List of practice Questions

Two dice are thrown together 4 times. The probability that both dice will show same numbers twice is:

Evaluate: \[ \int_0^{\pi/2} \frac{x}{\sqrt{4 - x^2}} \, dx \]

The area bounded by the curve \( y = \sin x \), x-axis and the ordinates \( x = 0 \) and \( x = \pi/2 \) is:

The differential equation whose solution is \( Ax^2 + Bx + C = 1 \) where A and B are arbitrary constants is of:

The unit vector perpendicular to the vectors \( 6i + 2j + 3k \) and \( 3i - 6j - 2k \) is:

If \( x = a \sin \theta \) and \( y = b \cos \theta \), then \( \frac{d^2y}{dx^2} \) is:

If \( f(x) = x^\alpha \log x \) and \( f(0) = 0 \), then the value of \( \alpha \) for which Rolle's theorem can be applied in \( [0, 1] \) is:

If the function \( f(x) = ax + b \), \( 2 < x < 4 \), is continuous at \( x = 2 \) and 4, then the values of \( a \) and \( b \) are:

If \( f(x) = a^2 - 1 \), \( x^3 - 3x + 5 \) is a decreasing function of \( x \in R \), then the set of possible values of \( a \) (independent of \( x \)) is:

The diagonal of a square is changing at the rate of \( 0.5 \, \text{cm/sec} \). Then the rate of change of area, when the area is 400 \( \text{cm}^2 \), is equal to:

If the normal to the curve \( y = f(x) \) at the point \( (3, 4) \) makes an angle \( 3\pi/4 \) with the positive x-axis, then \( f'(3) \) is:

The probability of simultaneous occurrence of at least one of two events A and B is \( p \). If the probability that exactly one of A, B occurs is \( q \), then \( P(A' \cup B') \) is equal to:

If \( f(x) \) is an even function and \( g(x) \) is an odd function, then the function \( f \circ g \) is:

\( \tan^{-1} \left( \frac{1}{4} \right) + \tan^{-1} \left( \frac{1}{9} \right) \) equals to:

If \( k \leq \sin^{-1} x + \cos^{-1} x + \tan^{-1} x \leq 5 \), then:

The equations \( 2x + 3y + 4 = 0, 3x + 4y + 6 = 0 \) and \( 4x + 5y + 8 = 0 \) are:

The nearest point on the line \( 3x + 4y = 12 \) from the origin is:

The length of the tangent drawn from any point on the circle \( x^2 + y^2 + 2\lambda x + \mu = 0 \) to the circle \( x^2 + y^2 + 2\gamma x + \lambda = 0 \), where \( \mu \geq \lambda \), is:

Find the eccentricity of the conic represented by \( x^2 - y^2 - 4x + 4y + 16 = 0 \):

\( \limₓ → ∞ \frac1 - \tan \left( \fracx2 \right)1 + \tan \left( \fracx2 \right) = ?

Let \( f(x + y) = f(x) \cdot f(y) \) for all \( x, y \), where \( f(0) = 0 \). If \( f(5) = 2 \) and \( f'(0) = 3 \), then \( f'(5) \) is equal to:

If sample A contains 100 observations 101, 102, .... 200 and sample B contains 100 observations 151, 152, .... 250, then the ratio of variance \( \frac{V_A}{V_B} \) is:

If \( x \to \infty \), then the value of \( x^4 + 3x^3 + 2x^2 - 11x - 6 \) is:

In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes?

The number of positive integral solutions of \( abc = 30 \) is: