Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance '$a$' from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$, is (A) $\frac{F}{2m}\frac{a}{\sqrt{a^2-x^2}}$ (B) $\frac{F}{2m}\frac{x}{\sqrt{a^2-x^2}}$ (C) $\frac{F}{2m}\frac{x}{a}$ (D) $\frac{F}{2m}\frac{\sqrt{a^2-x^2}}{x}$
STATEMENT-1 A block of mass $m$ starts moving on a rough horizontal surface with a velocity $v$. It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of $30^\circ$ with the horizontal and the same block is made to go up on the surface with the same initial velocity $v$. The decrease in the mechanical energy in the second situation is smaller than that in the first situation. because STATEMENT-2 The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
STATEMENT-1 In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision. because STATEMENT-2 In an elastic collision, the linear momentum of the system is conserved. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
Disc A has moment of inertia $I$ and angular velocity $\omega$. Disc B has moment of inertia $2I$ and is initially at rest. When disc B is brought in contact with disc A, they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is (A) $\frac{2I\omega}{3t}$ (B) $\frac{9I\omega}{2t}$ (C) $\frac{9I\omega}{4t}$ (D) $\frac{3I\omega}{2t}$
Disc A has moment of inertia $I$ and angular velocity $\omega$. Disc B has moment of inertia $2I$ and is initially at rest. When disc B is brought in contact with disc A, they acquire a common angular velocity. The loss of kinetic energy during the above process is (A) $\frac{I\omega^2}{2}$ (B) $\frac{I\omega^2}{3}$ (C) $\frac{I\omega^2}{4}$ (D) $\frac{I\omega^2}{6}$
Let $f(x) = 2 + \cos x$ for all real $x$. STATEMENT-1: For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f'(c) = 0$. because STATEMENT-2: $f(t) = f(t+2\pi)$ for each real $t$. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
Let $f(x) = x^x$ for $x > 0$. Then $f$ is (A) increasing on $(0, \infty)$ (B) decreasing on $(0, \infty)$ (C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$ (D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$
The number of distinct real values of $\lambda$ for which the system of linear equations $$x + y + z = 0$$ $$x + \lambda y + z = 0$$ $$x + y + \lambda z = 0$$ has a non-trivial solution is (A) 0 (B) 1 (C) 2 (D) 3
Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ is (A) $\frac{2}{9}(p-q)(2q-p)$ (B) $\frac{2}{9}(q-p)(2p-q)$ (C) $\frac{2}{9}(q-2p)(2q-p)$ (D) $\frac{2}{9}(2p-q)(2q-p)$
The number of solutions of the pair of equations $$2\sin^2\theta - \cos 2\theta = 0$$ $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $[0, 2\pi]$ is (A) 0 (B) 1 (C) 2 (D) 4
Let $O(0,0)$, $P(3,4)$, $Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR$, $PQR$, $OQR$ are of equal area. The coordinates of $R$ are (A) $\left(\frac{4}{3}, 3\right)$ (B) $(3, \frac{2}{3})$ (C) $(3, \frac{4}{3})$ (D) $\left(\frac{4}{3}, \frac{2}{3}\right)$
The tangent to the curve $y = e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$ (A) on the left of $x = c$ (B) on the right of $x = c$ (C) at no point (D) at all points
The number of distinct real values of $\lambda$ for which the vectors $-\lambda^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\lambda^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2\hat{k}$ are coplanar is (A) 0 (B) 1 (C) 2 (D) 3
If $|z| = 1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on (A) a line not passing through the origin (B) $|z| = \sqrt{2}$ (C) the $x$-axis (D) the $y$-axis
A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2 + 4y^2 = 12$. Then its equation is (A) $x^2\csc^2\theta - y^2\sec^2\theta = 1$ (B) $x^2\sec^2\theta - y^2\csc^2\theta = 1$ (C) $x^2\sin^2\theta - y^2\cos^2\theta = 1$ (D) $x^2\cos^2\theta - y^2\sin^2\theta = 1$
STATEMENT-1: The curve $y = \frac{-x^2}{2} + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2: A parabola is symmetric about its axis. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
Let $f: \{1,2,3,4\} \to \{1,2,3,4\}$ and $g: \{1,2,3,4\} \to \{1,2,3,4\}$ be invertible functions such that $f \circ g = $ identity. Then (A) $f = g^{-1}$ (B) $g = f^{-1}$ (C) $f \circ g \neq g \circ f$ (D) $f \circ g = g \circ f$
Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then (A) $f$ is an increasing function of $x$ (B) $f$ is a decreasing function of $x$ (C) $f$ is neither increasing nor decreasing function of $x$ (D) $f'$ is not a monotonic function of $x$
One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is (A) $\frac{1}{2}$ (B) $\frac{1}{3}$ (C) $\frac{2}{5}$ (D) $\frac{1}{5}$
The number of solutions of the equation $\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) = \sin^{-1}\left(\frac{x-1}{\sqrt{1+(x-1)^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+(x-1)^2}}\right)$ is (A) 0 (B) 1 (C) 2 (D) infinite
Let $f(x) = a_0 + a_1|x| + a_2|x|^2 + a_3|x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then $f'(x)$ exists at $x = 0$ if and only if (A) $a_1 = 0$ (B) $a_1 = 0$ and $a_2 = 0$ (C) $a_1 = 0$ and $a_3 = 0$ (D) $a_2 = 0$ and $a_3 = 0$
Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then (A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$ (B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$ (C) $f$ has local minima at $x = -1$ and at $x = 2$ (D) $f$ has local maxima at $x = -1$ and at $x = 2$
Let $I = \int_0^1 \frac{\sin x}{\sqrt{x}}\,dx$ and $J = \int_0^1 \frac{\cos x}{\sqrt{x}}\,dx$. Then which one of the following is true? (A) $I > \frac{2}{3}$ and $J > 2$ (B) $I < \frac{2}{3}$ and $J < 2$ (C) $I < \frac{2}{3}$ and $J > 2$ (D) $I > \frac{2}{3}$ and $J < 2$
The area of the region between the curves $y = \sqrt{\frac{1+\sin x}{\cos x}}$ and $y = \sqrt{\frac{1-\sin x}{\cos x}}$ bounded by the lines $x = 0$ and $x = \frac{\pi}{4}$ is (A) $\int_0^{\sqrt{2}-1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$ (B) $\int_0^{\sqrt{2}-1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$ (C) $\int_0^{\sqrt{2}+1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$ (D) $\int_0^{\sqrt{2}+1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$
Match the statements in Column I with the values in Column II. Column I (A) $\int_{-\pi}^{\pi} \cos^2 x\,\frac{1}{1+a^x}\,dx$, $a > 0$ (B) $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$ (C) $\int_{-2}^{2} \frac{x^2}{1+5^x}\,dx$ (D) $\int_1^2 \frac{\sqrt{\ln(3-x)}}{\sqrt{\ln(3-x)}+\sqrt{\ln(x+1)}}\,dx$ Column II (p) $\frac{1}{2}$ (q) $0$ (r) $\frac{\pi}{4}$ (s) $\frac{\pi}{2}$