List of practice Questions

If $ A(1,0,2) $, $ B(2,1,0) $, $ C(2,-5,3) $, and $ D(0,3,2) $ are four points and the point of intersection of the lines $ AB $ and $ CD $ is $ P(a,b,c) $, then $ a + b + c = ? $

If $\int \frac{\log(1+x^4)}{x^3} dx = f(x) \log(\frac{1}{g(x)}) + \tan^{-1}(h(x)) + c$, then $h(x) [f(x) + f(\frac{1}{x})] =$

The acceleration of a particle which moves along the positive $ x $-axis varies with its position as shown in the figure. If the velocity of the particle is $ 0.8 \, {m/s} $ at $ x = 0 $, then its velocity at $ x = 1.4 \, {m} $ is: \includegraphics[]{83.png}

The triangle $ PQR $ is inscribed in the circle \[ x^2 + y^2 = 25. \] If $ Q = (3,4) $ and $ R = (-4,3) $, then $ \angle QPR $ is:

If $\sin x = -\frac{3}{5}$, where $\pi<x<\frac{3\pi}{2}$, then $80(\tan^2 x - \cos x)$ is equal to:

If $ A(1,2,0), B(2,0,1), C(-3,0,2) $ are the vertices of $ \triangle ABC $, then the length of the internal bisector of $ \angle BAC $ is:

p-Methyl benzene nitrile can be prepared from which of the following?

If $ \theta $ is the angle between $ \vec{f} = i + 2j - 3k $ and $ \vec{g} = 2i - 3j + ak $ and $ \sin \theta = \frac{\sqrt{24}}{28} $, then $ 7a^2 + 24a = $ ?

Two blocks of masses $ m $ and $ 2m $ are connected by a massless string which passes over a fixed frictionless pulley. If the system of blocks is released from rest, the speed of the centre of mass of the system of two blocks after a time of 5.4 s is (Acceleration due to gravity = 10 ms$^{-2}$)

Two ideal gases A and B of the same number of moles expand at constant temperatures $ T_1 $ and $ T_2 $ respectively such that the pressure of gas A decreases by $ 50% $ and the pressure of gas B decreases by $ 75% $. If the work done by both the gases is the same, then the ratio $ T_1:T_2 $ is:

If all chords of the curve $ 2x^2 - y^2 + 3x + 2y = 0 $, which subtend a right angle at the origin, always pass through the point $ (a, \beta) $, then $ (a, \beta) = $:

If $ z_1 = 10 + 6i $, $ z_2 = 4 + 6i $ and $ z $ is any complex number such that the argument of $ \frac{z-z_1}{z-z_2} $ is $ \frac{\pi}{4} $, then the value of $ |z - 7 - 9i| $ is:

The gravitational potential energy of a body on the surface of the Earth is $E$. If the body is taken from the surface of the Earth to a height equal to $150%$ of the radius of the Earth, its gravitational potential energy is:

Two blocks of equal masses are tied with a light string passing over a massless pulley (Assuming frictionless surfaces). The acceleration of the centre of mass of the two blocks is (Given $ g = 10 \, \text{m/s}^2 $):

There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arrangements is:

A diving board is at a height of $ h $ from the water surface. A swimmer standing on this board throws a stone vertically upward with a velocity 16 ms$^{-1}$. It reaches the water surface in a time of 5 s. In the next 0.2s, the diver can hear the sound from the water surface. The speed of sound is (acceleration due to gravity $ g = 10 $ ms$^{-2} $)

The sum of the series $ 1 - \frac{2}{3} + \frac{2.4}{3.6} - \frac{2.4.6}{3.6.9} + \cdots \infty $ is:

For $|x|<\frac{1}{\sqrt{2}}$ the coefficient of $x$ in the expansion of $\frac{(1-4x)^2(1-2x^2)^{1/2}}{(4-x)^{3/2}}$ is

The circumference of a circle passing through the point $ (4, 6) $ with two normals represented by $ 2x - 3y + 4 = 0 $ and $ x + y - 3 = 0 $ is:

If $ z = x+iy $, $ x^2+y^2 = 1 $ and $ z_1 = e^{i\theta} $, then the expression $ \frac{z_1^{2n-1} - 1}{z_1^{2^n-1} + 1} $ simplifies to:

Let $ A, B, C, D, $ and $ E $ be $ n \times n $ matrices, each with non-zero determinant. If $ ABCDE = I $, then $ C^{-1} $ is:

If $2.4^{2n+1} + 3^{3n+1}$ is divisible by $k$ for all $n \in \mathbb{N}$, then $k$ is:

Two concentric thin circular rings of radii 50 cm and 40 cm each, carry a current of 3.5 A in opposite directions. If the two rings are coplanar, the net magnetic field due to the two rings at their centre is:

A boy throws a ball with a velocity $V_0$ at an angle $\alpha$ to the ground. At the same time, he starts running with uniform velocity to catch the ball before it hits the ground. To achieve this, he should run with a velocity of:

The square root of $ 7 + 24i $ is: