CUET Mathematics - Geometry - Trigonometry | Free Questions

Q1:

31st May Shift 2

Geometry > Trigonometry

Hard

The value of $\cos^{-1}\left\{\dfrac{\sin x+\cos x}{\sqrt{2}}\right\}, \dfrac{5\pi}{4} < x < \dfrac{9\pi}{4}$ is :

$x - \dfrac{\pi}{4}$

$\dfrac{\pi}{4} - x$

$\dfrac{9\pi}{4} - x$

$x - \dfrac{9\pi}{4}$

Correct Answer

Option 3

Correct Answer

Explanation

Q2:

30th May Shift 2

Geometry > Trigonometry

Easy

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Domain of $\sin^{-1}(2x-3)$ | (I) [-1, 0] | | (B) Domain of $\sin^{-1}(1+2x)$ | (II) [-1, 1] | | (C) Domain of $\cos^{-1}(2x-1)$ | (III) [1, 2] | | (D) Domain of $\sin^{-1}(x^2)$ | (IV) [0, 1] | Choose the correct answer from the options given below:

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (IV), (B) - (II), (C) - (III), (D) - (I)

Correct Answer

Option 2

Correct Answer

Explanation

Q3:

30th May Shift 1

Geometry > Trigonometry

Easy

The value of $2\sec^2(\tan^{-1}2)+3\text{cosec}^2(\cot^{-1}3)$ is:

15

30

40

10

Correct Answer

Option 3

Correct Answer

Explanation

Q4:

29th May Shift 2

Geometry > Trigonometry

Hard

Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | If $\cos^{-1}\alpha + \cos^{-1}\beta + \cos^{-1}\gamma = 3\pi$ then $\alpha(2\beta+\gamma) + \beta(2\gamma+\alpha) + \gamma(2\alpha+\beta)$ equals | I. | $1$ | | B. | The value of $\tan^2(\sec^{-1}4) + \cot^2(\text{cosec}^{-1}5)$ is | II. | $9$ | | C. | If $\sin\left(\cos^{-1}\dfrac{2}{5} + \sin^{-1}x\right) = 1$ then $10x$ equals | III. | $4$ | | D. | The length of the interval of domain of $\sin^{-1}2x$ is | IV. | $39$ | Choose the correct answer from the options given below:

A - II, B - I, C - III, D - IV

A - II, B - III, C - IV, D - I

A - II, B - IV, C - III, D - I

A - I, B - IV, C - III, D - II

Correct Answer

Option 3

Correct Answer

Explanation

Q5:

29th May Shift 1

Geometry > Trigonometry

Easy

The value of $\tan^{-1}(\sqrt3)-\sec^{-1}(-2)$ is

$\dfrac{\pi}{3}$

0

$-\dfrac{\pi}{3}$

$\dfrac{2\pi}{3}$

Correct Answer

Option 3

Correct Answer

Explanation

Q6:

26th May Shift 2

Geometry > Trigonometry

Easy

Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | The range of $\sin^{-1} x$ | I. | $[-1, 1]$ | | B. | Domain of $\cos^{-1} x$ | II. | $(-\infty, \infty)$ | | C. | The range of $\tan^{-1} x$ | III. | $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ | | D. | Domain of $\tan^{-1} x$ | IV. | $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ | Choose the correct answer from the options given below:

A-I, B-III, C-II, D-IV

A-III, B-I, C-IV, D-II

A-IV, B-III, C-II, D-I

A-I, B-II, C-III, D-IV

Correct Answer

Option 2

Correct Answer

Explanation

Q7:

25th May Shift 1

Geometry > Trigonometry

Hard

Match the LIST-I with LIST-II | LIST-I Function | LIST-II Value | |---|---| | A. $\tan[\cos^{-1}(\sin(\tan^{-1} 4/3))]$ | I. $1/\sqrt2$ | | B. $\sin\left(2\tan^{-1}\left(\dfrac{3}{4}\right)\right)$ | II. 1 | | C. $\cos\left[\tan^{-1}\left(\dfrac{1}{2}\right)+\tan^{-1}\left(\dfrac{1}{3}\right)\right]$ | III. 24/25 | | D. $\sin\left[\dfrac{\pi}{3} - \sin^{-1}(-1/2)\right]$ | IV. 3/4 | Choose the correct answer from the options given below:

A-I, B-II, C-IV, D-III

A-IV, B-III, C-II, D-I

A-II, B-I, C-III, D-IV

A-IV, B-III, C-I, D-II

Correct Answer

Option 4

Correct Answer

Explanation

Q8:

22nd May Shift 2

Geometry > Trigonometry

Medium

$\cos[\tan^{-1}\{\sin(\cot^{-1}x)\}]$ is equal to

$\sqrt{\frac{x^2+1}{x^2+2}}$

$\sqrt{\frac{x^2+2}{x^2+1}}$

$\sqrt{\frac{x}{x^2+2}}$

$\sqrt{\frac{x}{x^2+1}}$

Correct Answer

Option 1

Correct Answer

Explanation

Q9:

22nd May Shift 1

Geometry > Trigonometry

Easy

Match List-I with List-II | List-I (Function) | List-II (Principal value of y) | |---|---| | (A) $y=\sin^{-1}\left(-\frac{1}{2}\right)$ | (I) $\frac{3\pi}{4}$ | | (B) $y=\cot^{-1}\left(-\frac{1}{\sqrt3}\right)$ | (II) $-\frac{\pi}{4}$ | | (C) $y=cosec^{-1}(-\sqrt2)$ | (III) $-\frac{\pi}{6}$ | | (D) $y=\cos^{-1}\left(-\frac{1}{\sqrt2}\right)$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below:

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

Correct Answer

Option 4

Correct Answer

Explanation

Q10:

21st May Shift 2

Geometry > Trigonometry

Easy

Match the LIST-I with LIST-II | LIST-I (Function) | | LIST-II (Range of the principal value branch) | |---|---|---| | A. $\cos^{-1}x$ | | I. $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ | | B. $\text{cosec}^{-1}x$ | | II. $(0, \pi)$ | | C. $\tan^{-1}x$ | | III. $[0, \pi]$ | | D. $\cot^{-1}x$ | | IV. $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] - \{0\}$ | Choose the correct answer from the options given below:

A-II, B-I, C-III, D-IV

A-III, B-IV, C-I, D-II

A-III, B-II, C-IV, D-I

A-II, B-IV, C-I, D-III

Correct Answer

Option 2

Correct Answer

Explanation

Q11:

21st May Shift 1

Geometry > Trigonometry

Medium

Value of $\cot\left\{\frac{\pi}{4} - 2\cot^{-1}3\right\}$ is

5

6

7

9

Correct Answer

Option 3

Correct Answer

Explanation

Q12:

19th May Shift 2

Geometry > Trigonometry

Medium

Which one of the following is TRUE?

$\cot^{-1}1 < \cot 1$

$\cos^{-1}1 > \cos 1$

$\sin^{-1}1 < \sin 1$

$\tan^{-1}1 < \tan 1$

Correct Answer

Option 4

Correct Answer

Explanation

Q13:

18th May Shift 2

Geometry > Trigonometry

Easy

Match the LIST-I with LIST-II | LIST-I Functions | LIST-II Principal value | |---|---| | A. $\sin^{-1}\left(\dfrac{1}{2}\right)$ | I. $\dfrac{\pi}{4}$ | | B. $\sin^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)$ | II. $\dfrac{\pi}{6}$ | | C. $\tan^{-1}(1)$ | III. $-\dfrac{\pi}{4}$ | | D. $\cos^{-1}\left(-\dfrac{1}{2}\right)$ | IV. $\dfrac{2\pi}{3}$ | Choose the correct answer from the options given below:

A-I, B-IV, C-II, D-III

A-II, B-III, C-I, D-IV

A-IV, B-I, C-III, D-II

A-III, B-IV, C-II, D-I

Correct Answer

Option 2

Correct Answer

Explanation

Q14:

15th May Shift 1

Geometry > Trigonometry

Medium

The value of $\tan\left(\dfrac{1}{2}\cos^{-1}\dfrac{\sqrt5}{3}\right)$ is equal to

$\dfrac{5-\sqrt3}{2}$

$\dfrac{5+\sqrt3}{2}$

$\dfrac{3+\sqrt5}{2}$

$\dfrac{3-\sqrt5}{2}$

Correct Answer

Option 4

Correct Answer

Explanation

Q15:

14th May Shift 2

Geometry > Trigonometry

Medium

Match the LIST-I with LIST-II | | LIST-I (Functions) | | LIST-II (Principal Values) | |---|---|---|---| | A. | $\sin^{-1}\left(-\dfrac{1}{2}\right) + \cos^{-1}\left(-\dfrac{1}{2}\right)$ | I. | $-\dfrac{\pi}{3}$ | | B. | $\cos^{-1}\left(\dfrac{1}{2}\right) + 2\sin^{-1}\left(\dfrac{1}{2}\right)$ | II. | $\dfrac{3\pi}{4}$ | | C. | $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | III. | $\dfrac{\pi}{2}$ | | D. | $\cos^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)$ | IV. | $\dfrac{2\pi}{3}$ | Choose the correct answer from the options given below:

A-III, B-IV, C-I, D-II

A-IV, B-I, C-II, D-III

A-III, B-II, C-I, D-IV

A-IV, B-II, C-I, D-III

Correct Answer

Option 1

Correct Answer

Explanation

Q16:

12th May Shift 2

Geometry > Trigonometry

Easy

Match **List-I** with **List-II** | List-I (Functions) | List-II (Range, principal value) | |---|---| | (A) $\sin^{-1}x$ | (I) $[0,\pi]-\left\{\dfrac{\pi}{2}\right\}$ | | (B) $\tan^{-1}x$ | (II) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]-\{0\}$ | | (C) $cosec^{-1}x$ | (III) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ | | (D) $\sec^{-1}x$ | (IV) $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ | Choose the **correct** answer from the options given below:

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

Correct Answer

Option 2

Correct Answer

Explanation

Q17:

3 June Shift 2

Geometry > Trigonometry

Medium

Core

The value of $\tan^{-1}(2) + \tan^{-1}(3)$ is equal to

$\frac{\pi}{4}$

$-\frac{\pi}{4}$

$\frac{3\pi}{4}$

$-\frac{3\pi}{4}$

Correct Answer

Option 3

Correct Answer

Explanation

Q18:

3 June Shift 1

Geometry > Trigonometry

Medium

Core

The value of $-\cosec^2(\cot^{-1}y) + \sec^2( \tan^{-1}x)$ is equal to

$x^2 + y^2$

$x^2 - y^2$

$2 + x^2 + y^2$

$2 + x^2 - y^2$

Correct Answer

Option 2

Correct Answer

Explanation

Q19:

30 May Shift 2

Geometry > Trigonometry

Medium

Core

The simplified form of $\tan^{-1}\left(\frac{\cos x}{1+\sin x}\right)$, $-\frac{\pi}{2} < x < \frac{\pi}{2}$ is

$\frac{\pi}{4} + \frac{x}{2}$

$\frac{\pi}{4} - \frac{x}{2}$

$\frac{\pi}{2} + x$

$\frac{\pi}{4} + x$

Correct Answer

Option 2

Correct Answer

Explanation

Q20:

30 May Shift 1

Geometry > Trigonometry

Medium

Core

$\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ equals:

$\frac{7\pi}{6}$

$\frac{5\pi}{6}$

$-\frac{5\pi}{6}$

$\frac{\pi}{6}$

Correct Answer

Option 2

Correct Answer

Explanation

Q21:

30 May Shift 1

Geometry > Trigonometry

Medium

Applied

Consider $f(x) = \sin(3x) + 4, \forall x \in \mathbb{R}$. Then (A) Maximum value of $f(x)$ is 5 (B) Minimum value of $f(x)$ is 3 (C) Maximum value of $f(x)$ is attained at $x = \frac{\pi}{6}$ (D) Minimum value of $f(x)$ is attained at $x = 0$ Choose the correct answer from the options given below:

(A), (B) and (C) only

(A), (B) and (D) only

(C) and (D) only

(B) and (D) only

Correct Answer

Option 1

Correct Answer

Explanation

Q22:

29 May Shift 2

Geometry > Trigonometry

Medium

Core

The value of $\cos(2\cos^{-1}x + \sin^{-1}x)$ at $x = \frac{1}{5}$ is

$\sqrt{\frac{25}{24}}$

$-\sqrt{\frac{24}{25}}$

$\sqrt{\frac{24}{25}}$

$-\sqrt{\frac{25}{24}}$

Correct Answer

Option 2

Correct Answer

Explanation

Q23:

27 May Shift 1

Geometry > Trigonometry

Hard

Core

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24}$ | (I) $\frac{3\pi}{4}$ | | (B) $\tan^{-1}2 + \tan^{-1}3$ | (II) $\pi$ | | (C) $\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3$ | (III) $\tan^{-1}\frac{1}{2}$ | | (D) $\tan^{-1}\frac{1}{7} + \tan^{-1}\frac{1}{13}$ | (IV) $\tan^{-1}\frac{2}{9}$ | Choose the correct answer from the options given below:

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

(A) - (IV), (B) - (I), (C) - (III), (D) - (II)

Correct Answer

Option 2

Correct Answer

Explanation

Q24:

26 May Shift 2

Geometry > Trigonometry

Medium

Core

Arrange the principal values of the following functions in ascending order (A) $\cosec^{-1}(2)$ (B) $\tan^{-1}(-\sqrt{3})$ (C) $\tan^{-1}(1)$ (D) $\tan^{-1}\left(\cos\frac{3\pi}{7}\right)$ Choose the correct answer from the options given below:

(A), (B), (C), (D)

(A), (D), (B), (C)

(B), (D), (A), (C)

(D), (A), (B), (C)

Correct Answer

Option 3

Correct Answer

Explanation

Q25:

22 May Shift 2

Geometry > Trigonometry

Medium

Core

The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ is

$\frac{25}{24}$

$\frac{24}{25}$

$\frac{7}{25}$

$\frac{7}{24}$

Correct Answer

Option 4

Correct Answer

Explanation

Q26:

22 May Shift 1

Geometry > Trigonometry

Medium

Core

The value of $\tan^2(\sec^{-1} 2) + \cot^2(\cosec^{-1} 3)$ is equal to

6

11

13

15

Correct Answer

Option 2

Correct Answer

Explanation

Q27:

22 May Shift 1

Geometry > Trigonometry

Medium

Core

The minimum value of the function $f(x) = 3\sin x - 4\cos x, x \in [-4\pi, 4\pi]$ is equal to

0

-4

-5

-7

Correct Answer

Option 3

Correct Answer

Explanation

Q28:

21 May Shift 2

Geometry > Trigonometry

Medium

Core

$\sin^{-1}(\cos\frac{3\pi}{5})$ equals

$\frac{3\pi}{5}$

$\frac{-\pi}{10}$

$\frac{\pi}{10}$

$\frac{-3\pi}{10}$

Correct Answer

Option 2

Correct Answer

Explanation

Q29:

21 May Shift 1

Geometry > Trigonometry

Medium

Core

Match List-I with List-II | List-I | List-II | |---|---| | (Inverse Trigonometric Function) | (Principal Value) | | (A) $\sin^{-1}(-\frac{1}{2})$ | (I) ${\pi}/{6}$ | | (B) $\cos^{-1}(-\frac{1}{2})$ | (II) $-{\pi}/{6}$ | | (C) $\tan^{-1}(-\sqrt{3})$ | (III) ${2\pi}/{3}$ | | (D) $\cot^{-1}(\sqrt{3})$ | (IV) $-{\pi}/{3}$ | Choose the correct answer from the options given below:

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

Correct Answer

Option 3

Correct Answer

Explanation

Q30:

19 May Shift 1

Geometry > Trigonometry

Easy

Common

The maximum value of $\sin x \cdot \cos x$ is:

1

$\frac{1}{2}$

$\frac{1}{4}$

$\sqrt{2}$

Correct Answer

Option 2

Correct Answer

Explanation

Q31:

19 May Shift 1

Geometry > Trigonometry

Medium

Core

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\cos^{-1} x + \cos^{-1}(-x)$ | (I) $\frac{\pi}{3}$ | | (B) $\text{cosec}^{-1}(-x) + \sec^{-1}(-x)$ | (II) $-\frac{\pi}{3}$ | | (C) $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | (III) $\pi$ | | (D) $\tan^{-1}\left(\tan\frac{4\pi}{3}\right)$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (III), (B) - (II), (C) - (IV), (D) - (I)

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

Correct Answer

Option 2

Correct Answer

Explanation

Q32:

16 May Shift 1

Geometry > Trigonometry

Medium

Core

Match List-I with List-II | List-I | List-II | |---|---| | Inverse Trigonometric function | Principal values of arguments | | (A) $sin^{-1}\left(\frac{-1}{2}\right)$ | (I) $\frac{-\pi}{3}$ | | (B) $cos^{-1}\left(\frac{-1}{2}\right)$ | (II) $\frac{3\pi}{4}$ | | (C) $tan^{-1}(-\sqrt{3})$ | (III) $\frac{-\pi}{6}$ | | (D) $sec^{-1}(-\sqrt{2})$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below:

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (IV), (B) - (III), (C) - (III), (D) - (I)

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

Correct Answer

Option 1

Correct Answer

Explanation

Q33:

15 May Shift 1

Geometry > Trigonometry

Medium

Core

For the principal value branch, the value of $\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$ is

$\frac{1}{\sqrt{2}}$

$\frac{\sqrt{3}}{2}$

$\frac{1}{2}$

$-\frac{\sqrt{3}}{2}$

Correct Answer

Option 3

Correct Answer

Explanation

Q34:

14 May Shift 2

Geometry > Trigonometry

Medium

Core

For $x \in [-1,1]$, if $4\sin^{-1}x + \cos^{-1}x = \pi$ then $x$ is equal to

$-\frac{1}{2}$

$\frac{1}{2}$

$\frac{\sqrt{3}}{2}$

$\frac{-\sqrt{3}}{2}$

Correct Answer

Option 2

Correct Answer

Explanation

Q35:

14 May Shift 1

Geometry > Trigonometry

Medium

Core

for $|x| < 1$, $sin (tan^{-1}x)$ equal to

$\frac{1}{\sqrt{1 + x^2}}$

$\frac{1}{\sqrt{1 - x^2}}$

$\frac{x}{\sqrt{1 - x^2}}$

$\frac{x}{\sqrt{1 + x^2}}$

Correct Answer

Option 4

Correct Answer

Explanation

Q36:

13 May Shift 2

Geometry > Trigonometry

Medium

Core

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\sin^{-1}(-1)$ | (I) $\frac{5\pi}{6}$ | | (B) $\cot^{-1}(-1)$ | (II) $\frac{-\pi}{2}$ | | (C) $\sec^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ | (III) $\frac{\pi}{4}$ | | (D) $\tan^{-1}(1)$ | (IV) $\frac{3\pi}{4}$ | Choose the correct answer from the options given below:

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

(A) - (II), (B) - (III), (C) - (I), (D) - (IV)

(A) - (I), (B) - (IV), (C) - (II), (D) - (III)

(A) - (I), (B) - (III), (C) - (II), (D) - (IV)

Correct Answer

Option 1

Correct Answer

Explanation

Q37:

13 May Shift 1

Geometry > Trigonometry

Medium

Core

The value of $\tan^{-1}(1) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)$ is

$\frac{5\pi}{4}$

$\frac{\pi}{4}$

$\frac{5\pi}{12}$

$\frac{7\pi}{12}$

Correct Answer

Option 1

Correct Answer

Explanation

Q38:

16 May Shift 1

Geometry > Trigonometry

Medium

Core

If $\tan ^{-1}\left(\frac{2}{3^{-x}+1}\right)=\cot ^{-1}\left(\frac{3}{3^{x}+1}\right)$, then which one of the following is true ?

There is no real value of x satisfying the above equation.

There is one positive and one negative real value of $x$ satisfying the above equation.

There are two real positive values of x satisfying the above equation.

There are two real negative values of $x$ satisfying the above equation.

Correct Answer

Option 2

Correct Answer

Explanation

Q39:

23 May Shift 3

Geometry > Trigonometry

Easy

The principal value of $\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

$\frac{\pi}{3}$

$-\frac{\pi}{6}$

$\frac{1}{\sqrt{3}}$

$\frac{2\pi}{3}$

Correct Answer

Option 4

Correct Answer

Explanation

Q40:

23 May Shift 3

Geometry > Trigonometry

Medium

The simplest form of $\tan^{-1}\left\{\frac{x}{\sqrt{a^2 - x^2}}\right\}$ is, where $-a < x < a$.

$\tan^{-1}\frac{x}{a}$

$\tan^{-1}(ax)$

$a \tan^{-1}\frac{x}{a}$

$\sin^{-1}\frac{x}{a}$

Correct Answer

Option 4

Correct Answer

Explanation

Q41:

23 May Shift 3

Geometry > Trigonometry

Easy

The maximum value of $\sin x + \cos x, x \in R$ is:

2

$\sqrt{2}$

$\frac{1}{\sqrt{2}}$

Not known

Correct Answer

Option 2

Correct Answer

Explanation

Q42:

23 May Shift 3

Geometry > Trigonometry

Medium

The simplest form of $\tan^{-1} \frac{\sqrt{1+x^2} - 1}{x}$, $x \neq 0$ is:

$\tan^{-1} x$

$x$

$\frac{1}{2} \tan^{-1} x$

$\sqrt{1+x^2}$

Correct Answer

Option 3

Correct Answer

Explanation

Q43:

22 May Shift 3

Geometry > Trigonometry

Easy

The principal value of $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ is :

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

$\frac{2\pi}{3}$

$\frac{\pi}{3}$

$-\frac{\pi}{3}$

Correct Answer

Option 2

Correct Answer

Explanation

Q44:

30 May Shift 3

Geometry > Trigonometry

Easy

If $\sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3}$, then the value of $\cos^{-1} x + \cos^{-1} y$ is :

$\frac{\pi}{2}$

$\frac{\pi}{3}$

$\frac{2\pi}{3}$

$\frac{\pi}{6}$

Correct Answer

Option 2

Correct Answer

Explanation

Q45:

30 May Shift 3

Geometry > Trigonometry

Easy

The maximum value of $(\sin x)(\cos x)$ is :

1

$\frac{1}{2}$

$\frac{1}{4}$

$\sqrt{2}$

Correct Answer

Option 2

Correct Answer

Explanation

Q46:

30 May Shift 3

Geometry > Trigonometry

Medium

Let $a \leq \tan^{-1} x + \cot^{-1} x + \sin^{-1} x \leq b$. If $\alpha$ and $\beta$ denote the minimum and maximum possible values of a and b respectively, then :

$\alpha = 0, \beta = \pi$

$\alpha = 0, \beta = \frac{\pi}{2}$

$\alpha = \frac{\pi}{2}, \beta = \pi$

$\alpha = -\frac{\pi}{2}, \beta = \frac{\pi}{2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q47:

15 June Shift 2

Geometry > Trigonometry

Easy

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) Range of $y = \text{cosec}^{-1}x$ | (I) $R - (-1, 1)$ | | (B) Domain of $\sec^{-1}x$ | (II) $(0, \pi)$ | | (C) Domain of $\sin^{-1}x$ | (III) $[-1, 1]$ | | (D) Range of $y = \cot^{-1}x$ | (IV) $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | Choose the correct answer from the options given below :

(A)-(I), (B)-(II), (C)-(IV), (D)-(III)

(A)-(IV), (B)-(I), (C)-(III), (D)-(II)

(A)-(III), (B)-(IV), (C)-(II), (D)-(I)

(A)-(II), (B)-(III), (C)-(I), (D)-(IV)

Correct Answer

Option 2

Correct Answer

Explanation

Q48:

15 June Shift 2

Geometry > Trigonometry

Medium

Let $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Then $y$ is :

$\frac{3x - x^3}{1 - 3x^2}$

$\frac{3x + x^3}{1 - 3x^2}$

$\frac{3x - x^3}{1 + 3x^2}$

$\frac{3x + x^3}{1 + 3x^2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q49:

17 Aug Shift 2

Geometry > Trigonometry

Easy

The value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is :

$\frac{1}{2}$

$\frac{1}{\sqrt{2}}$

$\frac{\sqrt{3}}{2}$

$-\frac{\sqrt{3}}{2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q50:

17 Aug Shift 2

Geometry > Trigonometry

Medium

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | (I) $\frac{3\pi}{4}$ | | (B) $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ | (II) $-\frac{\pi}{3}$ | | (C) $\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)$ | (III) $\frac{\pi}{2}$ | | (D) $\cos^{-1}\left(\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below :

(A) - (IV), (B) - (I), (C) - (III), (D) - (II)

(A) - (III), (B) - (II), (C) - (IV), (D) - (I)

(A) - (I), (B) - (III), (C) - (IV), (D) - (II)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

Correct Answer

Option 4

Correct Answer

Explanation

Q51:

6 Aug Shift 2

Geometry > Trigonometry

Medium

The value of $\cos^{-1} \left( \sin \left( \cos^{-1} \frac{1}{2} \right) \right) + \tan^{-1}(1)$

$\frac{\pi}{6}$

$\frac{\pi}{3}$

$\frac{\pi}{2}$

$\frac{5\pi}{12}$

Correct Answer

Option 4

Correct Answer

Explanation

Q52:

6 Aug Shift 2

Geometry > Trigonometry

Easy

The Principal value of $\cos^{-1} \left( -\frac{1}{2} \right)$ is:

$\frac{\pi}{3}$

$\frac{2\pi}{3}$

$\frac{5\pi}{6}$

$\frac{3\pi}{4}$

Correct Answer

Option 2

Correct Answer

Explanation

Q53:

4 Aug Shift 1

Geometry > Trigonometry

Medium

Match List - I with List - II | List - I | List - II | |---|---| | A. $\tan^{-1}\left(\tan\frac{7\pi}{6}\right)$ | I. $\frac{5\pi}{6}$ | | B. $\tan^{-1}\left(\tan\frac{8\pi}{6}\right)$ | II. $\frac{\pi}{2}$ | | C. $\tan^{-1}\frac{1}{\sqrt{3}} + cosec^{-1}\frac{2}{\sqrt{3}}$ | III. $\frac{\pi}{6}$ | | D. $\cos^{-1}\left(\cos\frac{5\pi}{6}\right)$ | IV. $\frac{\pi}{3}$ | Choose the correct answer from the options given below:

A-III, B-II, C-IV, D-I

A-I, B-II, C-IV, D-III

A-III, B-IV, C-II, D-I

A-I, B-II, C-III, D-IV

Correct Answer

Option 3

Correct Answer

Explanation

Q54:

4 Aug Shift 1

Geometry > Trigonometry

Easy

The value of $\sec^2(\tan^{-1}2) + cosec^2(\cot^{-1}3)$ is:

9

5

10

15

Correct Answer

Option 4

Correct Answer

Explanation

Q55:

10 Aug Shift 1

Geometry > Trigonometry

Easy

Two men on either side of a pole of 30 m high observe its top at angles of elevation $\alpha$ and $\beta$ respectively. The distance between the two men is $40\sqrt{3}$ m and the distance between the first man at A and the pole is $30\sqrt{3}$ m. <img src="https://balti.afterboards.in/XTDaiKJaKRDRB95" width="300px"/> Based on the above information, answer the question:

The value of $\alpha$ is

$\frac{\pi}{3}$

$\frac{\pi}{4}$

$\frac{\pi}{6}$

$\frac{\pi}{8}$

Correct Answer

Option 3

Correct Answer

Explanation

Q56:

10 Aug Shift 1

Geometry > Trigonometry

Easy

Two men on either side of a pole of 30 m high observe its top at angles of elevation $\alpha$ and $\beta$ respectively. The distance between the two men is $40\sqrt{3}$ m and the distance between the first man at A and the pole is $30\sqrt{3}$ m. <img src="https://balti.afterboards.in/XTDaiKJaKRDRB95" width="300px"/> Based on the above information, answer the question:

$\tan(\alpha + \beta) =$

$\sqrt{3} + \frac{1}{\sqrt{3}}$

$\sqrt{3} - \frac{1}{\sqrt{3}}$

$2\sqrt{3} - 1$

Infinite (does not exist)

Correct Answer

Option 4

Correct Answer

Explanation

Q57:

10 Aug Shift 1

Geometry > Trigonometry

Easy

Two men on either side of a pole of 30 m high observe its top at angles of elevation $\alpha$ and $\beta$ respectively. The distance between the two men is $40\sqrt{3}$ m and the distance between the first man at A and the pole is $30\sqrt{3}$ m. <img src="https://balti.afterboards.in/XTDaiKJaKRDRB95" width="300px"/> Based on the above information, answer the question:

Area of $\triangle ABD$ is

$150\sqrt{3} \, m^2$

$900\sqrt{3} \, m^2$

$450\sqrt{3} \, m^2$

$100\sqrt{3} \, m^2$

Correct Answer

Option 3

Correct Answer

Explanation

Q58:

10 Aug Shift 1

Geometry > Trigonometry

Easy

Two men on either side of a pole of 30 m high observe its top at angles of elevation $\alpha$ and $\beta$ respectively. The distance between the two men is $40\sqrt{3}$ m and the distance between the first man at A and the pole is $30\sqrt{3}$ m. <img src="https://balti.afterboards.in/XTDaiKJaKRDRB95" width="300px"/> Based on the above information, answer the question:

$\angle ABC$ is equal to:

$\frac{\pi}{4}$

$\frac{\pi}{6}$

$\frac{\pi}{2}$

$\frac{\pi}{3}$

Correct Answer

Option 3

Correct Answer

Explanation

Q59:

