List of practice Questions

Find \( \int \sqrt{\frac{x + 2}{x - 2}} dx \)

If \( I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x} \) and \( I_2 = \int_{-1/2}^{1/2} |x| dx \), then show that \( I_1 - 4I_2 = 0 \).

Find the general solution of the differential equation \( (y^2 - x^2)dx = 2xy dy \)

Find the particular solution of the differential equation \( (1 + e^{2x})dy + (1 + y^2)e^x dx = 0 \), given that \( y(1) = 0 \).

Using vectors, find the area of \( \Delta ABC \) with vertices \( A(1, 2, 3) \), \( B(2, -1, 4) \) and \( C(4, 5, -1) \).

Vectors \( \vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{k} \) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

Evaluate : \( \int_{0}^{1} \log(1 + x^2)dx \)

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

The probability of simultaneous occurrence of atleast one of the two events X and Y is a. If the probability that exactly one of the events X, Y occurs is b, prove that \( P(X') + P(Y') = 2 - 2a + b \).

Simplify : \( \tan^{-1} \left( \frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x} \right) \), \( 0 < x < \frac{\pi}{4} \).

Evaluate : \( \tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right) \).

If \( \sqrt{3}(x^2 + y^2) = 4xy \), then find \( \frac{dy}{dx} \) at \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

Simplify : \( \sin^{-1} \sqrt{\frac{1 + \cos 2x}{2}} \), \( 0 < x < \frac{\pi}{2} \).

Assertion (A) : Lines given by \( x = py + q, z = ry + s \) and \( x = p'y + q', z = r'y + s' \) are perpendicular to each other when \( pp' + rr' + 1 = 0 \).
Reason (R) : Two lines \( \vec{r} = \vec{a}_1 + \lambda\vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu\vec{b}_2 \) are perpendicular to each other if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).

The area bounded by the curve \( y = x|x| \), x-axis and the ordinates \( x = -1 \) and \( x = 1 \) is given by

The integrating factor of differential equation \( R \frac{dx}{dy} + Px = Q \) where \( P, Q, R \) are functions of \( y \) is

The order and degree of the differential equation : \( \frac{d}{dx}(y')^3 + (y')^3 = 1 \) respectively are where \( y' = \frac{dy}{dx} \)

The value of \( p \) for which vectors \( \hat{i} + 2\hat{j} + 3\hat{k} \) and \( 2\hat{i} - p\hat{j} + \hat{k} \) are perpendicular to each other is

The value of \( m \) for which the points with position vectors \( -\hat{i} - \hat{j} + 2\hat{k} \), \( 2\hat{i} + m\hat{j} + 5\hat{k} \) and \( 3\hat{i} + 11\hat{j} + 6\hat{k} \) are collinear, is

One of the values of \( x \) for which \( \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1 \) is

If A and B are symmetric matrics of same order, then \( (AB - BA) \) is a

If \( \int \frac{3ax}{b^2 + c^2x^2} dx = A \log |b^2 + c^2x^2| + K \), then the value of \( A \) is

The value of \( \int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1} dx \) is

The degree of an objective function of a linear programming problem is

If \( \sin^{-1}x + \pi = y \), then