List of practice Questions

If the plane \[ 2x + 3y + 5z = 1 \] intersects the coordinate axes at the points A, B, C, then the centroid of \( \triangle ABC \) is

The angle between the lines \[ \frac{x - 1}{4} = \frac{y - 3}{8} = \frac{z - 2}{2} \quad \text{and} \quad \frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 4}{2} \] is

The p.d.f. of a continuous r.v. \( X \) is given by \[ f(x) = \frac{x}{8}, \quad 0<x<4 \quad \text{and} \quad f(x) = 0, \text{ otherwise,} \] then \( P(X \leq 2) \) is

Evaluate \[ \int \frac{\log x - 1}{1 + (\log x)^2} \, dx \]

If a die is thrown at random, then the expectation of the number on it is

The function \[ f(x) = \frac{x + 1}{9x + x^3} \] is

Evaluate \[ \tan A + 2 \tan 2A + 4 \tan 4A + 8 \cot 8A = \]

The particular solution of the differential equation \[ y \frac{dx}{dy} = x \log x \quad \text{at} \quad x = e \text{ and } y = 1 \] is

The odds in favour of drawing a king from a pack of 52 playing cards is

The eccentricity of the ellipse \[ y^2 + 4x^2 - 12x + 6y + 14 = 0 \] is

Evaluate the integral \[ \int \frac{dx}{\cos 2x - \cos^2 x} \]

The straight lines represented by the equation \( 9x^2 - 12xy + 4y^2 = 0 \) are

The length of the latus rectum of the parabola \( x^2 + 2y = 8x - 7 \) is

In a \( \triangle ABC \), if \( 2 \cos C = \sin B \cdot \csc A \), then

Evaluate \[ \left[ \sin \left( \tan^{-1} \frac{3}{4} \right) \right]^2 + \left[ \sin \left( \tan^{-1} \frac{4}{3} \right) \right]^2 \]

If \( p, q \) are true statements and \( r \) is a false statement, then which of the following statements is a true statement?

Evaluate the integral \[ \int_{-5}^{5} \log \left( \frac{7 - x}{7 + x} \right) \, dx \]

The direction cosines of the line which bisects the angle between the positive direction of Y and Z axes are

If \( f(x) = \frac{|x|}{x} \) for \( x \neq 0 \) and \( f(x) = 1 \) for \( x = 0 \), then the function is

Out of 100 people selected at random, 10 have common cold. If five persons are selected at random from the group, then the probability that at most one person will have common cold is

The matrix \( A = \begin{bmatrix} a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix} \) is not invertible only if \( a = \)

\( \cos(36^\circ - A) \cos(36^\circ + A) + \cos(54^\circ + A) \cos(54^\circ - A) = \)

The equation of normal to the curve \( y = \sin \left( \frac{\pi x}{4} \right) \) at the point (2, 5) is

For \( f(x) = [x] \), where \( [x] \) is the greatest integer function, which of the following is true, for every \( x \in \mathbb{R} \)?

Let \( G \) be the centroid of a triangle ABC and \( O \) be any other point in that plane, then \[ OA + OB + OC + OG = \]