List of practice Questions

The value of \[ \int_{-1}^{1} (x - \lfloor x \rfloor) dx \quad (\text{where} \, \lfloor \cdot \rfloor \, \text{denotes the greatest integer function}) \] is:

The correct evaluation of \[ \int_0^{\frac{\pi}{2}} \sin^4 x \, dx \] is:

The order and degree of the differential equation whose solution is \( y = cx + c^2 - 3c^3y^2 + 2 \), where \( c \) is a parameter, is:

The solution of \[ \frac{dv}{dt} + \frac{k}{m} v = -g \] is:

A unit vector perpendicular to the plane formed by the points \( (1, 0, 1) \), \( (0, 2, 2) \), and \( (3, 3, 0) \) is:

If \( \mathbf{a} = ( \hat{i} + \hat{j} + \hat{k} ) \), \( \mathbf{a} \times \mathbf{b} = \hat{i} - \hat{j} \), then \( \mathbf{b} \) is:

The mean and variance of a random variable \( X \) having binomial distribution are 4 and 2 respectively, then \( P(X = 1) \) is:

The radius of the sphere \[ x^2 + y^2 + z^2 = 49, \quad 2x + 3y - z - 5\sqrt{14} = 0 \] is:

One of the values of \[ \left( \frac{1 + i}{\sqrt{2}} \right)^{2/3} \] is:

The value of \( \lambda \) does the line \( y = x + \lambda \) touches the ellipse \( 9x^2 + 16y^2 = 144 \) is:

The combined equation of the asymptotes of the hyperbola \[ 2x^2 + 5xy + 2y^2 + 4x + 5y = 0 \] is:

The two curves \( x^3 - 3x^2 + 2 = 0 \) and \( 3x y - y^3 - 2 = 0 \) intersect at an angle of:

If at \( x = 1 \), the function \( x^4 - 62x^2 + ax + 9 \) attains its maximum value on the interval \( [0, 2] \), then the value of \( a \) is:

The value of \( c \) in Rolle's Theorem for the function \[ f(x) = e^x \sin x, \, x \in [0, \pi] \] is:

The area of the region bounded by the curve \( x = 2y + 3 \) and lines \( y = 1 \) and \( y = -1 \) is:

A signal which can be green or red with probability \( \frac{4}{5} \) and \( \frac{1}{5} \) respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is \( \frac{3}{4} \). If the signal received at station B is given, then the probability that the original signal is green is:

The value of the determinant \( \Delta = \begin{vmatrix} 1 & 4 & 3 \\ 0 & 12 & 9 \\ 1 & 2 & 2 \end{vmatrix} \) is:

If the equations \( x + ay = 0 \), \( 2x + y + az = 0 \), \( ax + y + 2z = 0 \) have non-trivial solutions, then \( a = \):

If the I.F. of the differential equation \[ \frac{dy}{dx} + 5y = \cos x \sin x e^{Adx}, \text{ then } A = \]

What is the length of the projection of \( 3\hat{i} + 4\hat{j} + 5\hat{k} \) on the xy-plane?

If \[ \int \frac{3x + 1}{(x-3)(x-5)} \, dx = \int \frac{-5}{(x-3)} \, dx + \int \frac{B}{(x-5)} \, dx, \] then the value of \( B \) is:

The vector equation of the symmetrical form of equation of straight line \( \mathbf{r} = (3\hat{i} + 7\hat{j} + 2\hat{k}) + \lambda (5\hat{i} + 4\hat{j} - 6\hat{k}) \) is:

Let the line \( \frac{x^2}{2} - \frac{y^2}{1} = 1 \) lie in the plane \( x + 3y - oz + \beta = 0 \). Then \( \beta \) equals:

The principal value of \( \sin^{-1} \left( \sin \frac{5\pi}{3} \right) \) is:

If \( \left[ 1 \quad x \quad 1 \right] \) is equal to \( \left[ 1 \quad 3 \quad 2 \right] \), then \( x \) is: