List of practice Questions

If three numbers \( a, b, c \) constitute both an (A)P and G.P, then

cos\(^6\) A - sin\(^6\) A is equal to

The distance between the foci of a hyperbola is 16 and its eccentricity is \( \sqrt{2} \). Then its equation is

The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between the lines \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is

If \( 2A + 3B = \begin{bmatrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{bmatrix} \) and \( A + 2B = \begin{bmatrix} 5 & 0 & 3 \\ 1 & 6 & 2 \end{bmatrix} \), then \( B = \)

If \( \csc(90^\circ + A) + x \cos A \cot(90^\circ + A) = \sin(90^\circ + A) \), then the value of

The scalar components of a unit vector which is perpendicular to each of the vectors \( \hat{i} + 2\hat{j} - \hat{k} \) and \( 3\hat{i} - \hat{j} + 2\hat{k} \) are:

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. The number of ways in which he can choose the 7 questions is:

Evaluate the integral \[ \int \sqrt{\csc x - 1} \, dx = \]

Evaluate the integral \[ \int_0^2 x^2 + 2x - 3 \, dx \]

Bag A contains 3 white and 2 red balls. Bag B contains only 1 white ball. A fair coin is tosse(D) If head appears then 1 ball is drawn at random from bag A and put into bag (B) However if tail appears then 2 balls are drawn at random from bag A and put into bag (B) Now one ball is drawn at random from bag (B) Given that the drawn ball from B is white, the probability that head appeared on the coin is

In a 12 storey house, 10 people enter a lift cabin. It is known that they will leave the lift in pre-decided groups of 2, 3, and 5 people at different storeys. The number of ways they can do so if the lift does not stop up to the second storey is

The minimum value of \( Z = 3x + 5y \), given subject to the constraints \( x + y \geq 2 \), \( x + 3y \geq 3 \), \( x, y \geq 0 \), is:

\[ \lim_{x \to 0} \frac{a^x - b^x}{x} \text{ is equal to:} \]

The coordinates of the vertices of the triangle are \( A(-2, 3, 6) \), \( B(-4, 4, 9) \), and \( C(0, 5, 8) \). The direction cosines of the median \( BE \) are:

How many factors of \( 2^5 \times 3^6 \times 5^2 \) are perfect squares?

The general solution of the differential equation \[ (1 + y^2) \, dx = ( \tan^{-1} y - x ) \, dy \] is:

The function \( f(x) = \frac{x}{2} + \frac{2}{x} \) has a local minimum at:

If the direction ratios of two lines are given by \( 3lm - 4ln + mn = 0 \) and \( l + 2m + 3n = 0 \), then the angle between the lines is:

Which of the following is a singleton set?

If the conjugate of \( (x + iy)(1 - 2i) \) be \( 1 + i \), then:

If the length of the major axis of an ellipse is 3 times the length of the minor axis, then its eccentricity is:

The particular solution of \( e^{\frac{dy}{dx}} = 2x + 1 \) given that \( y = 1 \) when \( x = 0 \) is:

If \( A = \left( \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix} \right), \, P = \left( \begin{matrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{matrix} \right), \, Q = P^T A P \), then \( PQ^{2014} \) is: