Two identical thin rods of mass M\text{ kg} and length L\text{ m} are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point
The heat generated in 1\text{ minute} between points A and B in the given circuit, when a battery of 9\text{ V} with internal resistance of 1\Omega is connected across these points
A collimated beam of light of diameter 2\text{ mm} is propagating along the x-axis. The beam is required to be expanded into a collimated beam of diameter 14\text{ mm} using a syst
10\text{ mole} of oxygen is heated at constant volume from 30^\circ\text{C} to 40^\circ\text{C}. The change in the internal energy of the gas is ____ \text{cal}. (The molecular spe
Given below are two statements : Statement I : The correct order in terms of bond dissociation enthalpy is \mathrm{Cl_2 > Br_2 > F_2 > I_2}. Statement II : The correct trend in the
The correct increasing order of \mathrm{C-H}(A), \mathrm{C-O}(B), \mathrm{C=O}(C) and \mathrm{C\equiv N}(D) bonds in terms of covalent bond length is :
The first and second ionization constants of \mathrm{H_2X} are 2.5 \times 10^{-8} and 1.0 \times 10^{-13} respectively. The concentration of \mathrm{X^{2-}} in 0.1\mathrm{M}\mathrm
If the line ax + 4y = \sqrt{7}, where a \in \mathbf{R}, touches the ellipse 3x^2 + 4y^2 = 1 at the point P in the first quadrant, then one of the focal distances of P is:
For a triangle \mathrm{ABC}, let \vec{p} = \vec{\mathrm{BC}}, \vec{q} = \vec{\mathrm{CA}} and \vec{r} = \vec{\mathrm{BA}}. If |\vec{p}| = 2\sqrt{3}, |\vec{q}| = 2 and \cos \theta =
Let a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \dots, \frac{a_{10}}{2^9} be a G.P. of common ratio \frac{1}{\sqrt{2}}. If a_1 + a_2 + \dots + a_{10} = 62, then a_1 is equal to :
A random variable X takes values 0, 1, 2, 3 with probabilities \frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b respectively, where \mathrm{a}, \mathrm{b} \in \mathbf{R}. Let \
Let f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3), x \in \mathbf{R}. Then the value of f'(5) is:
If \int_0^1 4\cot^{-1}(1 - 2x + 4x^2)\,\mathrm{d}x = a\tan^{-1}(2) - b\log_{\mathrm{e}}(5), where \mathrm{a}, \mathrm{b} \in \mathbf{N}, then (2\mathrm{a} + \mathrm{b}) is equal to
Let the maximum value of (\sin^{-1} x)^2 + (\cos^{-1} x)^2 for x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right] be \frac{m}{n}\pi^2, where \gcd(m, n) = 1. Then m + n is e
If the system of equations 3x + y + 4z = 3 2x + \alpha y - z = -3 x + 2y + z = 4 has no solution, then the value of \alpha is equal to:
The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are at
Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of
A capacitor C is first charged fully with potential difference of V_0 and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L.
Surface tension of two liquids (having same densities), T_1 and T_2, are measured using capillary rise method utilizing two tubes with inner radii of r_1 and r_2 where r_1 > r_2. T
The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176\text{rad/s}. The frequency of this simple harmonic oscillator is ___ \text{Hz}. [ ta
A river of width 200\text{ m} is flowing from west to east with a speed of 18\text{ km/h}. A boat, moving with speed of 36\text{ km/h} in still water, is made to travel one-round t
A large drum having radius R is spinning around its axis with angular velocity \omega, as shown in figure. The minimum value of \omega so that a body of mass M remains stuck to the
As compared with chlorocyclohexane, which of the following statements correctly apply to chlorobenzene? A. The magnitude of negative charge is more on chlorine atom. B. The \mathrm
Given below are two statements: Statement I: The halogen that makes longest bond with hydrogen in \mathrm{HX}, has the smallest covalent radius in its group. Statement II: A group
Given below are two statements: Statement I: Phenol on treatment with \mathrm{CHCl_3}/\text{aq. } \mathrm{KOH} under refluxing condition, followed by acidification produces p-hydro
Dissociation of a gas \mathrm{A_2} takes place according to the following chemical reaction. At equilibrium, the total pressure is 1\text{ bar} at 300\text{ K}. \mathrm{A_2(g)} \ri
If \int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}}\,dx = -\frac{p_1}{q_1} (\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2} (\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3} (\cot x)^{\frac{1}
Six point charges are kept 60^\circ apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the center of the circle is ___
A projectile is thrown upward at an angle 60^\circ with the horizontal. The speed of the projectile is 20\text{ m/s} when its direction of motion is 45^\circ with the horizontal. T
Two loudspeakers ( L_1 and L_2 ) are placed with a separation of 10\text{ m}, as shown in figure. Both speakers are fed with an audio input signal of same frequency with constant v