10 Aug Shift 1

Geometry > Trigonometry

Medium

Two men on either side of a pole of 30 m high observe its top at angles of elevation $\alpha$ and $\beta$ respectively. The distance between the two men is $40\sqrt{3}$ m and the distance between the first man at A and the pole is $30\sqrt{3}$ m. <img src="https://balti.afterboards.in/XTDaiKJaKRDRB95" width="300px"/> Based on the above information, answer the question:

The value of $\frac{1}{AB^2} + \frac{1}{BC^2}$ is:

$\frac{1}{BD^2}$

$\frac{1}{AD^2}$

$\frac{1}{CD^2}$

$\frac{1}{AC^2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q60:

30 Aug Shift 1

Geometry > Trigonometry

Medium

$\sin^{-1}(1 - x) - 2\sin^{-1}x = \frac{\pi}{2}$, than $x$ is equal to (a) $0$ (b) $1$ (c) $\frac{1}{2}$ (d) $2$ Choose the most appropriate answer from the options given below:

(a) and (b) only

(a) and (c) only

(a) only

(c) only

Correct Answer

Option 3

Correct Answer

Explanation

Q61:

30 Aug Shift 1

Geometry > Trigonometry

Medium

The value of $\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]$ is :

$\frac{120}{169}$

$\frac{-120}{169}$

$\frac{-60}{169}$

$\frac{60}{169}$

Correct Answer

Option 2

Correct Answer

Explanation

Q62:

16 July Shift 2

Geometry > Trigonometry

Easy

Match List I with List II | List I (Functions) | List II (Principal value branches) | |---|---| | A. $f(x) = \cos^{-1} x$ | I. $[0, \pi]$ | | B. $f(x) = \tan^{-1} x$ | II. $[0, \pi] - \left\{\frac{\pi}{2}\right\}$ | | C. $f(x) = \text{cosec}^{-1} x$ | III. $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | | D. $f(x) = \sec^{-1} x$ | IV. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ | Choose the correct answer from the options given below:

A-I, B-II, C-III, D-IV

A-I, B-IV, C-III, D-II

A-IV, B-III, C-II, D-I

A-III, B-IV, C-I, D-II

Correct Answer

Option 2

Correct Answer

Explanation

Q63:

16 July Shift 2

Geometry > Trigonometry

Easy

The graph of $\sin^{-1} x$ is represented by:

Correct Answer

Option 1

Correct Answer

Explanation

Q64:

16 July Shift 2

Geometry > Trigonometry

Medium

The value of $\text{cosec}^{-1}(-2) - 2\sec^{-1}(-2)$ is equal to:

$\frac{-\pi}{6}$

$\frac{2\pi}{3}$

$\frac{3\pi}{2}$

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

Correct Answer

Option 4

Correct Answer

Explanation

Q65:

23 Aug Shift 1

Geometry > Trigonometry

Medium

$\sin^{-1}(\cos x) = \frac{\pi}{2} - x$ is valid for :

$-\pi \leq x \leq 0$

$0 \leq x \leq \pi$

$-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

$-\frac{\pi}{4} \leq x \leq \frac{3\pi}{4}$

Correct Answer

Option 2

Correct Answer

Explanation

Q66:

25 May Shift 1

Geometry > Trigonometry

Medium

Match List I with List II | LIST I | LIST II | |---|---| | A. $\sin^{-1} x + \cos^{-1} x, x \in [-1,1]$ | I. $-\frac{\pi}{2}$ | | B. $\tan^{-1} \sqrt{3} - \cot^{-1}(-\sqrt{3})$ | II. $-\frac{\pi}{6}$ | | C. $\cos^{-1}\left(\cos\frac{13\pi}{6}\right)$ | III. $\frac{\pi}{2}$ | | D. $\sin^{-1}\left(-\frac{1}{2}\right)$ | IV. $\frac{\pi}{6}$ | Choose the correct answer from the options given below:

A-III, B-I, C-IV, D-II

A-IV, B-I, C-II, D-III

A-II, B-III, C-IV, D-I

A-I, B-II, C-III, D-IV

Correct Answer

Option 1

Correct Answer

Explanation

Q67:

25 May Shift 1

Geometry > Trigonometry

Easy

Value of $\frac{e^{\sin(\tan^{-1} x + \cot^{-1} x)}}{e^{\sin(\sin^{-1} x + \cos^{-1} x)}}, x \in [-1, 1]$, is:

$0$

$\frac{\pi}{2}$

$1$

$-\frac{\pi}{2}$

Correct Answer

Option 3

Correct Answer

Explanation

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