List of practice Questions

Let $X$ have a binomial distribution $B(6,p)$. If the sum of the mean and the variance of $X$ is $\dfrac{21}{8}$, then \[ \frac{P(2\le X<4)}{P(4<X<6)} \] is equal to:

The average energy released per fission for the nucleus of $^{235_{92}U$ is 190 MeV. When all the atoms of 47 g pure $^{235}_{92}U$ undergo fission process, the energy released is $\alpha \times 10^{23}$ MeV. The value of $\alpha$ is ___. (Avogadro Number $= 6 \times 10^{23}$ per mole)}

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by $\frac{x{256} Mr^2$. The value of x is ___.

A ball of radius r and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time t, given by $t = A \rho^a r^b \eta^c \sigma^d$, where A is a constant and a, b, c and d are integers. The value of $\frac{b+c{a+d}$ is ___.}

The velocity of sound in air is doubled when the temperature is raised from $0 ^\circ$C to $a ^\circ$C. The value of a is ___.

The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions 8 cm and 24 cm from the lens. The focal length of the lens is ___ cm.

If $ y = y(x) $ and \[ (1 + x^2)\,dy + (1 - \tan^{-1}x)\,dx = 0 \] and $ y(0) = 1 $, then $ y(1) $ is equal to:

The solution of the differential equation \[ x\,dy - y\,dx = \sqrt{x^2 + y^2}\,dx \] (where $c$ is the constant of integration) is

If xdy - ydx = $\sqrt{x^2 + y^2}$ dx. If y = y(x) $\&$ y(1) = 0 then y(3) is :

If \[ \sec x \frac{dy}{dx} - 2y = 2 + 3\sin x \] and \[ y(0) = -\frac{7}{4}, \] then $ y\left( \frac{\pi}{6} \right) $ is:

The solution of the differential equation \[ x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx \] is (where $ c $ is the integration constant):

If $ y = y(x) $ and $ (1 + x^2) \frac{dy{dx} + (1 - \tan^{-1}x)dx = 0 $ and $ y(0) = 1 $, then $ y(1) $ is}

If $ x^4 \, dy + (4x^3 y + 2 \sin x) \, dx = 0 $ and $ f\left( \frac{\pi}{2} \right) = 0 $, then find the value of $ \pi^4 f\left( \frac{\pi}{3} \right) $ (where $ y = f(x) $):

Given \[ f(t)=\left|\frac{t+1}{t^2}\right|,\ (t<0) \] is strictly decreasing in the interval $(2\alpha,\alpha)$, then the maximum value of \[ g(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha \] is:

If $y = y(x)$ satisfies
$(1+x^2)\frac{dy}{dx} + (2 - \tan^{-1}x) = 0$
and $y(0) = 0$, then the value of $y(1)$ is:

Let $f(x)$ be a differentiable function satisfying the equations $\lim_{t \to x} \dfrac{t^2 f(x)-x^2 f(t)}{t-x} = 3$ and $f(1)=2$. Find the value of $2f(2)$.

Let $y(x)$ be the solution of the differential equation \[ x\frac{dy}{dx}= y + x^2\cot x, \quad y\!\left(\frac{\pi}{2}\right)=\frac{\pi}{2}. \] The value of $6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right)$ equals:

Two positively charged particles $m_1$ and $m_2$ have been accelerated across the same potential difference of 200 keV. Given mass of $m_1 = 1 \,\text{amu}$ and $m_2 = 4 \,\text{amu}$. The de Broglie wavelength of $m_1$ will be $x$ times that of $m_2$. The value of $x$ is _______ (nearest integer).

A volume of x mL of 5 M NaHCO3 solution is mixed with 10 mL of 2 M H2CO3 solution to make an electrolytic buffer.
If the same buffer is used in the following electrochemical cell and the recorded cell potential is 253.5 mV, then the value of x (nearest integer) is ______ mL.

Electrochemical cell:
Sn(s) | Sn(OH)2(s) | HSnO2− (0.05 M) | OH− (0.05 M) || Bi2O3(s) | Bi(s)

Given:
Standard electrode potential of HSnO2− / Sn(OH)2 = −0.90 V
Standard electrode potential of Bi2O3 / Bi = −0.44 V

pKa of H2CO3 = 6.11
2.303 RT / F = 0.059 V
Antilog (1.29) = 19.5

The number of isoelectronic species among S2−, C4−, Mn2+, Co3+ and Fe3+ is ‘n’.

If ‘n’ moles of AgCl is formed during the reaction of the complex with formula
CoCl2(en)2NH3
with excess AgNO3 solution, then the number of electrons present in the t2g orbital of the complex is ________.

Consider the following two first-order reactions: A $\to$ B (first reaction) C $\to$ D (second reaction) The rate constant for first reaction at 500 K is double of the same at 300 K. At 500 K, 50% of the reaction becomes complete in 2 hours. The activation energy of the second reaction is half of that of first reaction. If the rate constant at 500 K of the second reaction becomes double of the rate constant of first reaction at the same temperature; then rate constant for the second reaction at 300 K is ______ $\times 10^{-3}\,\text{hour}^{-1}$ (nearest integer).

For a strong electrolyte, molar conductivity increases slowly with dilution and can be represented by the equation:
Lambda_m = Lambda_m(0) − A × square root of c

Molar conductivity values of solutions of strong electrolyte AB at 18 degree Celsius are given below:

Concentration (mol L-1): 0.04 , 0.09 , 0.16 , 0.25
Molar conductivity (S cm2 mol-1): 96.1 , 95.7 , 95.3 , 94.9

The value of constant A based on the above data [in S cm2 mol-1 per (mol L-1)1/2] is:

Consider the following aqueous solutions. I. 2.2 g Glucose in 125 mL of solution. II. 1.9 g Calcium chloride in 250 mL of solution. III. 9.0 g Urea in 500 mL of solution. IV. 20.5 g Aluminium sulphate in 750 mL of solution. The correct increasing order of boiling point of these solutions will be: [Given: Molar mass in g mol$^{-1}$: H = 1, C = 12, N = 14, O = 16, Cl = 35.5, Ca = 40, Al = 27 and S = 32]

Structures of four disaccharides are given below. Among the given disaccharides, the non-reducing sugar is: