence the only principal value of \tan^{-1}(-\sqrt3) when x = \frac{-\pi}{3} .

Question:5 Find the principal values of the following: \cos^{-1}\left(-\frac{1}{2} \right )

Answer:

Let us assume that \cos^{-1}\left(-\frac{1}{2} \right ) =y then,

Easily we have; \cos y = \left ( \frac{-1}{2} \right ) or we can write it as:

-\cos \left ( \frac{\pi}{3} \right ) = \cos \left ( \pi - \frac{\pi}{3} \right ) = \cos \left ( \frac{2\pi}{3} \right ).

as we know that the range of the principal values of \cos^{-1} is \left [ 0,\pi \right ] .

Hence \frac{2\pi}{3} lies in the range it is a principal solution.

Question:6 Find the principal values of the following : \tan^{-1}(-1)

Answer:

Given \tan^{-1}(-1) so we can assume it to be equal to 'z';

\tan^{-1}(-1) =z ,

\tan z = -1

or

-\tan (\frac{\pi}{4}) = \tan(\frac{-\pi}{4})= -1

And as we know the range of principal values of \tan^{-1} from \left ( \frac{-\pi}{2}, \frac{\pi}{2} \right ) .

As only one value z = -\frac{\pi}{4} lies hence we have only one principal value that is -\frac{\pi}{4} .

Question:7 Find the principal values of the following : \sec^{-1}\left (\frac{2}{\sqrt3}\right)

Answer:

Let us assume that \sec^{-1}\left (\frac{2}{\sqrt3}\right) = z then,

we can also write it as; \sec z = \left (\frac{2}{\sqrt3}\right) .

Or \sec (\frac{\pi}{6}) = \left (\frac{2}{\sqrt3}\right) and the principal values lies between \left [ 0, \pi \right ] - \left \{ \frac{\pi}{2} \right \} .

Hence we get only one principal value of \sec^{-1}\left (\frac{2}{\sqrt3}\right) i.e., \frac{\pi}{6} .

Question:8 Find the principal values of the following: \cot^{-1}(\sqrt3)

Answer:

Let us assume that \cot^{-1}(\sqrt3) = x , then we can write in other way,

\cot x = (\sqrt3) or

\cot (\frac{\pi}{6}) = (\sqrt3) .

Hence when x=\frac{\pi}{6} we have \cot (\frac{\pi}{6}) = (\sqrt3) .

and the range of principal values of \cot^{-1} lies in \left ( 0, \pi \right ) .

Then the principal value of \cot^{-1}(\sqrt3) is \frac{\pi}{6}

Question:9 Find the principal values of the following: \cos^{-1}\left(-\frac{1}{\sqrt2} \right )

Answer:

Let us assume \cos^{-1}\left(-\frac{1}{\sqrt2} \right ) = x ;

Then we have \cos x = \left ( \frac{-1}{\sqrt 2} \right )

or

-\cos (\frac{\pi}{4}) = \left ( \frac{-1}{\sqrt 2} \right ) ,

\cos (\pi - \frac{\pi}{4}) = \cos (\frac{3\pi}{4}) .

And we know the range of principal values of \cos^{-1} is [0,\pi] .

So, the only principal value which satisfies \cos^{-1}\left(-\frac{1}{\sqrt2} \right ) = x is \frac{3\pi}{4} .

Question:10 Find the principal values of the following: \textup{cosec}^{-1}(-\sqrt2)

Answer:

Let us assume the value of \textup{cosec}^{-1}(-\sqrt2) = y , then

we have cosec\ y = (-\sqrt 2) or

-cosec\ (\frac{\pi}{4}) = (-\sqrt 2) = cosec\ (\frac{-\pi}{4}) .

and the range of the principal values of \textup{cosec}^{-1} lies between \left [ \frac{-\pi}{2},\frac{\pi}{2} \right ] - \left \{ 0 \right \} .

hence the principal value of \textup{cosec}^{-1}(-\sqrt2) is \frac{-\pi}{4} .

Question:11 Find the values of the following: \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2} \right ) + \sin^{-1}\left(-\frac{1}{2} \right )

Answer:

To find the values first we declare each term to some constant ;

tan^{-1}(1) = x , So we have \tan x = 1 ;

or \tan (\frac{\pi}{4}) = 1

Therefore, x = \frac{\pi}{4}

cos^{-1}(\frac{-1}{2}) = y

So, we have

\cos y = \left ( \frac{-1}{2} \right ) = -\cos \left ( \frac{\pi}{3} \right ) = \cos(\pi - \frac{\pi}{3}) = \cos \left ( \frac{2\pi}{3} \right ) .

Therefore y = \frac{2\pi}{3} ,

\sin^{-1}(\frac{-1}{2}) = z ,

So we have;

\sin z = \frac{-1}{2} or -\sin (\frac{\pi}{6}) =\sin (\frac{-\pi}{6}) = \frac{-1}{2}

Therefore z = -\frac{\pi}{6}

Hence we can calculate the sum:

= \frac{\pi}{4}+\frac{2\pi}{3}-\frac{\pi}{6}

=\frac{3\pi + 8\pi -2\pi}{12} = \frac{9\pi}{12}=\frac{3\pi}{4} .

Question:12 Find the values of the following: \cos^{-1}\left(\frac{1}{2} \right ) + 2\sin^{-1}\left(\frac{1}{2} \right )

Answer:

Here we have \cos^{-1}\left(\frac{1}{2} \right ) + 2\sin^{-1}\left(\frac{1}{2} \right )

let us assume that the value of

\cos^{-1}\left ( \frac{1}{2} \right ) = x, \:and\:\sin^{-1}\left(\frac{1}{2} \right ) = y ;

then we have to find out the value of x +2y.

Calculation of x :

\Rightarrow \cos^{-1}\left ( \frac{1}{2} \right ) = x

\Rightarrow \cos x = \frac{1}{2}

\Rightarrow \cos \frac{\pi}{3} = \frac{1}{2} ,

Hence x = \frac{\pi}{3} .

Calculation of y :

\Rightarrow \sin^{-1}\left(\frac{1}{2} \right ) = y

\Rightarrow \sin y = \frac{1}{2}

\Rightarrow \sin \frac{\pi}{6} = \frac{1}{2} .

Hence y = \frac{\pi}{6} .

The required sum will be = \frac{\pi}{3}+2(\frac{\pi}{6}) = \frac{2\pi}{3} .

Question:13 If \sin^{-1}x = y then

(A) 0\leq y \leq \pi

(B) -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}

(C) 0 < y < \pi

(D) -\frac{\pi}{2} < y < \frac{\pi}{2}

Answer:

Given if \sin^{-1}x = y then,

As we know that the \sin^{-1} can take values between \left [ \frac{-\pi}{2}, \frac{\pi}{2} \right ].

Therefore, -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} .

Hence answer choice (B) is correct.

Question:14 \tan^{-1}(\sqrt3)-\sec^{-1}(-2) is equal to

(A) \pi

(B) -\frac{\pi}{3}

(C) \frac{\pi}{3}

(D) \frac{2\pi}{3}

Answer:

Let us assume the values of \tan^{-1}(\sqrt3) be 'x' and \sec^{-1}(-2) be 'y'.

Then we have;

\tan^{-1}(\sqrt3) = x or \tan x = \sqrt 3 or \tan \frac{\pi}{3} = \sqrt 3 or

x = \frac{\pi}{3} .

and \sec^{-1}(-2) = y or \sec y = -2

or -\sec (\frac{\pi}{3}) =\sec ({\pi - \frac{\pi}{3}}) = \sec{\frac{2\pi}{3}}

y = \frac{2\pi}{3}

also, the ranges of the principal values of \tan^{-1} and \sec^{-1} are (\frac{-\pi}{2},\frac{\pi}{2}) . and

[0,\pi] - \left \{ \frac{\pi}{2} \right \} respectively.

\therefore we have then;

\tan^{-1}(\sqrt3)-\sec^{-1}(-2)

= \frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3}

Inverse Trigonometric Functions Class 12 Solutions: Exercise 2.2

Question:1 Prove the following: 3\sin^{-1}x = \sin^{-1}(3x - 4x^3),\;\;x\in\left[-\frac{1}{2},\frac{1}{2} \right ]

Answer:

Given to prove: 3\sin^{-1}x = \sin^{-1}(3x - 4x^3)

where, x\:\epsilon \left[-\frac{1}{2},\frac{1}{2} \right ] .

Take \theta= \sin ^{-1}x or x = \sin \theta

Take R.H.S value

\sin^{-1}(3x - 4x^3)

= \sin^{-1}(3\sin \theta - 4\sin^3 \theta)

= \sin^{-1}(\sin 3\theta)

= 3\theta

= 3\sin^{-1}x = L.H.S

Question:2 Prove the following: 3\cos^{-1} x = \cos^{-1}(4x^3 - 3x), \;\;x\in\left[\frac{1}{2},1 \right ]

Answer:

Given to prove 3\cos^{-1} x = \cos^{-1}(4x^3 - 3x), \;\;x\in\left[\frac{1}{2},1 \right ] .

Take \cos^{-1}x = \theta or \cos \theta = x ;

Then we have;

R.H.S.

\cos^{-1}(4x^3 - 3x)

= \cos^{-1}(4\cos^3 \theta - 3\cos\theta) \left [ \because 4\cos^3 \theta - 3\cos\theta = \cos3 \theta \right ]

= \cos^{-1}(\cos3\theta)

= 3\theta

= 3\cos^{-1}x = L.H.S

Hence Proved.

Question:3 Prove the following: \tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24} = \tan^{-1}\frac{1}{2}

Answer:

Given to prove \tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24} = \tan^{-1}\frac{1}{2}

We have L.H.S

\tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24}

=\tan^{-1}\frac{\frac{2}{11} + \frac{7}{24} }{1 - \left ( \frac{2}{11}\times\frac{7}{24} \right ) } \left [ \because \tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x +y}{1 - xy} \right ]

=\tan^{-1}\frac{11\times 24 }{\frac{11\times24 -14}{11\times 24} }

=\tan^{-1}\frac{48 + 77}{264 -14}

=\tan^{-1}\left ( \frac{125}{250}\right ) = \tan^{-1}\left ( \frac{1}{2} \right )

= R.H.S

Hence proved.

Question:4 Prove the following: 2\tan^{-1} \frac{1}{2} + \tan^{-1}\frac{1}{7} = \tan^{-1}\frac{31}{17}

Answer:

Given to prove 2\tan^{-1} \frac{1}{2} + \tan^{-1}\frac{1}{7} = \tan^{-1}\frac{31}{17}

Then taking L.H.S.

We have 2\tan^{-1} \frac{1}{2} + \tan^{-1}\frac{1}{7}

=\tan^{-1} \frac{2.\frac{1}{2}}{1 - \left ( \frac{1}{2} \right )^2} + \tan^{-1} \frac{1}{7} \because 2\tan^{-1} x = \tan^{-1} \frac{2x}{1- x^2}

=\tan^{-1} \frac{1}{(\frac{3}{4})} + \tan^{-1} \frac{1}{7}

=\tan^{-1} \frac{4}{3} + \tan^{-1} \frac{1}{7}

=\tan^{-1} \frac{\frac{4}{3} + \frac{1}{7}}{1 - \frac{4}{3}.\frac{1}{7}} \left [ \because \tan^{-1}x + \tan^{-1} y = \tan^{-1} \frac{x +y}{1- xy}\right ]

=\tan^{-1} \left ( \frac{\frac{28+3}{21}}{\frac{21-4}{21}} \right )

=\tan^{-1} \frac{31}{17}

= R.H.S.

Hence proved.

Question:5 Write the following functions in the simplest form: \tan^{-1}\frac{\sqrt{1 + x^2}- 1}{x},\;\;x\neq 0

Answer:

We have \tan^{-1}\frac{\sqrt{1 + x^2}- 1}{x}

Take

\therefore \tan^{-1} \frac {\sqrt{1+x^2} - 1}{x} = \tan^{-1}\frac{\sqrt{1+\tan^2 \Theta - 1}}{\tan \Theta}

=\tan^{-1}(\frac{sec \Theta-1}{tan \Theta}) = \tan^{-1}\left ( \frac{1-cos \Theta}{sin \Theta} \right )

=\tan^{-1}\left ( \frac {2sin^2\left ( \frac{\Theta}{2} \right )}{2sin\frac{\Theta}{2}cos\frac{\Theta}{2}} \right )

=\tan^{-1}\left ( \tan\frac{\Theta}{2} \right ) = \frac{\Theta}{2} =\frac{1}{2}\tan^{-1}x

=\frac{1}{2}\tan^{-1}x is the simplified form.

Question:6 Write the following functions in the simplest form : \tan^{-1} \frac{1}{\sqrt{x^2 -1}},\;\; |x| > 1

Answer:

Given that \tan^{-1} \frac{1}{\sqrt{x^2 -1}},\;\; |x| > 1

Take x =cosec\ \Theta or \Theta = cosec ^{-1}x

\therefore tan^{-1}\frac{1}{\sqrt{x^2-1}}

=tan^{-1} \frac{1}{\sqrt{cosec^2 \Theta -1}}

=tan^{-1}(\frac{1}{\cot \Theta})

=tan^{-1}(\tan \Theta) = \Theta

= cosec^{-1}x

=\frac{\pi}{2}- \sec^{-1}x [\because cosec^{-1}x + \sec^{-1}x = \frac{\pi}{2}]

Question:7 Write the following functions in the simplest form: \tan^{-1}\left(\sqrt{\frac{1-\cos x}{1 + \cos x}} \right ),\;\; 0< x < \pi

Answer:

Given that \tan^{-1}\left(\sqrt{\frac{1-\cos x}{1 + \cos x}} \right ),\;\; 0< x < \pi

We have in inside the root the term : \frac{1-\cos x}{1 + \cos x}

Put 1-\cos x = 2\sin^2\frac{x}{2} and 1+\cos x = 2\cos^2\frac{x}{2} ,

Then we have,

=\tan^{-1}\left(\sqrt{\frac{2\sin^2\frac{x}{2}}{2\cos^2\frac{x}{2}}} \right )

=\tan^{-1}\left( \frac{\sin \frac{x}{2}}{\cos\frac{x}{2}} \right )

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

Hence the simplest form is \frac{x}{2}

Question:8 Write the following functions in the simplest form: \tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right ),\;\; \frac{-\pi}{4} < x < \frac{3\pi}{4}

Answer:

Given \tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right ) where x\:\epsilon\:( \frac{-\pi}{4} < x < \frac{3\pi}{4})

So,

=\tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right )

Taking \cos x common from numerator and denominator.

We get:

=\tan^{-1}\left(\frac{1 -(\frac{\sin x}{\cos x}) }{1+(\frac{\sin x}{\cos x}) } \right )

=\tan^{-1}\left(\frac{1 - \tan x }{1+\tan x } \right )

= \tan^{-1}(1) - \tan^{-1}(\tan x) as, \left [ \because \tan^{-1}x - \tan^{-1}y = \frac{x - y}{1 + xy} \right ]

= \frac{\pi}{4} - x is the simplest form.

Question:9 Write the following functions in the simplest form: \tan^{-1} \frac{x}{\sqrt{a^2 - x^2}}, \;\; |x| < a

Answer:

Given that \tan^{-1} \frac{x}{\sqrt{a^2 - x^2}}, \;\; |x| < a

Take x = a\sin \theta or

\theta = \sin^{-1}\left ( \frac{x}{a} \right ) and putting it in the equation above;

\tan^{-1} \frac{a\sin \theta}{\sqrt{a^2 - (a\sin \theta)^2}}

=\tan^{-1} \frac{a\sin \theta}{a\sqrt{1 - \sin^2 \theta}}

=\tan^{-1} \left ( \frac{\sin \theta}{\sqrt{\cos^2 \theta}} \right ) = \tan^{-1} \left ( \frac{\sin \theta}{{\cos \theta}} \right )

=\tan^{-1}\left ( \tan \theta \right )

=\theta = \sin^{-1}\left ( \frac{x}{a} \right ) is the simplest form.

Question:10 Write the following functions in the simplest form: \tan^{-1}\left(\frac{3a^2x -x^3}{a^3 - 3ax^2} \right ),\;\;a>0\;\;;\;\;\frac{-a}{\sqrt3} < x < \frac{a}{\sqrt3}

Answer:

Given \tan^{-1}\left(\frac{3a^2x -x^3}{a^3 - 3ax^2} \right )

Here we can take x = a\tan \theta \Rightarrow \frac{x}{a} = \tan \theta

So, \theta = \tan^{-1}\left ( \frac{x}{a} \right )

\tan^{-1}\left(\frac{3a^2x -x^3}{a^3 - 3ax^2} \right ) will become;

=\tan^{-1}\left(\frac{3a^2a\tan \theta -(a\tan \theta)^3}{a^3 - 3a(a\tan \theta)^2} \right ) = \tan^{-1}\left(\frac{3a^3\tan \theta -a^3\tan ^3 \theta}{a^3 - 3a^3\tan ^2 \theta} \right )

and as \left [ \because \left(\frac{3\tan \theta -\tan ^3 \theta}{ 1- 3\tan ^2 \theta} \right) =\tan 3\theta \right ] ;

=3 \theta

=3 \tan^{-1}(\frac{x}{a})

hence the simplest form is 3 \tan^{-1}(\frac{x}{a}) .

Question:11 Find the values of each of the following: \tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2} \right ) \right ]

Answer:

Given equation:

\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2} \right ) \right ]

So, solving the inner bracket first, we take the value of \sin x^{-1} \frac{1}{2} = x.

Then we have,

\sin x = \frac{1}{2} = \sin \left ( \frac{\pi}{6} \right )

Therefore, we can write \sin^{-1} \frac{1}{2} = \frac{\pi}{6} .

\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2} \right ) \right ] = \tan^{-1}\left[2\cos\left(2\times\frac{\pi}{6} \right ) \right ]

= \tan^{-1}\left[2\cos\left(\frac{\pi}{3} \right ) \right ] = \tan^{-1}\left[2\times\left(\frac{1}{2} \right ) \right ] = \tan^{-1}1 = \frac{\pi}{4} .

Question:12 Find the values of each of the following: \cot(\tan^{-1}a + \cot^{-1}a)

Answer:

We have to find the value of \cot(\tan^{-1}a + \cot^{-1}a)

As we know \left [\because \tan^{-1}x + \cot^{-1} x = \frac{\pi}{2} \right ] so,

Equation reduces to \cot(\frac{\pi}{2}) = 0 .

Question:13 Find the values of each of the following: \tan \frac{1}{2}\left[\sin^{-1}\frac{2x}{1+x^2} + cos^{-1}\frac{1-y^2}{1+y^2} \right ],\;\;|x|<1,\;y>0 and xy<1

Answer:

Taking the value x = \tan \Theta or \tan^{-1}x = \Theta and y = \tan \Theta or \tan^{-1} y = \Theta then we have,

= \tan \frac{1}{2}\left[\sin^{-1}\frac{2\tan \Theta}{1+(\tan \Theta)^2} + cos^{-1}\frac{1-\tan^2 \Theta}{1+(\tan \Theta)^2} \right ] ,

= \tan \frac{1}{2}\left[\sin^{-1}(\sin2\Theta) + cos^{-1} (\cos 2\Theta) \right ]

\because \left[\cos^{-1}(\frac{1-\tan^2 \Theta}{1+ \tan^2\Theta}) = \cos^{-1} (\cos2 \Theta) , \right ]

\because \left[\sin^{-1}(\frac{2\tan\Theta}{1+ \tan^2\Theta}) = \sin^{-1} (\sin2 \Theta) \right ]

Then,

=\tan \frac{1}{2}\left[2\tan^{-1}x + 2\tan^{-1}y \right ] \because \left[\tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1- xy} \right]

=\tan \left [ \tan^{-1}\frac{x+y}{1-xy} \right ]

=\frac{x+y}{1-xy} Ans.

Question:14 If \sin\left(\sin^{-1}\frac{1}{5} + \cos ^{-1}x \right ) =1 , then find the value of x .

Answer:

As we know the identity;

sin^{-1} x + cos^{-1} x = \frac {\pi}{2},\ x\ \epsilon\ [-1,1] . it will just hit you by practice to apply this.

So, \sin\left(\sin^{-1}\frac{1}{5} + \cos ^{-1}x \right ) =1 or \sin^{-1}\frac{1}{5} + \cos ^{-1}x =\sin^{-1}(1) ,

we can then write \cos^{-1}x = \frac{\pi}{2} - \sin^{-1}x ,

putting in above equation we get;

\sin^{-1}\frac{1}{5} + \frac{\pi}{2} - \sin^{-1}x =\frac{\pi}{2} \because \left [ \sin^{-1}(1)=\frac{\pi}{2} \right ]

= \sin^{-1}x = \sin^{-1} \frac{1}{5}

Ans. x = \frac{1}{5}

Question:15 If \tan^{-1}\frac{x-1}{x-2} + \tan^{-1}\frac{x+1}{x+2} =\frac{\pi}{4} , then find the value of x .

Answer:

Using the identity \tan^{-1}x+\tan^{-1} y = \tan^{-1}{\frac{x+y}{1-xy}} ,

We can find the value of x;

So, \tan^{-1}\frac{x-1}{x-2} + \tan^{-1}\frac{x+1}{x+2} =\frac{\pi}{4}

on applying,

= \tan^{-1}{\frac{\frac{x-1}{x-2}+\frac{x+1}{x+2}}{1- \left ( \frac{x-1}{x-2} \right )\left ( \frac{x+1}{x+2} \right )}}

=\tan^{-1}\frac{\frac{(x-1)(x+2)+(x-2)(x+1)}{x^2-4}}{1-\frac{x^2-1}{x^2-4}} = \tan^{-1} \left [ \frac{2x^2-4}{-3} \right ] = \frac{\pi}{4}

=\frac{2x^2-4}{-3} = \tan (\frac{\pi}{4})=1

= 2x^2=1 or x = \pm \frac{1}{\sqrt{2}} ,

Hence, the possible values of x are \pm \frac{1}{\sqrt{2}} .

Question:16 Find the values of each of the expressions in Exercises 16 to 18. \sin^{-1}\left (\sin\frac{2\pi}{3} \right )

Answer:

Given \sin^{-1}\left (\sin\frac{2\pi}{3} \right ) ;

We know that \sin^{-1}(\sin x) = x

If the value of x belongs to \left [ \frac{-\pi}{2},\frac{\pi}{2} \right ] then we get the principal values of \sin^{-1}x .

Here, \frac{2\pi}{3} \notin \left [ \frac{-\pi}{2},\frac{\pi}{2} \right ]

We can write \sin^{-1}\left (\sin\frac{2\pi}{3} \right ) is as:

= \sin^{-1}\left [ \sin\left ( \pi-\frac{2\pi}{3} \right ) \right ]

= \sin^{-1}\left [ \sin \frac{\pi}{3} \right ] where \frac{\pi}{3} \epsilon \left [ \frac{-\pi}{2}, \frac{\pi}{2} \right ]

\therefore \sin^{-1}\left (\sin\frac{2\pi}{3} \right )=\sin^{-1}\left [ \sin \frac{\pi}{3} \right ]=\frac{\pi}{3}

Question:17 Find the values of each of the expressions in Exercises 16 to 18. \tan^{-1}\left (\tan\frac{3\pi}{4} \right )

Answer:

As we know \tan^{-1}\left ( \tan x \right ) =x

If x \epsilon \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right ). which is the principal value range of \tan^{-1}x .

So, as in \tan^{-1}\left (\tan\frac{3\pi}{4} \right ) ;

\frac{3\pi}{4}\notin \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )

Hence we can write \tan^{-1}\left (\tan\frac{3\pi}{4} \right ) as :

\tan^{-1}\left (\tan\frac{3\pi}{4} \right ) = \tan^{-1}\left (\tan\frac{3\pi}{4} \right) = \tan^{-1}\left [ \tan(\pi - \frac{\pi}{4}) \right ] = \tan^{-1}\left [ \tan (\frac{-\pi}{4}) \right ]

Where -\frac{\pi}{4} \epsilon \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )

and \therefore \tan^{-1}\left (\tan\frac{3\pi}{4} \right )=\tan^{-1}\left [ \tan (\frac{-\pi}{4}) \right ]=-\frac{\pi}{4}

Question:18 Find the values of each of the expressions in Exercises 16 to 18. \tan\left(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2} \right )

Answer:

Given that \tan\left(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2} \right )

we can take \sin^{-1}\frac{3}{5} = x ,

then \sin x = \frac{3}{5}

or \cos x = \sqrt{1-\sin^{2}x}= \frac{4}{5}

\Rightarrow \tan x = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}

\Rightarrow \tan^{-1}\frac{3}{4}= x

We have similarly;

\cot^{-1} \frac{3}{2} = \tan^{-1} \frac{2}{3}

Therefore we can write \tan\left(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2} \right )

=\tan\left(\tan^{-1}\frac{3}{4}+\tan^{-1}\frac{2}{3} \right )

=\tan\left[\tan^{-1}\left ( \frac{\frac{3}{4}+\frac{2}{3}}{1- \frac{3}{4}.\frac{2}{3}} \right ) \right ] from As, \left [ \tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1- xy} \right ]

=\tan \left (\tan^{-1} \frac{9+8}{12-6} \right ) = \tan \left (\tan^{-1} \frac{17}{6} \right )= \frac{17}{6}

Question:19 \cos^{-1}\left(\cos\frac{7\pi}{6} \right ) is equal to

(A) \frac{7\pi}{6}

(B) \frac{5\pi}{6}

(C) \frac{\pi}{3}

(D) \frac{\pi}{6}

Answer:

As we know that \cos^{-1} (cos x ) = x if x\epsilon [0,\pi] and is principal value range of \cos^{-1}x .

In this case \cos^{-1}\left(\cos\frac{7\pi}{6} \right ) ,

\frac{7\pi}{6} \notin [0,\pi]

hence we have then,

\cos^{-1}\left(\cos\frac{7\pi}{6} \right ) = \cos^{-1} \left ( \cos \frac{-7\pi}{6} \right ) = \cos^{-1}\left [ \cos\left ( 2\pi - \frac{7\pi}{6} \right ) \right ]

\left [ \because \cos (2\pi + x) = \cos x \right ]

\therefore\ we\ have \cos^{-1}\left ( \cos \frac{7\pi}{6} \right ) = \cos^{-1}\left ( \cos \frac{5\pi}{6} \right ) = \frac{5\pi}{6}

Hence the correct answer is \frac{5\pi}{6} (B).

Question:20 \sin\left(\frac{\pi}{3} -\sin^{-1}\left(-\frac{1}{2} \right ) \right ) is equal to

(A) \frac{1}{2}

(B)

(C) \frac{1}{4}

(D) 1

Answer:

Solving the inner bracket of \sin\left(\frac{\pi}{3} -\sin^{-1}\left(-\frac{1}{2} \right ) \right ) ;

\left(\frac{\pi}{3} -\sin^{-1}\left(-\frac{1}{2} \right ) \right ) or

Take \sin^{-1}\left(-\frac{1}{2} \right ) = x then,

\sin x =-\frac{1}{2} and we know the range of principal value of \sin^{-1}x\ is\ \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ].

Therefore we have \sin^{-1}\left ( -\frac{1}{2} \right ) = -\frac{\pi}{6} .

Hence, \sin\left(\frac{\pi}{3} -\sin^{-1}\left(-\frac{1}{2} \right ) \right ) = \sin \left ( \frac{\pi}{3}+ \frac{\pi}{6} \right )= \sin \left ( \frac{3\pi}{6} \right ) = \sin\left ( \frac{\pi}{2} \right ) = 1

Hence the correct answer is D.

Question:21 \tan^{-1}\sqrt3 - \cot^{-1}(-\sqrt3) is equal to

(A) \pi

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

(C) 0

(D) 2\sqrt3

Answer:

We have \tan^{-1}\sqrt3 - \cot^{-1}(-\sqrt3) ;

finding the value of \cot^{-1}(-\sqrt3) :

Assume \cot^{-1}(-\sqrt3) =y then,

\cot y = -\sqrt 3 and the range of the principal value of \cot^{-1} is (0,\pi) .

Hence, principal value is \frac{5\pi}{6}

Therefore \cot^{-1} (-\sqrt3) = \frac {5\pi}{6}

and \tan^{-1} \sqrt3 = \frac{\pi}{3}

so, we have now,

\tan^{-1}\sqrt3 - \cot^{-1}(-\sqrt3)=\frac{\pi}{3} - \frac{5\pi}{6}

= \frac{2\pi - 5\pi}{6} = \frac{-3\pi}{6}

or, = \frac{ -\pi}{2}

Hence the answer is option (B).


NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions: Miscellaneous Exercise

Question:1 Find the value of the following: \cos^{-1}\left (\cos\left(\frac{13\pi}{6} \right ) \right )

Answer:

If x \epsilon [0,\pi] then \cos^{-1}(\cos x) = x , which is principal value of \cos^{-1} x .

So, we have \cos^{-1}\left (\cos\left(\frac{13\pi}{6} \right ) \right )

where \:\frac{13\pi}{6} \notin \left [ 0, \pi \right ].

Hence\: we \:can\: write\: \cos^{-1}\left (\cos\left(\frac{13\pi}{6} \right ) \right ) \:as

=\cos^{-1}\left (\cos\left(2\pi+\frac{\pi}{6} \right ) \right )

=\cos^{-1}\left (\cos\left(\frac{\pi}{6} \right ) \right )

\frac{\pi}{6}\ \epsilon \left [ 0, \pi \right ]

Therefore we have,

\cos^{-1}\left (\cos\left(\frac{13\pi}{6} \right ) \right ) = \cos^{-1}\left (\cos\left(\frac{\pi}{6} \right ) \right ) = \frac{\pi}{6} .

Question:2 Find the value of the following: \tan^{-1}\left(\tan\frac{7\pi}{6} \right )

Answer:

We have given \tan^{-1}\left(\tan\frac{7\pi}{6} \right ) ;

so, as we know \tan^{-1}\left(\tan x \right ) =x \:\:if\:\:x\epsilon \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )

So, here we have \frac{7\pi}{6} \notin \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right ) .

Therefore we can write \tan^{-1}\left(\tan\frac{7\pi}{6} \right ) as:

=\tan^{-1}\left(\tan \left (2\pi - \frac{5\pi}{6} \right ) \right ) \left [ \because \tan(2\pi - x) = -\tan x \right ]

=\tan^{-1}\left[-\tan \left ( \frac{5\pi}{6} \right ) \right ]

=\tan^{-1}\left[\tan \left (\pi- \frac{5\pi}{6} \right ) \right ]

=\tan^{-1}\left[\tan \left (\frac{\pi}{6} \right ) \right ]\:\:where\:\:\frac{\pi}{6} \epsilon \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )

\therefore \tan^{-1} \left ( \tan \frac{7\pi}{6} \right ) = \tan^{-1} \left ( \tan \frac{\pi}{6} \right ) = \frac{\pi}{6} .

