List of practice Questions

Evaluate the integral \[ \int \frac{e^x}{2^x} dx. \]

Ice is coated uniformly around a sphere of radius 15 cm. If ice is melting at the rate of \( 80 { cm}^3/{min} \) when the thickness is 5 cm, then the rate of change of thickness of ice is

If \[ f(x) = \sqrt[3]{x^2} + \sqrt{x}, \] then the value of \( f'(64) \) is equal to
If the function \[ f(x) = \begin{cases} x^2, & \text{for } x < 4 \\ 5x - k, & \text{for } x \geq 4 \end{cases} \] is continuous at \( x = 4 \), then the value of \( k \) is equal to
The angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \). If \( | \vec{a} \cdot \vec{b} |^2 = 15 \), then \( | \vec{a} \times \vec{b} |^2 \) is equal to
If \( \vec{a} = 5\hat{i} - 7\hat{j} + 9\hat{k} \) and \( \vec{b} = -5\hat{i} + 7\hat{j} - 9\hat{k} \), then \( \vec{a} \cdot (\vec{a} \times \vec{b}) + (\vec{a} + \vec{b}) \cdot \hat{b} \) is equal to
Let \( A(0,3,-3) \), \( B(1,1,1) \) and \( C(2,0,3) \) be three points in space. Then the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to
Let \( \alpha, \beta, \gamma \) be the direction cosines of a vector \( \vec{a} = x \hat{i} + y \hat{j} + z \hat{k} \), where \( z<0 \). If \( \alpha = \frac{-4}{\sqrt{105}} \) and \( \beta = \frac{\sqrt{5}}{\sqrt{21}} \), then \( \gamma \) is equal to
If \( (a,-6) \) lies on the perpendicular bisector of the line segment joining \( (-2,-1) \) and \( (4,-13) \), then the value of \( a \) is equal to
For an ellipse, the foci are \( F(3,0) \) and \( F'(-3,0) \). If the length of the minor axis is 8, then the length of the major axis is equal to
Find the equation of the parabola with focus at \( (3,1) \) and vertex at \( (5,1) \).
Find the values of \( \alpha \) for which the circle \[ x^2 + y^2 + \alpha x - 8y + 56 = 0 \] has radius 3.
Find the equation of the line perpendicular to the line \[ 7x - 5y = 11 \] and passing through the point \( (7, -9) \).

The value of \( \cosec x + \cot x \) is

Evaluate \[ \frac{\cosec^2(\theta) - 1}{\cosec^2(\theta)} - \frac{\sec^2(\theta) - 1}{\sec^2(\theta)} \]

If \[ 7 \cos^2 x + 3 \sin^2 x = 6, \] then the value of \( \cos 2x \) is equal to
Let \[ \sum_{k=1}^{15} \sin (t_k) = 0 \quad {and} \quad \sum_{k=1}^{15} \sin (3t_k) = \frac{-24}{5}, \] where \( t_1, t_2, t_3, \dots \) are real numbers. Then the value of the sum \[ \sum_{k=1}^{15} \sin^3 (t_k) \] is equal to
The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{2}{3} \right) + \cot^{-1} \left( \frac{9}{7} \right) \] is equal to
The value of \( \sin \left( 2 \cos^{-1} \left( \frac{5}{12} \right) + \sin^{-1} \left( \frac{5}{12} \right) \right) \) is equal to
If \( \sin t + \cos t = \sqrt{2} \), then \( \tan t + \cot t \) is equal to
The value of \( \tan \left( \cos^{-1} \left( \frac{-24}{25} \right) \right) \) is equal to

Simplify: \( \tan x - \cot x + \csc x \sec x \)

Let \( A \) and \( B \) be two events. If \( P(A) = 0.49 \), \( P(B) = 0.3 \) and \( P(A | B^c) = 0.4 \), then \( P(A | B) \) is equal to
If two dice are rolled simultaneously, then the probability that the difference of the numbers on the two dice equals to zero is
The standard deviation of a data set \( x_1, x_2, \dots, x_6 \) (\( x_i>0 \)) is 2. If \[ \sum_{i=1}^{9} x_i^2 = 360, \] then the mean of the data set is