List of practice Questions

For $ x<0 $, $ \frac{d}{dx} [|x|^x] $ is given by:

If the eleventh term in the binomial expansion of $ (x + a)^n $ is the geometric mean of the eighth and twelfth terms, then the greatest term in the expansion is:

A spring of $ 5 \times 10^3 $ Nm$^{-1} $ spring constant is stretched initially by 10 cm from the unstretched position. The work required to stretch it further by another 10 cm is

In the Young's double slit experiment, the 2nd bright fringe for red light coincides with the 3rd bright fringe for violet light. Then the value of 'n' is: (Given: wavelength of red light = 6300 $\text{Å}$ and wavelength of violet light = 4200 $\text{Å}$)

The set of all real values of $ a $ such that the function $ f(x) = x^3 + 2ax^2 + 3(a+1)x + 5 $ is strictly increasing in its entire domain is:

The length of the tangent drawn at the point $ P \left( \frac{\pi}{4} \right) $ on the curve $ x^{2/3} + y^{2/3} = 2^{2/3} $ is:

If A, B, C are the angles of a triangle, then $\frac{\sin A + \sin B + \sin C}{\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} - 1} =$

E, m, L, G represent energy, mass, angular momentum and gravitational constant respectively. The dimensions of \[ \frac{EL^2}{mG^2} \] will be that of

Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \cos x \cos 2x}{\sin^2 x} \]

If the distance between the planes $ 2x + y + z + 1 = 0 $ and $ 2x + y + z + \alpha = 0 $ is 3 units, then the product of all possible values of $ \alpha $ is:

The complex conjugate of $ (4 - 3i)(2 + 3i)(1 + 4i) $ is:

A running man has half the kinetic energy of a boy of half his mass. The man speeds up by $1 \, {m/s}^{-1}$ to have the same kinetic energy as the boy. The initial speed of the man is:

The line $ x - 2y - 3 = 0 $ cuts the parabola $ y^2 = 4ax $ at points P and Q. If the focus of this parabola is $ \left(\frac{1}{4}, k\right) $, then PQ is:

Heat is supplied at constant pressure to a diatomic gas. The part of this heat that was utilized to increase its internal energy is:

A projectile can have the same range $ R $ for two angles of projection. Their initial velocities are the same. If $ T_1 $ and $ T_2 $ are times of flight in two cases, then the product of two times of flight is directly proportional to:

A student writes an exam with 8 true/false questions. He passes if he answers at least 6 correctly. Find the probability that he fails.

In $ \triangle ABC $, if $ a = 13 $, $ b = 14 $, and $ \cos \frac{C}{2} = \frac{3}{\sqrt{13}} $, then $ 2r_1 = $:

Aqueous CuSO₄ solution was electrolysed by passing 2 amp of current for 10 min. What is the weight (in g) of copper deposited at cathode? (Cu = 63 u; F = 96500 C mol^-1)

If the matrix is:
$\left| \begin{array}{ccc} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{array} \right|>0, \text{ then } abc>?$

For any real value of $x$, if $ \frac{11x^{2}+12x+6}{x^{2}+4x+2} \not\in (a, b) $, then the value for $x$ for which $$ \frac{11x^{2}+12x+6}{x^{2}+4x+2} = b - a + 3 $$ is:

If \[ | \vec{P} + \vec{Q} | = | \vec{P} | = | \vec{Q} |, \] then the angle between $ \vec{P} $ and $ \vec{Q} $ is:

A 4 kg mass is suspended as shown in the figure. All pulleys are frictionless and spring constant $ K $ is $ 8 \times 10^3 $ Nm$^{-1}$. The extension in spring is ( $ g = 10 $ ms$^{-2}$ )

The partial vapor pressure of 0.1 molal solution of B in liquid A is 60 kPa at 300 K. The partial vapor pressure (in kPa) of a solution containing B with mole fraction of 0.1 in liquid A at 300 K is ______.
(Assume the solute B obeys Henry's law. The molar mass of A is 80 g mol^-1.)
(rounded off to three decimal places)

The total number of meiotic divisions that occurred in a flower of malvaceae, which has 60 stamens with 25 microspore mother cells in each sporangium and 25 ovules in each carpel:

If $ f(x) = \begin{cases \frac{\sqrt{a^2 - ax - x^2} - \sqrt{x^2 + ax + a^2}}{\sqrt{a + x} - \sqrt{a - x}}, & x \ne 0
K, & x = 0 \end{cases} $ is continuous at $ x = 0 $, then $ K = $}