List of practice Questions

The number of ways of forming the ordered pairs (p, q) such that p>q by choosing p and q from the first 50 natural numbers is:
Evaluate the limit: \[ \lim_{x \to 0} \frac{x^2 \sin^2(3x) + \sin^4(6x)}{(1 - \cos(3x))^2} \]
If \(y = \sqrt{\cosh x + \sqrt{\cosh x}}\), then \(\frac{dy}{dx} =\)
Two blocks A and B of masses 2 kg and 4 kg respectively are kept on a rough horizontal surface. If same force of 20 N is applied on each block, then the ratio of the accelerations of the blocks A and B is (Coefficient of kinetic friction between the surface and the blocks is 0.3 and acceleration due to gravity = \(10 \operatorname{ms}^{-2}\))
The domain of the function $f(x) = \ln\left(\frac{1}{\sqrt{x^2-4x+4}}\right) + \sin^{-1}(x^2-2)$ is
Tangents are drawn at three points P($t_1$), Q($t_2$), R($t_3$) on the parabola $y^2 = x$. Let these tangents intersect each other at the points L, M, N. If $t_1 = 2$, $t_2 = -4$, $t_3 = 6$, then the area of the triangle LMN is
The energy stored in a capacitor of capacitance \( 10 \, \mu\text{F} \) when charged to a potential of 6 kV is
At T(K), the vapor pressure of x molal aqueous solution containing a non-volatile solute is 12.078 kPa. The vapor pressure of pure water at T(K) is 12.3 kPa. What is the value of x?
The differential equation of the family of circles passing through the origin and having centre on X-axis is
In \( \text{Fe}_x[\text{Fe}_y(\text{CN})_6]_3 \), x, y respectively, are
Out of 8 students in a classroom, 4 of them are chosen and they are arranged around a table. If the remaining 4 are arranged in a row, then the total number of arrangements that can be made with those 8 students is
If 3 squares are chosen at random from the 64 squares of a chessboard, then the probability that all of them lie along the same diagonal line is:
If the number of diagonals of a regular polygon is 35, then the number of sides of the polygon is
Two bodies A and B of masses 20 kg and 5 kg respectively are at rest. Due to the action of a force of 40 N separately, if the two bodies acquire equal kinetic energies in times \( t_A \) and \( t_B \) respectively, then \( t_A : t_B = \)
If the maximum velocity and maximum acceleration of a particle executing simple harmonic motion are respectively \( 5 \, \text{m s}^{-1} \) and \( 10 \, \text{m s}^{-2} \), then the time period of oscillation of the particle is
P and Q are the ends of a diameter of the circle \( x^2+y^2=a^2(a>\frac{1}{\sqrt{2}}) \). \( s \) and \( t \) are the lengths of the perpendiculars drawn from P and Q onto the line \( x+y=1 \) respectively. When the product \( st \) is maximum, the greater value among \( s, t \) is:
If a wheel with 24 metallic spokes each \( 40 \) cm long is rotated with a speed of \( 180 \) rev/min in a plane normal to the horizontal component of Earth's magnetic field, the emf induced between the axle and the rim of the wheel is \( E \). If the number of spokes is made \( 12 \) and the wheel is rotated with a speed of \( 90 \) rev/min in the same field, the induced emf is:
At \( 27^\circ C \) kinetic energy of 4 g of \( H_2 \) is \( x \) J. What is the kinetic energy (in J) of 6.4 g of oxygen at \( 127^\circ C \)?
If $x = 3 - 2\sqrt{3}i$, then evaluate $x^4 - 12x^3 + 54x^2 - 108x - 54 = $
Evaluate \[ \lim_{x \to 0} \sqrt{\frac{x + 2 \sin x + 3 \tan x - \tan^3 x}{x^2 + 2 \sin x + \tan x + 3 - \sqrt{\sin^2 x - 2 \tan x - x + 3}}} = ? \]
If \(f(x) = \sec^{-1} \left(\frac{1}{2x^2 -1}\right)\) and \(g(x) = \tan^{-1} \left(\frac{\sqrt{1 + x^2} - 1}{x}\right)\), then the derivative of \(f(x)\) with respect to \(g(x)\) is?
A circular disc of diameter 0.8 m and mass 4 kg is rolling on a smooth horizontal plane. If 2.56 N torque is acting on the disc, then its angular acceleration is:
What are the major products X and Y respectively in the following set of reactions?
Fe (dark)
\( \int x \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \, dx \, (x>0) = \)
A car accelerates uniformly from rest to a speed of 20 m/s in 8 seconds. What is the distance covered by the car during this time?