List of practice Questions

If \[ A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix} \] and \[ kA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix}, \] then the values of \( k \), \( a \), and \( b \) respectively are:
If the sum of an infinite GP \( a, ar, ar^2, ar^3, \dots \) is 15 and the sum of the squares of each term is 150, then the sum of the series \( ar^2, a r^4, ar^6, \dots \) is:
The interval in which the function \( f(x) = 4x^2 + x \) is decreasing is:
If \( \int \frac{e^x (1 + \sin x)}{1 + \cos x} \, dx = e^x f(x) + C \), then \( f(x) \) is equal to:
The curve given by \( x + y = e^{xy} \) has a tangent parallel to the Y-axis at the point:
The area enclosed between the curve \( y = \log_e(x + e) \) and the coordinate axes is:
If \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{a} \cdot \vec{b} = 1 \) and \( \vec{a} \times \vec{b} = \hat{j} - \hat{k} \), then \( \vec{b} \) is:
Find: \( \lim_{x \to 0} \frac{| \sin x |}{x} \)
The lines \[ \frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{-k} \] and \[ \frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1} \] are coplanar if:
The maximum value of \( z = 5x + 2y \) subject to the constraints: \[ x + y \leq 7, \quad x + 2y \leq 10, \quad x, y \geq 0 \]
Find the mean deviation about the mean for the data: \( 4, 7, 8, 9, 10, 12, 13, 17 \)
Bag P contains 6 red and 4 blue balls, and bag Q contains 5 red and 6 blue balls. A ball is transferred from bag P to bag Q and then a ball is drawn from bag Q. What is the probability that the ball drawn is blue?
Find the value of \[ \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) \]
The middle term in the expansion of \( \left(\frac{10}{x} + \frac{x}{10}\right)^{10} \) is:
The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x - 2)^2 \) is:
The function \( f(x) = \tan^{-1}(\sin x + \cos x) \) is an increasing function in:
Simplify \( i^{57} + \frac{1}{i^{25}} \) and find its value:
Evaluate the integral: \[ I = \int \frac{x + 3}{(x + 4)^2} e^x \,dx \]
The shortest distance between the lines \[ \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{-1} \] and \[ \frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3} \] is:
If \( P(B) = \frac{3}{5} \), \( P(A \mid B) = \frac{1}{2} \), and \( P(A \cup B) = \frac{4}{5} \), then the value of \( P(A \cup B)' + P(A' \cup B) \) is:
In a current carrying coil of inductance 60 mH,the current is changed from 2.5 A in one direction to 2.5 A in the opposite direction in 0.10 sec. The average induced EMF in the coil will be:
Arrange the following events relating to India in the correct time line sequence.
A. First 'Industrial Policy Resolution'
B. "Modern Bread", a bread manufacturing firm sold to private sector
C. First phase of green revolution
D. Setting up of planning commission
Choose the correct answer from the options given below:
If \( n(A) = 4 \) and \( n(B) = 7 \), then the difference between the maximum and minimum value of \( n(A \cup B) \) is:
The domain of the function \[ f(x) = \frac{1}{\sqrt{9 - x^2}} \] is:
Evaluate: \[ \cos^{-1} \frac{1}{2} + \sin^{-1} (1) + \tan^{-1} \frac{1}{\sqrt{3}} \]