Recent questions tagged past papers

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out out at random and then transferred to the other. The number of ways in which this can be done is

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Let $f : (-1, 1) \to R$ be a differentiable function with $f(0) = -1$ and $f'(0)=1$. Let $g(x) = [f(2f(x) + 2)]^2$. Then $g'(0)=$

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Let $p(x)$ be a function defined on $R$ such that $p'(x) = p'(1-x)$, for all $x \in [0,1], \; p(0)=1$ and $p(1) =41$. Then $\begin{align} \int_0 ^1p(x) \; dx\end{align}$ equals

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Let $f : R \to R$ be a positive increasing function with $\displaystyle\lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1$ Then $\displaystyle\lim_{x \to \infty} \frac{f(2x)}{f(x)} = $

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A person is to count 4500 currency notes. Let $a_n$ denote the number of notes the counts in the $n^{th} $ minute. If $a_1 = a_2 = .......= a_{10} = 150$ and $a_{10}, a_{11}, ....$ are in A.P. with common difference $-2$, then the time taken by him to count all notes is

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The line L given by $\frac{x}{5}+ \frac{y}{b} = 1$ passes through the poin (13, 32). The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

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A line AB in three-dimensional space makes angle $45^{\circ}$ and $120^{\circ}$ with the positive $x-axis$ and the positive $y-axis$ respectively. If AB makes an acute $\theta$ with the positive $z-axis$, then $\theta$ equals

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The number of complex number $z$ such that $|z-1| = | z+ 1| = |z-i|$ equals

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If $\alpha$ and $\beta$ are the roots of the equation $x^2 - x + 1 =0$, then $\alpha^{2009} + \beta^{2009}$ =

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For a regular polygon, let $r$ and $R$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is

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Let $f : R \to R$ be a continuous function defined by $f(x) = \frac{1}{e^x + 2e^{-x}}$. Statement-1 : $f(c) = \frac{1}{3}$, for some $c \in R$. Statement-2 : $0 < f(x) \leq \frac{1}{2 \sqrt{2}}$, for all $x \in R$

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Let $A$ be a $ 2 \times 2$ matrix with non-zero entries and let $A^2 =I$, where $I$ is $2 \times 2$ identity matrix. Define $Tr(A)$ = sum of diagonal elements of $A$ and $|A|$ = determinant of matrix A. Assertion : Statement -1: $Tr(A) = 0$ Reason : Statement-2 : $|A| =1$

asked Dec 11, 2018 by pady_1

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Let $S_1 = \displaystyle\sum_{j=1}^{10} j (j - 1)^{10} C_j, S_2 = \displaystyle\sum_{j=1}^{10} j \;^{10} C_j $ and $S_3 = \displaystyle\sum_{j=1}^{10} j^2 \;^{10} C_j $ Statement-1 : $S_3 = 55 \times 2^9$ Statement-2 : $S_1 = 90 \times 2^8$ and $S_2 = 10 \times 2^8$.

asked Dec 11, 2018 by pady_1

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Statement-1: The point $A(3, 1, 6)$ is the mirror image of the point $B (1, 3, 4)$ in the plane $x - y +z =5$. Statement-2 : The plane $x-y +z=5$ bisects the line segment joining $A(3, 1, 6)$ and $B(1, 3, 4)$

asked Dec 11, 2018 by pady_1

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Four numbers are chosen at random (without replacement) from the set $\{1, 2, 3, ..., 20\}$. Statement-1 : The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$. Statement-2 : If the four chosen numbers from an AP, then the set of all possible values of common difference is $\{ \pm1, \pm 2, \pm 3, \pm 4, \pm 5 \}$.

asked Dec 11, 2018 by pady_1

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The number of $3 \times 3$ non-singular matrices, with four entries as 1 and all other entries as 0, is

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Let $f : R \to R$ be defined by $f(x) = \begin{cases} k-2x, & \quad \text{if } \text{ x $\leq$ -1}\\ 2x + 3, & \quad \text{if } \text{ x > -1} \end{cases} $. If $f$ has a local minimum at $x=-1$, then a possible value of $k$ is

asked Dec 11, 2018 by pady_1

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Consider the following relations : $R = \{ (x, y) | x, \; y$ are real number and $x = wy$ for some rational number $w \}$; $S = \{ (\frac{m}{n}, \frac{p}{q} ) | m, \; n, \; p $ and $q$ are integers such that $n, \;q \neq 0$ and $qm = pn \}$. Then

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If the vectors $\overrightarrow{a} = \hat{i} - \hat{j} + 2 \hat{k}, \; \overrightarrow{b} = 2 \hat{i} + 4 \hat{j} + \hat{k}$ and $\overrightarrow{c} = \lambda \hat{i} + \hat{j} + \mu \hat{k}$ are mutually orthogonal, then $(\lambda, \mu)$ =

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If two tangets drawn from a point P to the parabola $y^2 = 4x$ are at right angles, then the locus of P is

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The area bounded by the curves $y = \cos x$ and $ y = \sin x$ between the ordinates $x=0$ and $x =\frac{3 \pi}{2}$ is

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Solution of the differential equation $\cos x\; dy = y (\sin x - y) dx, \; 0 < x < \frac{\pi}{2}$ is

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The equation of the tangent to the curve $y = x + \frac{4}{x^2}$, that is parallel to the $x-axis$, is

