List of practice Questions

Let $y=y(x)$ be the solution of the differential equation $dy = e^{\alpha x+y}dx; \alpha \in \mathbb{N}$. If $y(\log_e 2) = \log_e 2$ and $y(0)=\log_e(\frac{1}{2})$, then the value of $\alpha$ is equal to ________.
If $\int_0^\pi (\sin^3 x) e^{-\sin^2 x} dx = \alpha - \frac{\beta}{e} \int_0^1 \sqrt{t} e^t dt$, then $\alpha+\beta$ is equal to ________.
Let $\vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k}$, $\vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k}$ and $\vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b} = -1$ and $\vec{b} \cdot \vec{c} = 10$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to ________.
Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, -4), one focus at (4, -4) and one vertex at (5, -4). If $mx-y=4, m>0$ is a tangent to the ellipse E, then the value of $5m^2$ is equal to ________.
Let $A = \{n \in \mathbb{N} | n^2 \le n+10000\}$, $B = \{3k+1 | k \in \mathbb{N}\}$ and $C = \{2k | k \in \mathbb{N}\}$. Then the sum of all the elements of the set $A \cap (B-C)$ is equal to ________.
The number of real roots of the equation $e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$ is equal to ________.
The half-life of $^{198}\text{Au}$ is 3 days. If atomic weight of $^{198}\text{Au}$ is 198 g/mol then the activity of 2 mg of $^{198}\text{Au}$ is [in disintegration/second] :
A parallel plate capacitor with plate area 'A' and distance of separation 'd' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as : \[ \epsilon(x) = \epsilon_0 + kx, \text{ for } \left( 0<x \le \frac{d}{2} \right) \] \[ \epsilon(x) = \epsilon_0 + k(d-x), \text{ for } \left( \frac{d}{2} \le x \le d \right) \]
Water droplets are coming from an open tap at a particular rate. The spacing between a droplet observed at 4th second after its fall to the next droplet is 34.3 m. At what rate the droplets are coming from the tap? (Take $g = 9.8 \text{ m/s}^2$)
In Amplitude Modulation, the message signal $V_m(t) = 10 \sin(2\pi \times 10^5 t)$ volts and Carrier signal $V_c(t) = 20 \sin(2\pi \times 10^7 t)$ volts. The modulated signal now contains the message signal with lower side band and upper side band frequency, therefore the bandwidth of modulated signal is $\alpha \text{ kHz}$. The value of $\alpha$ is :
Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as : (where $\lambda$ is the decay constant)
What should be the order of arrangement of de-Broglie wavelength of electron ($\lambda_e$), an $\alpha$-particle ($\lambda_\alpha$) and proton ($\lambda_p$) given that all have the same kinetic energy?
A particle of mass 4M at rest disintegrates into two particles of mass M and 3M respectively having non zero velocities. The ratio of de-Broglie wavelength of particle of mass M to that of mass 3M will be :
In the Young's double slit experiment, the distance between the slits varies in time as $d(t) = d_0 + a_0 \sin \omega t$; where $d_0, \omega$ and $a_0$ are constants. The difference between the largest fringe width and the smallest fringe width obtained over time is given as :
A ray of laser of a wavelength 630 nm is incident at an angle of $30^\circ$ at the diamond-air interface. It is going from diamond to air. The refractive index of diamond is 2.42 and that of air is 1. Choose the correct option.
A linearly polarized electromagnetic wave in vacuum is \[ E = 3.1 \cos [(1.8)z - (5.4 \times 10^8)t] \hat{i} \text{ N/C} \] is incident normally on a perfectly reflecting wall at $z=a$. Choose the correct option :

In the given figure, there is a circuit of potentiometer of length AB = 10 m. The resistance per unit length is 0.1 $\Omega$ per cm. Across AB, a battery of emf E and internal resistance 'r' is connected. The maximum value of emf measured by this potentiometer is : 

Identify the logic operation carried out. 

For a gas $C_p - C_v = R$ in a state P and $C_p - C_v = 1.10 R$ in a state Q. $T_P$ and $T_Q$ are the temperatures in two different states P and Q respectively. Then

Two different metal bodies A and B of equal mass are heated at a uniform rate under similar conditions. The variation of temperature of the bodies is graphically represented as shown in the figure. The ratio of specific heat capacities is : 

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Moment of inertia of a circular disc of mass 'M' and radius 'R' about X, Y axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be $I_x, I_y, & I_z$, respectively. The respective radii of gyration about all the three axes will be the same.
Reason R : A rigid body making rotational motion has fixed mass and shape.
In the light of the above statements, choose the most appropriate answer from the options given below :

Two billiard balls of equal mass 30 g strike a rigid wall with same speed of 108 kmph (as shown) but at different angles. If the balls get reflected with the same speed then the ratio of the magnitude of impulses imparted to ball 'a' and ball 'b' by the wall along 'X' direction is : 

Match List I with List II. 

Choose the correct answer from the options given below : 
 

Two wires of same length and radius are joined end to end and loaded. The Young's moduli of the materials of the two wires are $Y_1$ and $Y_2$. The combination behaves as a single wire then its Young's modulus is :
The minimum and maximum distances of a planet revolving around the Sun are $x_1$ and $x_2$. If the minimum speed of the planet on its trajectory is $v_0$ then its maximum speed will be :