List of top Mathematics Questions asked in CBSE CLASS XII

Find the absolute maximum and absolute minimum values of the function \( f(x) = \frac{x}{2} + \frac{2}{x} \) on the interval \( [1, 2] \).
Find the intervals in which the function \( f(x) = \frac{\log x}{x} \) is strictly increasing or strictly decreasing.
Find the value of \( a \) and \( b \) so that the function \( f \) defined as: \[ f(x) = \begin{cases} \frac{x - 2}{|x - 2|} + a, & x < 2, \\ a + b, & x = 2, \\ \frac{x - 2}{|x - 2|} + b, & x > 2, \end{cases} \] is a continuous function.
The chances of \( P \), \( Q \), and \( R \) getting selected as CEO of a company are in the ratio \( 4 : 1 : 2 \), respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, \( P, Q, \) or \( R \), are \( 0.3, 0.8, \) and \( 0.5 \), respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of \( R \) as CEO.
Check the differentiability of \( f(x) \) at \( x = 1 \), where: \[ f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1, \\ 3 - x, & 1 \leq x \leq 2. \end{cases} \]
If \( x = e^{y^2} \), prove that \( \frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2} \).
Given \[ \frac{d}{dx} F(x) = \frac{1}{\sqrt{2x - x^2}} \] and \( F(1) = 0 \), find \( F(x) \).
Assertion (A): Every scalar matrix is a diagonal matrix.
Reason (R): In a diagonal matrix, all the diagonal elements are 0.
If \( A = [a_{ij}] \) is an identity matrix, then which of the following is true?
If \( g(x) \) is a continuous function satisfying \( g(-x) = -g(x) \), then \( \int_{0}^{2a} g(x) \, dx \) is equal to: {5pt}
The point of inflexion of a function \( f(x) \) is the point where: {5pt}
Let \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] be a square matrix such that \[ \text{adj } A = A. \] Then, \( (a + b + c + d) \) is equal to:
A function \( f : \mathbb{R} \to A \) defined as \( f(x) = x^2 + 1 \) is onto, if \( A \) is: {5pt}
For any square matrix \( A \), \( (A - A') \) is always: {5pt}
If \[ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \] then \( A^{-1} \) is:

Of the following, which group of constraints represents the feasible region given below?
 

The derivative of \( 5^x \) w.r.t. \( e^x \) is: {5pt}

If \[ \begin{bmatrix} 1 & 3 & 1 \\ k & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \] has a determinant of \( \pm 6 \), then the value of \( k \) is:
It is given that function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.
Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \hat{i} + 2\hat{j} - \hat{k} \) and \( -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( | \overrightarrow{AB} | : | \overrightarrow{BC} | \).
Evaluate: \[ \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx \]
Find: \[ \int \frac{2 + \sin 2x}{1 + \cos 2x} e^x \, dx \]
If \( x \cos(p + y) + \cos p \sin(p + y) = 0 \), prove that \( \cos p \frac{dy}{dx} = -\cos^2(p + y) \), where \( p \) is a constant.
Assertion (A): Projection of \( \vec{a} \) on \( \vec{b} \) is the same as the projection of \( \vec{b} \) on \( \vec{a} \).
Reason (R): The angle between \( \vec{a} \) and \( \vec{b} \) is the same as the angle between \( \vec{b} \) and \( \vec{a} \) numerically.
Let \( E \) and \( F \) be two events such that \( P(E) = 0.1, P(F) = 0.3, P(E \cup F) = 0.4 \). Then \( P(F \,|\, E) \) is: