List of practice Questions

Let $\vec{c}$ be the projection vector of $\mathbf{b} = \lambda \hat{i} + 4 \hat{k}, \lambda > 0$ , on the vector $\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If $|\vec{a} + \vec{c}| = 7$ , then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is:

A gun fires a lead bullet of temperature 300 K into a wooden block. The bullet having melting temperature of 600 K penetrates into the block and melts down. If the total heat required for the process is 625 J, then the mass of the bullet is grams. Given Data: Latent heat of fusion of lead = $2.5 \times 10^4 \, \text{J kg}^{-1}$ and specific heat capacity of lead = 125 J kg $^{-1}$ K $^{-1}$ .

A solid sphere of mass $m$ and radius $r$ is allowed to roll without slipping from the highest point of an inclined plane of length $L$ and makes an angle of $30^\circ$ with the horizontal. The speed of the particle at the bottom of the plane is $v_1$ . If the angle of inclination is increased to $45^\circ$ while keeping $L$ constant, the new speed of the sphere at the bottom of the plane is $v_2$ . The ratio of $v_1^2 : v_2^2$ is:

If the system of equations

(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5

\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7

(\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9

has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to:

Identify the valid statements relevant to the given circuit at the instant when the key is closed.

$\text{A}$ : There will be no current through resistor R.
$\text{B}$ : There will be maximum current in the connecting wires.
$\text{C}$ : Potential difference between the capacitor plates A and B is minimum.
$\text{D}$ : Charge on the capacitor plates is minimum.
Choose the correct answer from the options given below:

If $\sum_{r=0}^5 \frac{{}^{11}C_{2r+1}}{2r+2} = \frac{m}{n}$ , gcd(m, n) = 1, then $m - n$ is equal to:

Consider a circular disc of radius 20 cm with center located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of the center of mass of the residual or remaining disc from the origin will be:

Given a thin convex lens (refractive index $\mu_2$ ), kept in a liquid (refractive index $\mu_1, \mu_1<\mu_2$ ) having radii of curvature $|R_1|$ and $|R_2|$ . Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

The motion of an airplane is represented by the velocity-time graph as shown below. The distance covered by the airplane in the first 30.5 seconds is km.

If $A$ , $B$ , and $\left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right)$ are non-singular matrices of the same order, then the inverse of $A \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) B$ is equal to:

A particle is projected at an angle of $30^\circ$ from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is $h_0$ and height traversed in the last second, before it reaches the maximum height, is $h_1$ . The ratio $\frac{h_0}{h_1}$ is __________. [Take $g = 10 \, \text{m/s}^2$ ]

Let $A$ be a square matrix of order 3 such that $\text{det}(A) = -2$ and $\text{det}(3 \cdot \text{adj}(-6 \cdot \text{adj}(3A))) = 2^{m+n} \cdot 3^{mn}$ , where $m > n$ . Then $4m + 2n$ is equal to:

Let a curve $y = f(x)$ pass through the points $(0,5)$ and $(\log 2, k)$ . If the curve satisfies the differential equation:

2(3+y)e^{2x}dx - (7+e^{2x})dy = 0,

then $k$ is equal to:

In the given circuit the sliding contact is pulled outwards such that the electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 Ω, the value of the current in the circuit will be A.

Let the function, $f(x)$ = $\begin{cases} -3ax^2 - 2, & x < 1 \\a^2 + bx, & x \geq 1 \end{cases}$ Be differentiable for all $x \in \mathbb{R}$ , where $a > 1$ , $b \in \mathbb{R}$ . If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta\sqrt{3}$ , where $\alpha, \beta \in \mathbb{Z}$ , then the value of $\alpha + \beta$ is:

Let $L_1 : \frac{x - 1}{3} = \frac{y - 1}{-1} = \frac{z + 1}{0}$ and $L_2 : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{\alpha}$ , where $\alpha \in \mathbb{R}$ , be two lines which intersect at the point $B$ . If $P$ is the foot of the perpendicular from the point $A(1, 1, -1)$ on $L_2$ , then the value of $26 \alpha (PB)^2$ is:

Let the position vectors of the vertices A, B, and C of a tetrahedron ABCD be $\hat{i} + 2\hat{j} + \hat{k}$ , $\hat{i} + 3\hat{j} - 2\hat{k}$ , and $2\hat{i} + \hat{j} - \hat{k}$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is $\frac{\sqrt{10}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt{2}}$ , then the position vector of E is:

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}} (x+15)^{\frac{15}{13}}}$ If $I(37) - I(24) = \frac{1}{4} \left( b^{\frac{1}{13}} - c^{\frac{1}{13}} \right)$ where $b, c \in \mathbb{N}$ , then $3(b + c)$ is equal to:

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness $h$ in terms of the angle of incidence $i$ and angle of refraction $r$ , if the glass slab is placed in air medium?

A light hollow cube of side length 10 cm and mass 10g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y \pi \times 10^{-2}$ s, where the value of $y$ is:
(Acceleration due to gravity,

g = 10 \, \text{m/s}^2

, density of water =

10^3 \, \text{kg/m}^3

)

A point particle of charge $Q$ is located at $P$ along the axis of an electric dipole 1 at a distance $r$ as shown in the figure. The point $P$ is also on the equatorial plane of a second electric dipole 2 at a distance $r$ . The dipoles are made of opposite charge $q$ separated by a distance $2a$ . For the charge particle at $P$ not to experience any net force, which of the following correctly describes the situation?

Three conductors of same length having thermal conductivity $k_1$ , $k_2$ , and $k_3$ are connected as shown in figure. Area of cross sections of 1st and 2nd conductor are same and for 3rd conductor it is double of the 1st conductor. The temperatures are given in the figure. In steady state condition, the value of θ is ________ °C. (Given: $k_1$ = 60 Js⁻¹m⁻¹K⁻¹, $k_2$ = 120 Js⁻¹m⁻¹K⁻¹, $k_3$ = 135 Js⁻¹m⁻¹K⁻¹)