The number of relations, defined on the set \{a, b, c, d\}, which are both reflexive and symmetric, is equal to:
Let (\alpha, \beta, \gamma) be the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line \vec{r} = (-\hat{i} + 3\hat{j} + \hat{k}) + \lambda(2\hat
Let f : \mathbf{R} \rightarrow \mathbf{R} be a twice differentiable function such that the quadratic equation f(x)m^2 - 2f'(x)m + f''(x) = 0 in m, has two equal roots for every x \
Let a_1 = 1 and for n \ge 1, a_{n+1} = \frac{1}{2}a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}. Then \left|\sum_{n=1}^{\infty} \left(a_n - \frac{2}{n^2}\right)\right| is equal to ____.
For some \alpha, \beta \in \mathbf{R}, let A = \begin{bmatrix} \alpha & 2 \\ 1 & 2 \end{bmatrix} and B = \begin{bmatrix} 1 & 1 \\ 1 & \beta \end{bmatrix} be such that A^2 - 4A + 2I
6 \int_{0}^{\pi} |(\sin 3x + \sin 2x + \sin x)|\, dx is equal to ____.
Let S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}. If the number of elements (m, n) in S such that 6^m + 9^n is a multiple of 5 is p and the number of elements (m, n) in S such
Let \vec{c} and \vec{d} be vectors such that |\vec{c} + \vec{d}| = \sqrt{29} and \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d}.
The area of the region, inside the ellipse x^2 + 4y^2 = 4 and outside the region bounded by the curves y = |x| - 1 and y = 1 - |x|, is :
If the coefficient of x in the expansion of (ax^2 + bx + c)(1 - 2x)^{26} is -56 and the coefficients of x^2 and x^3 are both zero, then a + b + c is equal to:
Let \mathrm{O} be the vertex of the parabola x^2 = 4y and \mathrm{Q} be any point on it. Let the locus of the point \mathrm{P}, which divides the line segment \mathrm{OQ} internall
Let the foci of a hyperbola coincide with the foci of the ellipse \frac{x^2}{36} + \frac{y^2}{16} = 1. If the eccentricity of the hyperbola is 5, then the length of its latus rectu
Initially a satellite of 100\text{ kg} is in a circular orbit of radius 1.5R_E. This satellite can be moved to a circular orbit of radius 3R_E by supplying \alpha \times 10^6\text{
If an alpha particle with energy 7.7\text{ MeV} is bombarded on a thin gold foil, the closest distance from nucleus it can reach is ____ \text{m}. (Atomic number of gold = 79 and \
The electric field in a plane electromagnetic wave is given by : E_y = 69 \sin[0.6 \times 10^3x - 1.8 \times 10^{11}t]\text{ V/m}. The expression for magnetic field associated with
Consider a modified Bernoulli equation. \left(P + \frac{A}{Bt^2}\right) + \rho g(h + Bt) + \frac{1}{2}\rho V^2 = \text{constant} If t has the dimension of time then the dimensions
A 4\text{ kg} mass moves under the influence of a force \vec{F} = (4t^3\hat{i} - 3t\hat{j})\text{ N} where t is the time in second. If mass starts from origin at t = 0, the velocit
A parallel plate capacitor has capacitance C, when there is vacuum within the parallel plates. A sheet having thickness \left(\frac{1}{3}\right)^{\text{rd}} of the separation betwe
In a double slit experiment the distance between the slits is 0.1\text{ cm} and the screen is placed at 50\text{ cm} from the slits plane. When one slit is covered with a transpare
A light wave described by E = 60[\sin(3 \times 10^{15})t + \sin(12 \times 10^{15})t] (in SI units) falls on a metal surface of work function 2.8\text{ eV}. The maximum kinetic ener
A gas based geyser heats water flowing at the rate of 5.0\text{ litres} per minute from 27^\circ\text{C} to 87^\circ\text{C}. The rate of consumption of the gas is ____ \text{g/s}.
A current carrying solenoid is placed vertically and a particle of mass m with charge Q is released from rest. The particle moves along the axis of solenoid. If g is acceleration d
A point charge of 10^{-8}\text{C} is placed at origin. The work done in moving a point charge 2\mu\text{C} from point A(4, 4, 2)\text{m} to B(2, 2, 1)\text{m} is ____ \text{J}. (\f
The given circuit works as :
A uniform rod of mass m and length l suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is
An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120\text{ cm} at 30^\circ\text{C}. The coefficient of linear expansion of alum
Two strings (A, B) having linear densities \mu_A = 2 \times 10^{-4}\text{ kg/m} and, \mu_B = 4 \times 10^{-4}\text{ kg/m} and lengths L_A = 2.5\text{ m} and L_B = 1.5\text{ m} resp
Potential energy (V) versus distance (x) is given by the graph. Rank various regions as per the magnitudes of the force (F) acting on a particle from high to low.
A 1\text{ m} long metal rod \mathrm{AB} completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10\text{ T}. If the resistance of
Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5\text{ cm} and the area of cross-sections at