List of practice Questions

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:

The coefficient of \( x^{20} \) in the expansion of \( (1 + x^2)^{40} \left( x^2 + 2 + \frac{1}{x^2} \right)^{-5} \) is:

If \( x \) is positive then the sum to infinity of the series \[ \frac{1}{1+3x} - \frac{1}{1+3x^2} + \frac{1}{1+3x^3} - \dots \, \infty \] is:

Which of the following is soluble in yellow ammonium sulphide?

Let A and B be two sets then \( (A \cup B) \cup (A \cap B) \) is equal to:

Let \( x \) and \( y \) be two natural numbers such that \( x \cdot y = 12(x + y) \) and \( x \leq y \). Then the total number of pairs \( (x, y) \) is:

In \( \sin \theta + \sin^2 \theta = 1/2 \), cos\( 2\theta + \) cos\( \theta = 3/2 \), then cos\( \theta - \phi \) is equal to:

Let \( T(k) \) be the statement \( 1 + 3 + 5 + \dots + (2k - 1) = k^2 + 10 \). Which of the following is correct?

The amplitude of \( \sin \frac{\pi}{5} + i \left( 1 - \cos \frac{\pi}{5} \right) \) is:

The helical structure of protein is stabilized by:

Complete hydrolysis of cellulose gives: