Question 1: Palette 1/58

Objective Physics Vol-1 MCQ

If $\sin \theta = \frac{4}{5}$, where $\theta$ lies in the first quadrant, then find all the other T-ratios.

A

$\cos \theta = \frac{3}{5}$, $\tan \theta = \frac{4}{3}$, $\cot \theta = \frac{3}{4}$, $\sec \theta = \frac{5}{3}$, $\csc \theta = \frac{5}{4}$

B

$\cos \theta = \frac{4}{5}$, $\tan \theta = \frac{3}{4}$, $\cot \theta = \frac{4}{3}$, $\sec \theta = \frac{5}{4}$, $\csc \theta = \frac{5}{3}$

C

$\cos \theta = \frac{3}{4}$, $\tan \theta = \frac{5}{3}$, $\cot \theta = \frac{3}{5}$, $\sec \theta = \frac{4}{3}$, $\csc \theta = \frac{5}{4}$

D

$\cos \theta = \frac{3}{5}$, $\tan \theta = \frac{3}{4}$, $\cot \theta = \frac{4}{3}$, $\sec \theta = \frac{5}{4}$, $\csc \theta = \frac{5}{3}$

Question 2: Palette 2/58

Objective Physics Vol-1 MCQ

Find the values of:
(i) $\sin(-45^\circ)$
(ii) $\tan 225^\circ$
(iii) $\cos 300^\circ$
(iv) $\sec 120^\circ$

A

(i) $\frac{1}{\sqrt{2}}$, (ii) $-1$, (iii) $\frac{\sqrt{3}}{2}$, (iv) $2$

B

(i) $-\frac{1}{\sqrt{2}}$, (ii) $1$, (iii) $\frac{1}{2}$, (iv) $-2$

C

(i) $-\frac{1}{2}$, (ii) $1$, (iii) $\frac{1}{\sqrt{2}}$, (iv) $-2$

D

(i) $-\frac{1}{\sqrt{2}}$, (ii) $-1$, (iii) $\frac{1}{2}$, (iv) $2$

Question 3: Palette 3/58

Objective Physics Vol-1 MCQ

Find the value of:
(i) $\sin 15^\circ$
(ii) $\tan 75^\circ$

A

(i) $\frac{\sqrt{3}+1}{2\sqrt{2}}$, (ii) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

B

(i) $\frac{\sqrt{3}-1}{2\sqrt{2}}$, (ii) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

C

(i) $\frac{\sqrt{3}-1}{2}$, (ii) $\frac{\sqrt{3}+1}{\sqrt{3}}$

D

(i) $\frac{1}{2\sqrt{2}}$, (ii) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

Question 4: Palette 4/58

Objective Physics Vol-1 MCQ

Find $\sin \theta$ and $\tan \theta$, if $\cos \theta = -\frac{12}{13}$ and $\theta$ lies in the third quadrant.

A

$\sin \theta = \frac{5}{13}$, $\tan \theta = -\frac{5}{12}$

B

$\sin \theta = -\frac{5}{13}$, $\tan \theta = -\frac{5}{12}$

C

$\sin \theta = -\frac{5}{13}$, $\tan \theta = \frac{5}{12}$

D

$\sin \theta = \frac{5}{13}$, $\tan \theta = \frac{5}{12}$

Question 5: Palette 5/58

Objective Physics Vol-1 MCQ

Find the values of other five T-ratios, if $\tan \theta = -\frac{3}{4}$ and $\theta$ lies in II quadrant.

A

$\sin \theta = \frac{3}{5}$, $\cos \theta = -\frac{4}{5}$, $\csc \theta = \frac{5}{3}$, $\sec \theta = -\frac{5}{4}$, $\cot \theta = -\frac{4}{3}$

B

$\sin \theta = -\frac{3}{5}$, $\cos \theta = \frac{4}{5}$, $\csc \theta = -\frac{5}{3}$, $\sec \theta = \frac{5}{4}$, $\cot \theta = -\frac{4}{3}$

C

$\sin \theta = \frac{4}{5}$, $\cos \theta = -\frac{3}{5}$, $\csc \theta = \frac{5}{4}$, $\sec \theta = -\frac{5}{3}$, $\cot \theta = -\frac{3}{4}$

D

$\sin \theta = \frac{3}{5}$, $\cos \theta = \frac{4}{5}$, $\csc \theta = \frac{5}{3}$, $\sec \theta = \frac{5}{4}$, $\cot \theta = \frac{4}{3}$

Question 6: Palette 6/58

Objective Physics Vol-1 MCQ

Find the values of the following T-ratios:
(i) $\csc 315^\circ$
(ii) $\cos 210^\circ$
(iii) $\sin(-330^\circ)$

A

(i) $\sqrt{2}$, (ii) $\frac{\sqrt{3}}{2}$, (iii) $-\frac{1}{2}$

B

(i) $-\sqrt{2}$, (ii) $-\frac{\sqrt{3}}{2}$, (iii) $\frac{1}{2}$

C

(i) $-\frac{1}{\sqrt{2}}$, (ii) $-\frac{1}{2}$, (iii) $\frac{\sqrt{3}}{2}$

D

(i) $-\sqrt{2}$, (ii) $\frac{1}{2}$, (iii) $-\frac{\sqrt{3}}{2}$

Question 7: Palette 7/58

Objective Physics Vol-1 MCQ

Find the value of:
(i) $\sec 165^\circ$
(ii) $\cot 105^\circ$

A

(i) $\sqrt{2} + \sqrt{6}$, (ii) $\frac{1+\sqrt{3}}{1-\sqrt{3}}$

B

(i) $\sqrt{2} - \sqrt{6}$, (ii) $\frac{1-\sqrt{3}}{1+\sqrt{3}}$

C

(i) $\sqrt{6} - \sqrt{2}$, (ii) $\frac{\sqrt{3}-1}{1+\sqrt{3}}$

D

(i) $-\sqrt{2} - \sqrt{6}$, (ii) $\frac{\sqrt{3}+1}{1-\sqrt{3}}$

Question 8: Palette 8/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = x^{-3}$ with respect to $x$.

A

$-3x^{-4}$

B

$-3x^{-2}$

C

$3x^{-4}$

D

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

Question 9: Palette 9/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = 6x^5 + 4x^3 - 3x^2 + 2x - 7$ with respect to $x$.

A

$30x^5 + 12x^3 - 6x^2 + 2x$

B

$30x^4 + 12x^2 - 6x + 2$

C

$30x^4 + 12x^2 - 6x$

D

$24x^4 + 12x^2 - 6x + 2$

Question 10: Palette 10/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = (x + 2)(x^2 + 1)$ with respect to $x$.

A

$3x^2 + 2x + 1$

B

$2x^2 + 4x + 1$

C

$3x^2 + 4x + 1$

D

$3x^2 + 4x$

Question 11: Palette 11/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = \frac{x^3 + 4}{x + 1}$ with respect to $x$.

A

$\frac{3x^3 + 3x^2 - 4}{(x + 1)^2}$

B

$\frac{2x^3 + 3x^2 + 4}{(x + 1)^2}$

C

$\frac{x^3 + 3x^2 - 4}{(x + 1)^2}$

D

$\frac{2x^3 + 3x^2 - 4}{(x + 1)^2}$

Question 12: Palette 12/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = \sin x + e^x$ with respect to $x$.

A

$\cos x + e^x$

B

$-\cos x + e^x$

C

$\cos x - e^x$

D

$\sin x + e^x$

Question 13: Palette 13/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = 3x^2 + \log x + 4e^x + 5$ with respect to $x$.

A

$6x + \frac{1}{x} + e^x$

B

$6x + \frac{1}{x} + 4e^x + 5$

C

$6x + \frac{1}{x} + 4e^x$

D

$3x + \frac{1}{x} + 4e^x$

Question 14: Palette 14/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = e^x \cdot \tan x$ with respect to $x$.

A

$e^x (\sec^2 x - \tan x)$

B

$e^x \sec^2 x$

C

$e^x (\tan x - \sec^2 x)$

D

$e^x (\sec^2 x + \tan x)$

Question 15: Palette 15/58

Objective Physics Vol-1 MCQ

Find the derivative of $y = \sin(x^2 + 5)$ with respect to $x$.

A

$2x \sin(x^2 + 5)$

B

$2x \cos(x^2 + 5)$

C

$\cos(x^2 + 5)$

D

$-2x \cos(x^2 + 5)$

Question 16: Palette 16/58

Objective Physics Vol-1 MCQ

Divide the number $1000$ into two parts such that their product is maximum.