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Let $\overrightarrow{a} = \hat{j} - \hat{k}$ and $\overrightarrow{c} = \hat{i} - \hat{j} - \hat{k}$. Then vector $\overrightarrow{b}$ satisfying $\overrightarrow{a} \times \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{0}$ and $\overrightarrow{a}.\overrightarrow{b} = 3$ is

asked Dec 11, 2018 by pady_1

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Let S be a non-empty subset of R. Consider the following statement : P : There is a rational number $x \in S$ such that $x > 0$. Which of the following statements is the negation of the statement P ?

asked Dec 11, 2018 by pady_1

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Let $\cos (\alpha + \beta) = \frac{4}{5}$ and let $\sin (\alpha - \beta) = \frac{5}{13}$, where $0 \leq \alpha, \; \beta \leq \frac{\pi}{4}$, then $\tan 2 \alpha = $

asked Dec 11, 2018 by pady_1

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The equation of a wave on a string of linear mass density $0.04\; kg\;m^{-1}$ is given by$ y = 0.02 (m) \sin [ 2 \pi (\frac{t}{0.04 (s)} - \frac{x}{0.50(m)} ) ]$. The tension in the string is

asked Dec 11, 2018 by pady_1

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A small particleof mass m is projected at an angle $\theta$ with the x-axis with a initial velocity $v_0$ in the x-yplane as shown in the figure. At a time $t < \frac{v_0 \sin \theta}{g}$, the angular momentum of the particle is where $\hat{i}, \hat{j} $ and $\hat{k}$ are unit vectors along $x, \; y$ and $z-axis$ respectively.

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The speed of light in the medium is

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An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) = \mu_0 + \mu_2I$, where $\mu_0 $ and $\mu_2$ are positive constants and $I$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. QUESTION : The initial shape of the wave front of the beam is

asked Dec 11, 2018 by pady_1

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An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) = \mu_0 + \mu_2I$, where $\mu_0 $ and $\mu_2$ are positive constants and $I$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. Question : As the beam enters the medium, it will

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For a particle in uniform circular motion the acceleration $\overrightarrow{a}$ at a point $P (R, \theta)$ on the circle of radius R is (here $\theta$ is measured from the x-axis)

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Two fixed frictionless inclined plane making an angle $30^{\circ}$ and $60^{\circ}$ with the vertical are shown in the figure. Two block A and B are placed on the two planes. What is the relative vertical acceleration of A with respect to B ?

asked Dec 11, 2018 by pady_1

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A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of $'P'$ is such that it sweeps out a length $s = t^3 +5$, where $s$ is metres and $t$ is in seconds. The radius of the path is $20 \;m$. The acceleration of $'P'$ when $t = 2\;s$ is nearly

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Two conductors have the same resistance at $0^{\circ}C$ but their temperature coefficients of resistance are $\alpha_1$ and $\alpha_2$. The respective temperature coefficients of their series and parallel combinations are nearly

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Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $30^{\circ}$ with each other. When suspended in a liquid of density $0.8\;g\;cm^{-3}$, the angle remains the same. If density of the material of the sphere is $16\;g\; cm^{-3}$, the dielectric constant of the liquid is

asked Dec 11, 2018 by pady_1

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The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U(x) = \frac{a}{x^{12}} - \frac{b}{x^6}$, where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is $D = [U_{(x = \infty)} - U_{at\; equilibrium}], \; D$ is

asked Dec 11, 2018 by pady_1

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Let there be a spherically symmetric charge distribution with charge density varying as $\rho(r) = \rho_{_0}(\frac{5}{4} - \frac{r}{R})$ upto $r = R$ and $\rho (r) = 0$ for $r > R$, where $r$ is the distance from the origin. The electric field at a distance $r (r < R) $ from the origin is given by

asked Dec 11, 2018 by pady_1

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In a series LCR circuit $R = 200 \Omega$ and the voltage and the frequency of the main supply is 220 V and 50 Hz respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $30^{\circ}$. On taking out the inductor from the circuit leads the voltage by $30^{\circ}$. The power dissipiated in the LCR circuit is

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The respective number of significant figures for the numbers $23.023, 0.0003$ and $2.1 \times 10^{–3}$ are

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If a source of power 4 kW produces $10^{20}$ photons/second, the radiation belong to a part of the spectrum called

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A diatomic ideal gas is used in a Car engine as the working substance. If during the adiabatic expansion part of the cycle, volume of the gas increases from V to 32V the efficiency of the engine is

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The combination of gates shown below yields

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A thin semi-circular ring of radius r has a positive charge q distributed uniformly over it. The net field E at the centre O is

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A radioactive nucleus (initial mass number A and atomic number Z) emits 3 $\alpha$ particles and 2 positrons. The ratio of number of neutrons to that of protons in the final nucleus will be

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The speed of daughter nuclei is

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The binding energy per nucleon for the parent nucleus is $E_1$ and that for the daughter nuclei is $E_2$.Then

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The figure shows the position â€“ time (x â€“ t) graph of one-dimensional motion of a body of mass 0.4 kg. The magnitude of each impulse is

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In the circuit shown below, the key K is closed at t = 0. The current through the battery is

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Two long parallel wires are at a distance 2d apart. They carry steady equal current flowing out of the plane of the paper as shown. The variation of the magnetic field along the line XX ' is given by

asked Dec 11, 2018 by pady_1

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