List of practice Questions

Solve: \[ 1 + \frac{dy}{dx} = {cosec}(x + y); \quad {put } x + y = u. \]
Find the area of the region bounded by the curve \( y = x^2 \), and the lines \( x = 1 \), \( x = 2 \), and \( y = 0 \).
Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx. \]
Evaluate: \[ \int \log x \,dx. \]
Find \( \frac{dy}{dx} \), if \( y = (\log x)^x \).
Find the vector equation of the line passing through the point having position vector \[ 4\hat{i} - \hat{j} + 2\hat{k} \] and parallel to the vector \[ -2\hat{i} - \hat{j} + \hat{k}. \]
Find \( k \), if the sum of the slopes of the lines represented by \[ x^2 + kxy - 3y^2 = 0 \] is twice their product.

Check whether the matrix 


 is invertible or not.

A particle is moving along the X-axis. Its acceleration at time \( t \) is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]
If the vectors \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \( p\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, then find the value of \( p \).
Write the compound statement ‘Nagpur is in Maharashtra and Chennai is in Tamilnadu’ symbolically.

The integrating factor of the linear differential equation \[ x \frac{dy}{dx} + 2y = x^2 \log x \] is __________.

If \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin^4 x \, dx = k \), then \( k \) is ____________.

If \( \alpha, \beta, \gamma \) are direction angles of a line and \( \alpha = 60^\circ, \beta = 45^\circ \), then \( \gamma \) is _________.

The principal solutions of the equation \( \cos\theta = \frac{1}{2} \) are _________.

The dual of statement \( t \lor (p \lor q) \) is _________.

Solve by matrix method the system of equations \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \] \
The value of \( \int \sin^2 x \, dx \) is:
For a zero-order reaction P→Q, the concentration of P becomes half of its initial concentration in 30 minutes after starting the reaction.
The concentration of P becomes zero at ____ minutes. (rounded off to the nearest integer)
Name and explain any two problems related to sustainable development.

Interpret the given diagrams A and B. Enlist the changes occurring during inspiration and expiration. 

Describe the nervous system in Planaria with a well-labeled diagram.
What are chromosomal disorders? Describe Turner’s syndrome and Klinefelter’s syndrome.
Describe the histological structure of the testis with a well-labeled diagram.