List of practice Questions

In the experiment to determine the internal resistance of a cell ($E_1$) using potentiometer, the resistance drawn from the resistance box is $R$. The potential difference across the balancing length of the wire is equal to the terminal potential difference $(V)$ of the cell. The value of internal resistance $(r)$ of the cell is

Seven capacitors each of capacitance $2\,\mu F$ are to be connected to obtain equivalent capacitance of $\dfrac{10{11}\,\mu F$. Which of the following combination is possible?}

If \( x^2 + y^2 = 1 \), then find \( \dfrac{d^2x}{dy^2} \).

The general solution of the differential equation \[ \sec^2 x \tan y\,dx + \sec^2 y \tan x\,dy = 0 \] is

If \( \sec\theta = \dfrac{13}{12} \) and \( \theta \) lies in the fourth quadrant, then find \[ \tan\theta \times \csc\theta \times \sin\theta \times \cos\theta \]

The acute angle included between the lines \[ x\sin\theta - y\cos\theta = 5 \] and \[ x\sin\alpha - y\cos\alpha + 11 = 0 \] is

Water at \(100^\circ\text{C}\) cools in 15 minutes to \(75^\circ\text{C}\) in a room temperature of \(25^\circ\text{C}\). Then the temperature of water after 30 minutes is

If \[ A=\begin{bmatrix} 1 & 2 & i\\ 1 & 1 & 1\\ 1 & 1 & 0 \end{bmatrix}, \] then \( [\operatorname{adj}(\operatorname{adj}A)]^{-1} \) is

The length of the latus rectum of the parabola whose focus is at \( (1,-2) \) and directrix is the line \( x + y + 3 = 0 \) is

If \( A=\begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix} \) and \( B=\begin{bmatrix}2 & -3 \\-1 & 2\end{bmatrix} \), then \( (B^{-1}A^{-1})^{-1} \) is

If \( \log_{10}\!\left(\dfrac{x^3-y^3}{x^3+y^3}\right)=2 \), then find \( \dfrac{dx}{dy} \).

The integrating factor of the differential equation \[ \frac{dy}{dx} + \frac{1}{x}y = x^3 - 3 \] is

The distance of the point \( (3,4,5) \) from the point of intersection of the line \[ \frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2} \] and the plane \( x+y+z=2 \) is

Evaluate the integral \[ \int \frac{x^2+1}{x^4+x^2+1}\,dx \]

If \( a, b, c \) are lengths of the sides \( BC, CA, AB \) respectively of \( \triangle ABC \) and \( a\vec{AH} + b\vec{BH} + c\vec{CH} = \vec{0} \), then point \( H \) is the

Which of the following statement patterns is a tautology?
\[ \begin{aligned} S_1 &: (\sim q \land p) \land q
S_2 &: [p \land (p \rightarrow q)] \rightarrow q
S_3 &: (p \land q) \land (\sim p \lor \sim q)
S_4 &: (p \land q) \rightarrow r \end{aligned} \]

If \( \vec{AB} = 3\hat{i} + 5\hat{j} + 4\hat{k} \), \( \vec{AC} = 5\hat{i} - 5\hat{j} + 2\hat{k} \) represent the sides of triangle \( ABC \), then the length of the median through \( A \) is

A plane \( E_1 \) makes intercepts \( 1, -3, 4 \) on the coordinate axes. The equation of a plane parallel to \( E_1 \) and passing through \( (2,6,-8) \) is

If a line in octant OXYZ makes equal angles with the coordinate axes, then

The radius of a circle is increasing at the rate of \( 2 \,\text{cm/sec} \). Find the rate at which its area is increasing when the radius of the circle is \( 5 \) decimeters.

Evaluate the integral \[ \int_{0}^{1} \tan^{-1}\!\left( \frac{2x - 1}{1 + x - x^2} \right) dx \]

In a certain culture of bacteria, the rate of increase is proportional to the number present. It is found that the number doubles in 4 hours. Then the number of times the bacteria are increased in 12 hours is

If one of the lines given by the equation \( x^2 + kxy + 2y^2 = 0 \) is \( x + 2y = 0 \), then the value of \( k \) is

The area bounded by the parabola \( x^2 = 4y \), the lines \( y = 2 \), \( y = 4 \) and the Y-axis is

Evaluate the integral \[ \int \frac{\sin 2x}{\sin^2 x \cos^2 x}\, dx \]