Question:3 Prove that 2\sin^{-1}\frac{3}{5} = \tan^{-1}\frac{24}{7}

Answer:

To prove: 2\sin^{-1}\frac{3}{5} = \tan^{-1}\frac{24}{7} ;

L.H.S=2\sin^{-1}\frac{3}{5}

Assume that \sin^{-1}\frac{3}{5} = x

then we have \sin x = \frac{3}{5} .

or \cos x = \sqrt{1-\left (\frac{3}{5} \right )^2} = \frac{4}{5}

Therefore we have

\tan x = \frac{3}{4}\:\:or\:\:x = \sin^{-1} \frac{3}{5} = \tan^{-1} \frac{3}{4}

Now,

We can write L.H.S as

2\sin^{-1}\frac{3}{5} = 2\tan^{-1}\frac{3}{4}

=\tan^{-1} \left [\frac{2\times\frac{3}{4}}{1- \left ( \frac{3}{4} \right )^2} \right ] as we know \left [2\tan^{-1} x = \tan^{-1} \frac{2x}{1-x^2} \right ]

=\tan^{-1} \left [\frac{\frac{3}{2}}{\left ( \frac{16-9}{16} \right )} \right ] = \tan^{-1}\left ( \frac{3}{2}\times \frac{16}{7} \right )

=\tan^{-1} \frac{24}{7}=R.H.S

L.H.S = R.H.S

Question:4 Prove that \sin^{-1}\frac{8}{17} + \sin^{-1}\frac{3}{5} =\tan^{-1}\frac{77}{36}

Answer

Taking \sin ^{-1} \frac{8}{17} = x

then,

\sin x = \frac{8}{17} \Rightarrow \cos x = \sqrt{1- \left ( \frac{8}{17} \right )^2} = \sqrt {\frac{225}{289}} = \frac{15}{17}.

Therefore we have-

\tan^{-1} x = \frac{8}{15} \Rightarrow x = \tan^{-1} \frac{8}{15}

\therefore \sin ^{-1} \frac{8}{17} = \tan ^{-1} \frac{8}{15} .............(1).

Now, let\:\sin ^{-1} \frac{3}{5} = y ,

Then,

\sin ^{-1} \frac{3}{5} = \tan ^{-1} \frac{3}{4} .............(2).

So, we have now,

L.H.S.

\sin^{-1}\frac{8}{17} + \sin^{-1}\frac{3}{5}

using equations (1) and (2) we get,

=\tan ^{-1} \frac{8}{15} + \tan^{-1} \frac{3}{4}

=\tan^{-1} \frac{\frac{8}{15}+ \frac{3}{4}}{1-\frac{8}{15}\times \frac{3}{4}} [\because \tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1 - xy}]

=\tan^{-1} (\frac{32+45}{60-24})

=\tan^{-1} (\frac{77}{36})

= R.H.S.

Question:5 Prove that \cos^{-1}\frac{4}{5} + \cos^{-1}\frac{12}{13} = \cos^{-1}\frac{33}{65}

Answer:

Take \cos^{-1}\frac{4}{5} = x and \cos^{-1}\frac{12}{13} = y and \cos^{-1}\frac{33}{65} = z

then we have,

\cos x = \frac{4}{5}

\sin x = \sqrt {1- \left ( \frac {4}{5} \right )^2} = \frac {3}{5}

Then we can write it as:

\tan x = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} or x= \tan^{-1} \frac{3}{4}

\therefore \cos ^{-1} \frac{4}{5} = \tan^{-1} \frac{3}{4} ...............(1)

Now, \cos^{-1}\frac{12}{13} = y

\cos y = \frac{12}{13} \Rightarrow \sin y =\frac{5}{13}

\therefore \tan y = \frac{5}{12} \Rightarrow y = \tan^{-1} \frac{5}{12}

So, \cos^{-1}\frac{12}{13} = \tan^{-1} \frac{5}{12} ...................(2)

Also we have similarly;

\cos^{-1}\frac{33}{65} = z

Then,

\cos^{-1}\frac{33}{65} = \tan^{-1} \frac{56}{33} ...........................(3)

Now, we have

L.H.S

\cos^{-1}\frac{4}{5} + \cos^{-1}\frac{12}{13} so, using (1) and (2) we get,

=\tan^{-1}\frac{3}{4} + \tan^{-1}\frac{5}{12}

=\tan^{-1}\left ( \frac{\frac{3}{4}+ \frac{5}{12}}{1-\left ( \frac{3}{4}\times \frac{5}{12} \right )} \right ) \because \left [ \tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1- xy} \right ]

=\tan^{-1}\left ( \frac{36+20}{48-15} \right )

=\tan^{-1}\left ( \frac{56}{33} \right ) or we can write it as;

=\cos^{-1}\frac{33}{65}

= R.H.S.

Hence proved.


Question:6 Prove that \cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5} = \sin^{-1}\frac{56}{65}

Answer:

Converting all terms in tan form;

Let \cos^{-1}\frac{12}{13} = x , \sin^{-1}\frac{3}{5} = y and \sin^{-1}\frac{56}{65} = z .

now, converting all the terms:

\cos^{-1}\frac{12}{13} = x or \cos x = \frac{12}{13}

We can write it in tan form as:

\cos x = \frac{12}{13} \Rightarrow \sin x = \frac{5}{13} .

\therefore \tan x = \frac{5}{12} \Rightarrow x = \tan^{-1} \frac{5}{12}

or \cos^{-1}\frac{12}{13} = \tan^{-1} \frac{5}{12} ................(1)

\sin^{-1}\frac{3}{5} = y or \sin y = \frac{3}{5}

We can write it in tan form as:

\sin y = \frac{3}{5} \Rightarrow \cos y = \frac{4}{5}

\therefore \tan y =\frac{3}{4} \Rightarrow y = \tan^{-1} \frac{3}{4}

or \sin^{-1}\frac{3}{5} = \tan^{-1} \frac{3}{4} ......................(2)

Similarly, for \sin^{-1}\frac{56}{65} = z ;

we have \sin^{-1}\frac{56}{65} = \tan^{-1} \frac{56}{33} .............(3)

Using (1) and (2) we have L.H.S

\cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5}

= \tan^{-1} \frac{5}{12} + \tan^{-1} \frac{3}{4}

On applying \tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1-xy}

We have,

=\tan^{-1} \frac{\frac{5}{12}+\frac{3}{4}}{1-(\frac{5}{12}.\frac{3}{4})}

=\tan^{-1} (\frac{20+36}{48-15})

=\tan^{-1} (\frac{56}{33})

=\sin^{-1} (\frac{56}{65}) ...........[Using (3)]

=R.H.S.

Hence proved.

Question:7 Prove that \tan^{-1}\frac{63}{16} = \sin^{-1}\frac{5}{13} + \cos^{-1}\frac{3}{5}

Answer:

Taking R.H.S;

We have \sin^{-1}\frac{5}{13} + \cos^{-1}\frac{3}{5}

Converting sin and cos terms in tan forms:

Let \sin^{-1}\frac{5}{13} = x and \cos^{-1}\frac{3}{5} = y

now, we have \sin^{-1}\frac{5}{13} = x or \sin x = \frac{5}{13}

\sin x = \frac{5}{13} \:or\: \cos x =\frac{12}{13}\:or\:\tan x = \frac{5}{12}

\tan x = \frac{5}{12} \Rightarrow x =\tan^{-1} \frac{5}{12}

\therefore \sin^{-1} \frac{5}{13} = \tan^{-1} \frac{5}{12} ............(1)

Now, \cos^{-1}\frac{3}{5} = y\Rightarrow \cos y = \frac{3}{5}

\cos y = \frac{3}{5} \:or\: \sin y = \frac{4}{5}\:or\:\tan y = \frac{4}{3}

\tan y = \frac{4}{3} \Rightarrow y = \tan^{-1} \frac{4}{3}

\therefore \cos^{-1}\frac{3}{5} = \tan^{-1} \frac{4}{5} ................(2)

Now, Using (1) and (2) we get,

R.H.S.

\sin^{-1}\frac{5}{13} + \cos^{-1}\frac{3}{5} = \tan^{-1} \frac{5}{12} + \tan^{-1} \frac{4}{3}

=\tan^{-1}\left ( \frac{\frac{5}{12}+\frac{4}{3}}{1- \frac{5}{12}\times \frac{4}{3}} \right ) as we know \left [ \tan^{-1} x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1-xy} \right ]

so,

= \tan^{-1} \frac{63}{16}

equal to L.H.S

Hence proved.