A

$400$ and $600$

B

$500$ and $500$

C

$300$ and $700$

D

$200$ and $800$

Question 17: Palette 17/58

Objective Physics Vol-1 MCQ

The displacement of a particle as a function of time $t$ is given by $s = \alpha + \beta t + \gamma t^2 + \delta t^4$, where $\alpha, \beta, \gamma$ and $\delta$ are constants. Find the ratio of the initial velocity to the initial acceleration.

A

$\frac{\beta}{2\gamma}$

B

$\frac{2\gamma}{\beta}$

C

$\frac{\beta}{\gamma}$

D

$\frac{\alpha}{2\beta}$

Question 18: Palette 18/58

Objective Physics Vol-1 MCQ

The position of a particle moving along the X-axis varies with time $t$ as $x = 6t - t^2 + 4$. Find the time interval during which the particle is moving along the positive x-direction.

A

$t = 0$ to $t = 6\text{ s}$

B

$t = 0$ to $t = 3\text{ s}$

C

$t = 3\text{ s}$ to $t = 6\text{ s}$

D

$t > 3\text{ s}$

Question 19: Palette 19/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = 3x^4 + 2\frac{1}{x^2} + \log x$ with respect to $x$.

A

$12x^3 - \frac{4}{x^3} + \frac{1}{x}$

B

$12x^3 + \frac{4}{x^3} + \frac{1}{x}$

C

$12x^3 - \frac{2}{x^3} + \frac{1}{x}$

D

$3x^3 - \frac{4}{x^3} + \log x$

Question 20: Palette 20/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = (x^2 + 1)(x + 2)$ with respect to $x$.

A

$3x^2 + 2x + 1$

B

$3x^2 + 4x + 1$

C

$2x^2 + 4x + 1$

D

$3x^2 + 4x$

Question 21: Palette 21/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = \frac{3x^2}{x + 1}$ with respect to $x$.

A

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

B

$\frac{6x^2 + 6x}{(x + 1)^2}$

C

$\frac{3x^2 + 6x}{(x + 1)^2}$

D

$\frac{3x^2 - 6x}{(x + 1)^2}$

Question 22: Palette 22/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = \sin x$ with respect to $x$.

A

$-\cos x$

B

$\cos x$

C

$\tan x$

D

$-\sin x$

Question 23: Palette 23/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = \tan(x^2 + 3x + 1)$ with respect to $x$.

A

$(2x + 3) \sec^2(x^2 + 3x + 1)$

B

$\sec^2(x^2 + 3x + 1)$

C

$(2x + 3) \tan(x^2 + 3x + 1)$

D

$(x + 3) \sec^2(x^2 + 3x + 1)$

Question 24: Palette 24/58

Objective Physics Vol-1 MCQ

Differentiate the function $y = 3\cos x + 7e^x + \log(x^2 + 1)$ with respect to $x$.

A

$3\sin x + 7e^x + \frac{2x}{x^2 + 1}$

B

$-3\sin x + 7e^x + \frac{1}{x^2 + 1}$

C

$-3\sin x + 7e^x + \frac{2x}{x^2 + 1}$

D

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

Question 25: Palette 25/58

Objective Physics Vol-1 MCQ

A particle is moving with velocity $v = t^3 - 6t^2 + 4$, where $v$ is in $\text{m/s}$ and $t$ is in seconds. At what time will the velocity be maximum/minimum and what is it equal to?

A

$v_{\text{max}} = 4\text{ m/s}$ at $t = 4\text{ s}$ and $v_{\text{min}} = -28\text{ m/s}$ at $t = 0\text{ s}$

B

$v_{\text{max}} = 4\text{ m/s}$ at $t = 0\text{ s}$ and $v_{\text{min}} = -28\text{ m/s}$ at $t = 4\text{ s}$

C

$v_{\text{max}} = 6\text{ m/s}$ at $t = 0\text{ s}$ and $v_{\text{min}} = -20\text{ m/s}$ at $t = 4\text{ s}$

D

$v_{\text{max}} = 4\text{ m/s}$ at $t = 2\text{ s}$ and $v_{\text{min}} = -16\text{ m/s}$ at $t = 4\text{ s}$

Question 26: Palette 26/58

Objective Physics Vol-1 MCQ

If the time $t$ and displacement $x$ of a particle moving along the positive X-axis are related as $t = (x^2 - 1)^{\frac{1}{2}}$, then find the acceleration of the particle in terms of $x$.

A

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

B

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

C

$\frac{1}{x^4}$

D

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

Question 27: Palette 27/58

Objective Physics Vol-1 MCQ

Evaluate the integral $\int \left( e^x + \frac{1}{x} + 2x^2 + 3 \right) dx$.

A

$e^x + \log_e x + \frac{2}{3}x^3 + 3x + C$

B

$e^x - \log_e x + \frac{2}{3}x^3 + 3x + C$

C

$e^x + \log_e x + 2x^3 + 3x + C$

D

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

Question 28: Palette 28/58

Objective Physics Vol-1 MCQ

Evaluate the integral $\int \left( \cos x + 3x^{1/2} + \frac{3}{x} + \frac{4}{x^2} \right) dx$.

A

$\sin x + 2x^{3/2} + 3\log_e x + \frac{4}{x} + C$

B

$\sin x + 3x^{3/2} + 3\log_e x - \frac{4}{x} + C$

C

$\sin x + 2x^{3/2} + 3\log_e x - \frac{4}{x} + C$

D

$-\sin x + 2x^{3/2} + 3\log_e x - \frac{4}{x} + C$

Question 29: Palette 29/58

Objective Physics Vol-1 MCQ

Evaluate the integral $\int (2x + 1)^3 dx$.

A

$\frac{(2x + 1)^4}{4} + C$

B

$\frac{(2x + 1)^4}{8} + C$

C

$\frac{(2x + 1)^3}{6} + C$

D

$\frac{(2x + 1)^4}{2} + C$

Question 30: Palette 30/58

Objective Physics Vol-1 MCQ

Evaluate the integral $\int \left( \frac{1}{a - x} \right) dx$.