Question:8 Prove that \tan^{-1}\frac{1}{5} + \tan^{-1}\frac{1}{7} +\tan^{-1}\frac{1}{3} +\tan^{-1}\frac{1}{8} = \frac{\pi}{4}

Answer:

Applying the formlua:

\tan^{-1}x + \tan^{-1}y = \tan^{-1} \frac{x+y}{1-xy} on two parts.

we will have,

=\tan^{-1}\left (\frac{\frac{1}{5}+ \frac{1}{7}}{1- \frac{1}{5}\times \frac{1}{7}} \right ) + \tan^{-1}\left (\frac{\frac{1}{3}+ \frac{1}{8}}{1- \frac{1}{3}\times \frac{1}{8}} \right )

= \tan^{-1} \left ( \frac{7+5}{35-1} \right ) + \tan^{-1} \left ( \frac{8+3}{24-1} \right )

= \tan^{-1} \left ( \frac{12}{34} \right ) + \tan^{-1} \left ( \frac{11}{23} \right )

= \tan^{-1} \left ( \frac{6}{17} \right ) + \tan^{-1} \left ( \frac{11}{23} \right )

= \tan^{-1}\left [ \frac{\frac{6}{17}+\frac{11}{23}}{1-\frac{6}{17}\times\frac{11}{23}} \right ]

= \tan^{-1}\left [ \frac{325}{325} \right ] = \tan^{-1} 1

=\frac{\pi}{4}

Hence it s equal to R.H.S

Proved.

Question:9 Prove that \tan^{-1} \sqrt{x} = \frac{1}{2}\cos^{-1}\frac{1-x}{1+x},\;\;x\in [0,1]

Answer:

By observing the square root we will first put

x= \tan^2 \theta .

Then,

we have \tan^{-1} \sqrt{\tan^2 \theta} = \frac{1}{2}\cos^{-1}\frac{1-\tan^2 \theta}{1+\tan^2 \theta}

or, R.H.S.

\frac{1}{2}\cos^{-1}\frac{1-\tan^2 \theta}{1+\tan^2 \theta} = \frac{1}{2}\cos^{-1}(cos2 \theta)

= \frac{1}{2}\times 2\theta = \theta .

L.H.S. \tan^{-1} \sqrt{\tan^2 \theta} = \tan^{-1}(\tan \theta) = \theta

hence L.H.S. = R.H.S proved.

Question:10 Prove that \cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1 - \sin x}}{\sqrt{1+\sin x} - \sqrt{1 - \sin x}} \right ) = \frac{x}{2},\;\;x\in\left(0,\frac{\pi}{4} \right )

Answer:

Given that \cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1 - \sin x}}{\sqrt{1+\sin x} - \sqrt{1 - \sin x}} \right )

By observing we can rationalize the fraction

\left(\frac{\sqrt{1+\sin x} + \sqrt{1 - \sin x}}{\sqrt{1+\sin x} - \sqrt{1 - \sin x}} \right )

We get then,

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

= \left(\frac{{1+\sin x} +{1 - \sin x} + 2\sqrt{(1+\sin x)(1-\sin x)} }{{1+\sin x} - {1 + \sin x}} \right )

= \frac{2(1+\sqrt{1-\sin^2 x})}{2\sin x} = \frac{1+\cos x}{\sin x} = \frac{2\cos^2 \frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}

= \cot \frac{x}{2}

Therefore we can write it as;

\cot^{-1}\left ( \cot \frac{x}{2} \right ) = \frac{x}{2}

As L.H.S. = R.H.S.

Hence proved.

Question:11 Prove that \tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} \right ) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x,\;\;-\frac{1}{\sqrt2}\leq x\leq 1

[Hint: Put x = \cos 2\theta ]

Answer:

By using the Hint we will put x = \cos 2\theta ;

we get then,

=\tan^{-1}\left(\frac{\sqrt{1+\cos 2\theta} - \sqrt{1-\cos2\theta}}{\sqrt{1+\cos 2\theta} + \sqrt{1-\cos 2\theta}} \right )

=\tan^{-1}\left(\frac{\sqrt{2\cos^2 \theta} - \sqrt{2\sin^2\theta}}{\sqrt{2\cos^2 \theta} + \sqrt{2\sin^2\theta}} \right )

=\tan^{-1}\left(\frac{\sqrt2{\cos \theta} - \sqrt2{\sin\theta}}{\sqrt2{\cos \theta} + \sqrt2{\sin\theta}} \right )

=\tan^{-1}\left(\frac{{\cos \theta} - {\sin\theta}}{{\cos \theta} + {\sin\theta}} \right ) dividing numerator and denominator by \cos \theta ,

we get,

= \tan^{-1}\left ( \frac{1-\tan \theta}{1+\tan \theta} \right )

= \tan^{-1} 1 - \tan^{-1} (\tan \theta) using the formula \left [ \tan^{-1}x - \tan^{-1} y = \tan^{-1} \frac{x-y}{1+xy} \right ]

= \frac{\pi}{4} - \theta = \frac{\pi}{4}- \frac{1}{2}\cos^{-1}x

As L.H.S = R.H.S

Hence proved

Question:12 Prove that \frac{9\pi}{8} - \frac{9}{4}\sin^{-1}\frac{1}{3} = \frac{9}{4}\sin^{-1}\frac{2\sqrt2}{3}

Answer:

We have to solve the given equation:

\frac{9\pi}{8} - \frac{9}{4}\sin^{-1}\frac{1}{3} = \frac{9}{4}\sin^{-1}\frac{2\sqrt2}{3}

Take \frac{9}{4} as common in L.H.S,

=\frac{9}{4}\left [ \frac{\pi}{2}- \sin^{-1}\frac{1}{3} \right ]

or =\frac{9}{4}\left [ \cos^{-1}\frac{1}{3} \right ] from \left [ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \right ]

Now, assume,

\left [ \cos^{-1}\frac{1}{3} \right ] = y

Then,

\cos y = \frac{1}{3} \Rightarrow \sin y = \sqrt{1-(\frac{1}{3})^2} = \frac{2.\sqrt2}{3}

Therefore we have now,

y = \sin^{-1} \frac{2.\sqrt2}{3}

So we have L.H.S then = \frac{9}{4}\sin^{-1} \frac{2.\sqrt2}{3}

That is equal to R.H.S.

Hence proved.

Question:13 Solve the following equations: 2\tan^{-1}(\cos x) = \tan^{-1}(2\textup{cosec}x)

Answer:

Given equation 2\tan^{-1}(\cos x) = \tan^{-1}(2\textup{cosec}x) ;

Using the formula:

\left [ 2\tan^{-1}z = \tan^{-1} \frac{2z}{1-z^2} \right ]

We can write

2\tan^{-1}(\cos x) = \tan^{-1}\left [ \frac{2\cos x}{1- (\cos x )^2 }\right ]

\tan^{-1}\left [ \frac{2\cos x}{1- (\cos x )^2 }\right ] = \tan^{-1}\left [2cosec x \right ]

So, we can equate;

=\left [ \frac{2\cos x}{1- (\cos x )^2 }\right ] = \left [2cosec x \right ]

=\left [ \frac{2\cos x}{\sin^2 x }\right ] = \left [ \frac{2}{sinx } \right ]

that implies that \cos x = \sin x .

or \tan x =1 or x = \frac{\pi}{4}

Hence we have solution x = \frac{\pi}{4} .

Question:14 Solve the following equations: \tan^{-1} \frac{1-x}{1+x} = \frac{1}{2}\tan^{-1}x,\;(x>0)

Answer:

Given equation is

\tan^{-1} \frac{1-x}{1+x} = \frac{1}{2}\tan^{-1}x :

L.H.S can be written as;

\tan^{-1} \frac{1-x}{1+x} = \tan^{-1}1 - \tan^{-1}x

Using the formula \left [ \tan^{-1}x -\tan^{-1}y = \tan^{-1} \frac{x-y}{1+xy} \right ]

So, we have \tan^{-1}1 - \tan^{-1}x = \frac{1}{2} \tan^{-1}x

\Rightarrow \tan^{-1}1= \frac{3}{2} \tan^{-1}x

\Rightarrow\frac{\pi}{4}= \frac{3}{2} \tan^{-1}x

\Rightarrow \tan^{-1}x = \frac{\pi}{6}

\Rightarrow x= \tan \frac{\pi}{6} = \frac{1}{\sqrt3}

Hence the value of x= \frac{1}{\sqrt3} .

Question:15 \sin(\tan^{-1}x),\;|x|<1 is equal to

(A) \frac{x}{\sqrt{1-x^2}}

(B) \frac{1}{\sqrt{1-x^2}}

(C) \frac{1}{\sqrt{1+x^2}}

(D) \frac{x}{\sqrt{1+x^2}}

Answer:

Let \tan^{-1}x = y then we have;

\tan y = x or

y=\sin^{-1} \left ( \frac{x}{\sqrt{1+x^2}} \right ) \Rightarrow \tan^{-1} x = \sin^{-1} \left ( \frac{x}{\sqrt{1+x^2}} \right)

\Rightarrow \sin \left ( \tan^{-1} x \right ) = \sin\left ( \sin^{-1}\left ( \frac{x}{\sqrt{1+x^2}} \right ) \right ) = \frac{x}{\sqrt {1+x^2}}

Hence the correct answer is D.

Question:16 \sin^{-1}(1-x) - 2\sin^{-1}x = \frac{\pi}{2} then x is equal to

(A) 0,\frac{1}{2}

(B) 1,\frac{1}{2}

(C) 0

(D) \frac{1}{2}

Answer:

Given the equation: \sin^{-1}(1-x) - 2\sin^{-1}x = \frac{\pi}{2}

we can migrate the \sin^{-1}(1-x) term to the R.H.S.

then we have;

- 2\sin^{-1}x = \frac{\pi}{2} - \sin^{-1}(1-x)

or - 2\sin^{-1}x =\cos^{-1}(1-x) ............................(1)

from \left [\because \cos^{-1}(1-x) + \sin^{-1}(1-x) = \frac{\pi}{2} \right ]

Take \sin^{-1}x = \Theta \Rightarrow \sin \Theta = x or \cos \Theta = \sqrt{1-x^2} .

So, we conclude that;

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

Therefore we can put the value of \sin^{-1}x in equation (1) we get,

- 2\cos^{-1}\left ( \sqrt{1-x^2} \right ) =\cos^{-1}(1-x)

Putting x= sin y , in the above equation; we have then,

\Rightarrow - 2\cos^{-1}\left ( \sqrt{1-(\sin y)^2} \right ) =\cos^{-1}(1-\sin y )

\Rightarrow - 2\cos^{-1}\left ( \sqrt{\cos^2 y} \right ) =\cos^{-1}(1-\sin y )

\Rightarrow - 2\cos^{-1}\left ( \cos y \right ) =\cos^{-1}(1-\sin y )

\Rightarrow \cos(-2y) = 1-\sin y

\Rightarrow - 2y=\cos^{-1}(1-\sin y )

\Rightarrow 1- 2\sin^2 y = 1-\sin y

\Rightarrow 2\sin^2 y - \sin y = 0

\Rightarrow \sin y(2 \sin y -1) = 0

So, we have the solution;

\sin y = 0\ or\ \frac{1}{2} Therefore we have x = 0\ or\ x= \frac{1}{2} .

When we have x= \frac{1}{2} , we can see that :

L.H.S. = \sin ^{-1}\left ( 1 - \frac{1}{2} \right ) - 2\sin^{-1}\frac{1}{2} = - \sin^{-1}\frac{1}{2} = -\frac{\pi}{6}

So, it is not equal to the R.H.S. -\frac{\pi}{6} \neq \frac{\pi}{2}

Thus we have only one solution which is x = 0

Hence the correct answer is (C).

Question:17 \tan^{-1}\left (\frac{x}{y} \right )-\tan^{-1}\frac{x-y}{x+y} is equal to

(A) \frac{\pi}{2}

(B) \frac{\pi}{3}

(C) \frac{\pi}{4}

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

Answer:

Applying formula: \left [ \tan^{-1} x - \tan^{-1}y = \tan^{-1} \left ( \frac{x-y}{1+xy} \right ) \right ] .

We get,

\tan^{-1}\left (\frac{x}{y} \right )-\tan^{-1} \left ( \frac{x-y}{x+y} \right ) = \tan^{-1} \left [\frac{ \frac{x}{y} - \frac{x-y}{x+y}}{1+\left ( \frac{x}{y} \right ) \left ( \frac{x-y}{x+y} \right ) } \right ]

= \tan^{-1} \left [\frac{ \frac{x}{y} - \frac{x-y}{x+y}}{1+\left ( \frac{x}{y} \right ) \left ( \frac{x-y}{x+y} \right ) } \right ] = \tan^{-1} \left [ \frac{\frac{x(x+y)-y(x-y)}{y(x+y)}}{\frac{y(x+y) + x(x-y)}{y(x+y)}} \right ]

= \tan^{-1}\left ( \frac{x^2+xy - xy + y^2}{xy + y^2 + x^2 - xy} \right )

= \tan^{-1}\left ( \frac{x^2 + y^2}{ y^2 + x^2 } \right ) = \tan^{-1} 1 = \frac{\pi}{4}

Hence, the correct answer is C.

If you are looking for inverse trigonometric functions class 12 exercises then these are listed below.

Inverse Trigonometric Functions Class 12 Exercise 2.1

Inverse Trigonometric Functions Class 12 Exercise 2.2

Inverse Trigonometric Functions Class 12 Miscellaneous Exercise

NCERT Solutions for Class 12 Maths Chapter 2 Introduction

In class 11 Maths you have already learnt about trigonometric functions. It won't take much effort to command on inverse trigonometric functions if you have good knowledge of trigonometric functions. You just need to practice NCERT questions including examples and miscellaneous exercises given in inverse trigonometric functions class 12. You may find some difficulties in solving the problems, so you can take the help of inverse trigonometry class 12 solutions maths chapter 2 notes.

There are important applications of ITF in geometry, navigation, science, and engineering. Also, inverse trigonometric functions play an important role in the calculus part of mathematics to define many integrals. Many students have a misconception in class 12 maths ch 2 question answer function like \sin^-^1x=\frac{1}{\sin x} .But the inverse function f^-^1 is not the same as \frac{1}{f} for example \sin^-^1x\neq \frac{1}{\sin x} .

For inverse to exist, the function must be one-one and onto but trigonometric functions are neither one-one and onto over their domain and natural ranges. So, to ensure the existence of their inverse we restrict domains and ranges of trigonometric functions. And this range is known as principal value.

Class 12 inverse trigonometric functions NCERT solutions - Topics

The important topics covered in the class 12 maths ch 2 question answer are

  • Basic concepts

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This chapter of class 12 NCERT includes concepts of domain and range of trigonometric function in which their inverse exists. the chapter comprehensively discusses graphical representation, the elementary property of inverse trigonometry function.

ch 2 maths class 12 discusses the basic concepts of the inverse of the sine function, cosine function, secant function, cosecant function, tangent function, and cotangent function. Also, we will be familiar with their range and domain, concepts related to one-one, and onto function. Problems related to these concepts are in detail in NCERT solutions for class 12 maths chapter 2.

  • Properties of inverse trigonometry function

Maths class 12 chapter 2 includes important property of inverse trigonometry function that might be valid in principal value branch and corresponding inverse trigonometric function. To get a good hold on these concepts you can solve the NCERT solution for class 12 maths chapter 2.

Topics mentioned in class 12 ncert are very important to obtain decent marks in board exams and students are recommended to proceed through all the concepts cited in the topics. Questions from all the above topics are covered in the NCERT solutions for class 12 maths chapter 2.

NCERT Exemplar Class 12 Solutions

Class 12 Maths Question Answer - Chapter Wise

Chapter 1NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
Chapter 2NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions
Chapter 3NCERT solutions for class 12 maths chapter 3 Matrices
Chapter 4NCERT solutions for class 12 maths chapter 4 Determinants
Chapter 5NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability
Chapter 6NCERT solutions for class 12 maths chapter 6 Application of Derivatives
Chapter 7NCERT solutions for class 12 maths chapter 7 Integrals
Chapter 8NCERT solutions for class 12 maths chapter 8 Application of Integrals
Chapter 9NCERT solutions for class 12 maths chapter 9 Differential Equations
Chapter 10NCERT solutions for class 12 maths chapter 10 Vector Algebra
Chapter 11NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry
Chapter 12NCERT solutions for class 12 maths chapter 12 Linear Programming
Chapter 13NCERT solutions for class 12 maths chapter 13 Probability

More about NCERT Solutions for Class 12 Maths Chapter 2

  • In class 11 maths you have already learnt about trigonometric functions. It won't take much effort to command inverse trigonometric functions if you have good knowledge of trigonometric functions.

  • For this, you just need to practice NCERT questions including examples and miscellaneous exercise.

  • You may find some difficulties in solving the problems, so you can take the help of these NCERT Class 12 Maths solutions chapter 2.

  • NCERT Solutions for Class 12 Maths Chapter 2 PDF Download will also be available soon to help you with the questions offline.

Inverse Trigonometric Functions Class 12 Solutions - Topics

2.1 Introduction

2.2 Basic Concepts

2.3 Properties of Inverse Trigonometric Functions

Class 12 Maths Question Answer - Subject Wise

NCERT Books and NCERT Syllabus

Tips to use class 12 maths chapter 2 question answer

  • If you wish to perform well in the CBSE 12 board examination, NCERT Class 12 Maths solutions chapter 2 inverse trigonometric functions are very helpful but here are some tips to make command on Inverse Trigonometric Functions.

  • The inverse trigonometric function is inverse of the trigonometric function, so if you have a command on the trigonometric function then it will be easy for you to understand inverse trigonometric functions, use class 12 maths chapter 2 question answer pdf to learn easily.

  • Try to relate trigonometric functions formulas with inverse trigonometric functions formulas, so that memorizing the formulae becomes easier.

  • Before starting to solve an exercise, first solve the examples that are given in the NCERT class 12 maths textbook.

  • Also, try to solve every exercise including inverse trigonometric functions class 12 miscellaneous, and examples, miscellaneous examples on your own, if you are finding difficulties, you can take the help of Class 12 Maths Chapter 2 NCERT solutions.

  • If you have solved all NCERT then you can solve previous years paper CBSE board to get familiar with the pattern of the board exam question paper.

Also, check NCERT solutions for class 12 for other chapter and subjects.

Frequently Asked Questions (FAQs)

1. What are the important topics in the chapter on inverse trigonometric functions?

Some basic concepts of inverse trigonometry and properties of inverse trigonometric functions are important topics in this chapter. these concepts are very important as they are used in solving or simplifying differential calculus, integral calculus, and other topics like differential equations therefore having a command on inverse trigonometry concepts becomes very important. students can refer to class 12 maths chapter 2 question answer for getting a good hold on the concepts.

2. Do inverse trigonometric functions class 12 solutions aid in preparing for the board exams?

NCERT class 12 maths ch 2 solutions provide students with a strong foundation in basic Maths topics. These exercise-wise class 12 maths ch 2 ncert solutions are crafted by experienced and knowledgeable subject experts at Carresr360. By using these resources, students can improve their focus and achieve better scores on their board exams. The ultimate goal is to enhance students' confidence and ability to efficiently solve difficult problems in a timely manner.

3. What is the weightage of the chapter inverse trigonometry for the CBSE board exam?

Relation and function and inverse trigonometry combined have 10 % weightage in the CBSE class 12 board examination. According to priority students can give time to chapters but if you want full marks then it demand practice and in depth understanding of concepts therefor ncert textbook are suggested to students.  

4. Which are the most difficult chapters of NCERT Class 12 Maths syllabus?

Students consider Integration and it's application as the most difficult unit in CBSE class 12 maths. With regular practice, you will get conceptual clarity and will be able to have a strong grip on Integration. if you want then inverse trigonometric functions solutions are recommended to students for getting command and in depth understanding of concepts.

5. Which is the best book for CBSE class 12 Maths?

NCERT textbook is the best book for CBSE class 12 maths. Most of the questions in the CBSE class 12 board exam are directly asked from the NCERT textbook. All you need to do is practice of the questions given in the NCERT textbook that is best way to get command on concepts. if you have any difficulty then you can read solutions of ncert exercise also help you in getting hold on the chapter.

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Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?
quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

Read Complete Answer
Jefferson 25 September,2024
i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

  1. Re-evaluate Your Study Strategies:

    • Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
    • Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
    • Practice Regularly: Consistent practice is key to mastering chemistry.

  2. Consider Professional Help:

    • Tutoring: A tutor can provide personalized guidance and support.
    • Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.

  3. Explore Alternative Options:

    • Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
    • Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.

  4. Focus on NEET 2025 Preparation:

    • Stay Dedicated: Continue your NEET preparation with renewed determination.
    • Utilize Resources: Make use of study materials, online courses, and mock tests.

  5. Seek Support:

    • Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
    • Join Study Groups: Collaborating with peers can create a supportive learning environment.

Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.







Read Complete Answer
Kanishka kaushikii 17 September,2024
i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or 11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.

Scholarship Details:

Type A: For candidates scoring 60% or above in the exam.

Type B: For candidates scoring between 50% and 60%.

Type C: For candidates scoring between 40% and 50%.

Cash Scholarship:

Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).

Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to maximize your chances of receiving a higher scholarship.

Hope you find this useful!

Read Complete Answer
Yuvan 30 August,2024
i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Read Complete Answer
Ashvadha 12 July,2024
Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

Read Complete Answer
kumardurgesh1802 10 July,2024
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Popular CBSE Class 12th Questions

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

 Option 2)

0.16\; J
Option 3)

1.00\; J

 Option 4)

0.67\; J
Read More

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

 Option 2)

 6.45×10−3 kg
Option 3)

 9.89×10−3 kg

 Option 4)

12.89×10−3 kg

 
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An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

 Option 2)

200 \, \, J - 500 \, \, J
Option 3)

2\times 10^{5}J-3\times 10^{5}J

 Option 4)

20,000 \, \, J - 50,000 \, \, J
Read More

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

 Option 2)

\; K\;
Option 3)

zero\;

 Option 4)

K/4
Read More

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

 Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced
Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

 Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .
Read More

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

 Option 2)

3.125 × 10-2
Option 3)

1.25 × 10-2

 Option 4)

2.5 × 10-2
Read More

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

 Option 2)

increase two fold
Option 3)

remain unchanged

 Option 4)

be a function of the molecular mass of the substance.
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With increase of temperature, which of these changes?

Option 1)

Molality

 Option 2)

Weight fraction of solute
Option 3)

Fraction of solute present in water

 Option 4)

Mole fraction.
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Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

 Option 2)

6.023 × 1022
Option 3)

half that in 8 g He

 Option 4)

558.5 × 6.023 × 1023
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A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

 Option 2)

more than 3 but less than 6
Option 3)

more than 6 but less than 9

 Option 4)

more than 9
Read More

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News and Notifications
October 23, 2024 - 04:01 PM IST :
No changes in exam, IA system: CBSE rebuts claims of syllabus reduction, open book exams in 2025
November 14, 2024 - 07:31 PM IST
Fact-check: CBSE hasn't reduced syllabus by 15%, internal assessments will not be 40% of score in 2025
November 14, 2024 - 02:32 PM IST
CBSE date sheet 2025 for Classes 10, 12 likely in November last week: Reports
November 08, 2024 - 06:29 PM IST
CBSE allows Class 10, 12 students from disaffiliated schools to appear for board exam 2025 from same school
November 07, 2024 - 05:46 PM IST
CBSE withdraws affiliation of 21 schools, downgrades 6 schools in two states; most in Delhi
November 06, 2024 - 04:22 PM IST
CBSE board exams 2025 from February 15; Class 10, 12 subject-wise marks distribution
October 24, 2024 - 11:43 AM IST
CBSE Date Sheet 2025: Class 10, 12 board exam dates soon on cbse.gov.in; previous trends, sample papers
October 23, 2024 - 03:21 PM IST

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