A

$\log_e(a - x) + C$

B

$-\frac{1}{(a - x)^2} + C$

C

$-\log_e(a - x) + C$

D

$\frac{1}{a}\log_e(a - x) + C$

Question 31: Palette 31/58

Objective Physics Vol-1 MCQ

Evaluate $\int (x^2 + 3x + 4)^4 dx$

A

$\frac{(x^2 + 3x + 4)^5}{5} + C$

B

$\frac{(x^2 + 3x + 4)^5}{2x + 3} + C$

C

$\frac{(x^2 + 3x + 4)^5}{5(2x + 3)} + C$

D

$\frac{(x^2 + 3x + 4)^4}{5(2x + 3)} + C$

Question 32: Palette 32/58

Objective Physics Vol-1 MCQ

Evaluate $\int \sin(2x^2) dx$

A

$\frac{-\cos(2x^2)}{4x} + C$

B

$\frac{\cos(2x^2)}{4x} + C$

C

$\frac{-\sin(2x^2)}{4x} + C$

D

$\frac{-\cos(2x^2)}{2x} + C$

Question 33: Palette 33/58

Objective Physics Vol-1 MCQ

Evaluate $\int_0^2 (4x^3 + 2x^2 + 2x + 1) dx$

A

$\frac{80}{3}$

B

$\frac{82}{3}$

C

$\frac{76}{3}$

D

$\frac{85}{3}$

Question 34: Palette 34/58

Objective Physics Vol-1 MCQ

Evaluate $\int_0^{\pi/4} (\sin x + \cos x) dx$

A

$\sqrt{2} - 1$

B

$\sqrt{2} + 1$

C

$1$

D

$0$

Question 35: Palette 35/58

Objective Physics Vol-1 MCQ

Evaluate $\int_2^4 \frac{dx}{x}$

A

$\log_e 4$

B

$\log_e 3$

C

$\log_e 2$

D

$\log_e 8$

Question 36: Palette 36/58

Objective Physics Vol-1 MCQ

Evaluate $\int_1^2 e^{(x + 4)} dx$

A

$e^5 (e - 1)$

B

$e^4 (e - 1)$

C

$e^6 - e^4$

D

$e^5 (e + 1)$

Question 37: Palette 37/58

Objective Physics Vol-1 MCQ

Evaluate $\int_0^{\pi/4} \cos(2x^2 + x) dx$

A

$\frac{1}{\pi + 1} \sin\left( \frac{\pi^2 + 8\pi}{16} \right)$

B

$\frac{1}{\pi + 1} \sin\left( \frac{\pi^2 + 2\pi}{8} \right)$

C

$\frac{1}{\pi + 1} \sin\left( \frac{\pi^2 + 4\pi}{8} \right)$

D

$\frac{1}{\pi + 2} \sin\left( \frac{\pi^2 + 2\pi}{8} \right)$

Question 38: Palette 38/58

Objective Physics Vol-1 MCQ

Evaluate $\int_1^2 \frac{dx}{3x + 4}$

A

$\frac{1}{3} \log_e\left(\frac{10}{7}\right)$

B

$\frac{1}{3} \log_e\left(\frac{7}{10}\right)$

C

$\log_e\left(\frac{10}{7}\right)$

D

$\frac{1}{4} \log_e\left(\frac{10}{7}\right)$

Question 39: Palette 39/58

Objective Physics Vol-1 MCQ

Evaluate $\int_0^4 (3x^2 + 4x + 5) dx$

A

$112$

B

$120$

C

$116$

D

$124$

Question 40: Palette 40/58

Objective Physics Vol-1 MCQ

A particle is moving under constant acceleration $a = 3t + 4t^2$. If the position and velocity of the particle at start (i.e. $t = 0$) are $x_0$ and $v_0$ respectively, find the displacement $x - x_0$ as a function of time $t$.

A

$v_0 t + t^3 + \frac{1}{3} t^4$

B

$v_0 t + \frac{3}{2} t^2 + \frac{1}{3} t^4$

C

$v_0 t + \frac{3}{2} t^2 + \frac{4}{3} t^3$

D

$v_0 t + \frac{1}{2} t^3 + \frac{1}{3} t^4$

Question 41: Palette 41/58

Objective Physics Vol-1 MCQ

The velocity of a particle is given by $v = v_0 \sin\omega t$, where $v_0$ is constant and $\omega = \frac{2\pi}{T}$. Find the average velocity in the time interval $t = 0$ to $t = \frac{T}{2}$.

A

$\frac{v_0}{\pi}$

B

$\frac{2v_0}{\pi}$

C

$\frac{v_0}{2\pi}$

D

$\frac{3v_0}{\pi}$

Question 42: Palette 42/58

Objective Physics Vol-1 MCQ

Evaluate $\int \left(\sin x + \frac{1}{x} + 2\frac{1}{x^2} + 3x^3\right) dx$

A

$-\cos x + \log_e x - \frac{2}{x} + \frac{3}{4} x^4 + C$

B

$\cos x + \log_e x - \frac{2}{x} + \frac{3}{4} x^4 + C$

C

$-\cos x - \log_e x + \frac{2}{x} + \frac{3}{4} x^4 + C$

D

$-\cos x + \log_e x + \frac{2}{x} + 3x^4 + C$

Question 43: Palette 43/58

Objective Physics Vol-1 MCQ

Evaluate $\int \left(3\cos x + e^x + 4x^2 + x + 5\right) dx$

A

$3\sin x + e^x + 4x^3 + \frac{x^2}{2} + 5x + C$

B

$-3\sin x + e^x + \frac{4}{3} x^3 + \frac{x^2}{2} + 5x + C$

C

$3\sin x + e^x + \frac{4}{3} x^3 + \frac{x^2}{2} + 5x + C$

D

$3\sin x + e^x + \frac{4}{3} x^3 + x^2 + 5x + C$

Question 44: Palette 44/58

Objective Physics Vol-1 MCQ

Evaluate $\int (2x^2 + 4x + 1)^2 dx$

A

$\frac{1}{12} \frac{(2x^2 + 4x + 1)^3}{(x + 1)} + C$

B

$\frac{4}{5} x^5 + 4x^4 + \frac{20}{3} x^3 + 4x^2 + x + C$

C

$\frac{1}{12} (2x^2 + 4x + 1)^3 + C$

D

$\frac{4}{5} x^5 + 2x^4 + \frac{20}{3} x^3 + 2x^2 + x + C$

Question 45: Palette 45/58

Objective Physics Vol-1 MCQ

Evaluate $\int \frac{1}{x^2 + 2} dx$

A

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

B

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

C

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

D

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

Question 46: Palette 46/58

Objective Physics Vol-1 MCQ

Evaluate $\int \cos(x + 2) dx$

A

$-\sin(x + 2) + C$

B

$\cos(x + 2) + C$

C

$\sin(x + 2) + C$

D

$\frac{\sin(x + 2)}{2} + C$

Question 47: Palette 47/58

Objective Physics Vol-1 MCQ

Evaluate $\int_1^3 \left(4x + \frac{1}{x} + 1\right) dx$

A

$16 + \log_e 3$

B

$18 + \log_e 3$

C

$18 - \log_e 3$

D

$20 + \log_e 3$

Question 48: Palette 48/58

Objective Physics Vol-1 MCQ

Evaluate $\int_0^{\pi/4} (\sin x - \cos x) dx$

A

$\sqrt{2} - 1$

B

$1 + \sqrt{2}$

C

$1 - \sqrt{2}$

D

$-\sqrt{2}$

Question 49: Palette 49/58

Objective Physics Vol-1 MCQ

Evaluate $\int_0^2 \frac{dx}{x^2 + 4x + 1}$

A

$\frac{1}{2\sqrt{3}} \log_e\left(\frac{13}{8}\right)$

B

$\frac{1}{\sqrt{3}} \log_e\left(\frac{13}{8}\right)$

C

$\frac{1}{2\sqrt{3}} \log_e 13$

D

$\frac{1}{2} \log_e\left(\frac{13}{8}\right)$

Question 50: Palette 50/58

Objective Physics Vol-1 MCQ

Evaluate $\int_1^3 (4x^3 + 3x^2 + 2x + 1) dx$

A

$112$

B

$116$

C

$120$

D

$124$

Question 51: Palette 51/58

Objective Physics Vol-1 MCQ

A particle is moving in a straight line under acceleration $a = kt$, where $k$ is a constant. Find the velocity in terms of $t$, if the motion starts from rest.

A

$kt^2$

B

$\frac{kt^2}{2}$

C

$\frac{kt^3}{3}$

D

$\frac{kt}{2}$

Question 52: Palette 52/58

Objective Physics Vol-1 MCQ

A particle is moving in a straight line such that its velocity varies as $v = v_0 e^{-\lambda t}$, where $\lambda$ is a constant. Find the average velocity during the time interval in which the velocity decreases from $v_0$ to $\frac{v_0}{2}$.

A

$\frac{v_0}{2 \log_e 2}$

B

$\frac{v_0}{\log_e 2}$

C

$\frac{v_0}{2 \lambda \log_e 2}$

D

$\frac{2 v_0}{\log_e 2}$

Question 53: Palette 53/58

Objective Physics Vol-1 MCQ

Find the area of the region in the first quadrant enclosed by the X-axis, the line $y = x$, and the circle $x^2 + y^2 = 32$.

A

$4\pi \text{ sq units}$

B

$2\pi \text{ sq units}$

C

$8\pi \text{ sq units}$

D

$16\pi \text{ sq units}$

Question 54: Palette 54/58

Objective Physics Vol-1 MCQ

Find the area of the region bounded by the curve $y^2 = 4x$ and the line $x = 4$.

A

$\frac{32}{3} \text{ sq units}$

B

$\frac{64}{3} \text{ sq units}$

C

$16 \text{ sq units}$

D

$\frac{128}{3} \text{ sq units}$

Question 55: Palette 55/58

Objective Physics Vol-1 MCQ

Find the area of the region bounded by the line $y = 3x + 2$, the X-axis and the ordinates $x = -1$ and $x = 1$.

A

$\frac{11}{3} \text{ sq units}$

B

$\frac{13}{3} \text{ sq units}$

C

$\frac{25}{6} \text{ sq units}$

D

$\frac{14}{3} \text{ sq units}$

Question 56: Palette 56/58

Objective Physics Vol-1 MCQ

Find the area of the region bounded by the curve $y = x^3$, $y = x + 6$, and $x = 0$.

A

$10 \text{ sq units}$

B

$8 \text{ sq units}$

C

$12 \text{ sq units}$

D

$14 \text{ sq units}$

Question 57: Palette 57/58

Objective Physics Vol-1 MCQ

Find the area of the region included between the parabola $y = \frac{3x^2}{4}$ and the line $3x - 2y + 12 = 0$.

A

18 sq units

B

27 sq units

C

36 sq units

D

45 sq units

Question 58: Palette 58/58

Objective Physics Vol-1 MCQ

For the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$, evaluate the area of the region under the curve and above the $X$-axis.

A

$\pi$ sq units

B

$2\pi$ sq units

C

$3\pi$ sq units

D

$6\pi$ sq units