List of top Mathematics Questions asked in CBSE CLASS XII

If $y = \sin^{-1} x$, then $(1 - x^2) \frac{d^2y}{dx^2}$ is equal to :
Find the value of $\lambda$ if the following lines are perpendicular to each other:
\[ l_1: \frac{1 - x}{-3} = \frac{3y - 2}{2\lambda} = \frac{z - 3}{3}, \quad l_2: \frac{x - 1}{3\lambda} = \frac{1 - y}{1} = \frac{2z - 5}{3} \]
Find the point on the line \( \frac{x-1}{3} = \frac{y+1}{2} = \frac{z-4}{3} \) at a distance of \( \sqrt{2} \) units from the point \( (-1, -1, 2) \).
Find: \[ J = \int \frac{\sqrt{x^2 + 1} \left[ \log(x^2 + 1) - 2 \log x \right]}{x^2} \, dx \]
Solve the differential equation \( (x - \sin y) \, dy + (\tan y) \, dx = 0 \), given \( y(0) = 0 \).
Evaluate \( \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \)
Find \( \int \frac{3x + 1}{(x - 2)^2 (x + 2)} \, dx \)
Show that of all rectangles with a fixed perimeter, the square has the greatest area.
Differentiate \( \log(x^2 + \csc^2 x) \) with respect to \( x \).
If \( A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} \), then find the value of \( K \) if \( A^2 = 6A + K I_2 \), where \( I_2 \) is the identity matrix.
If A and B are two events such that $P(B) = \frac{1}{5}$, $P(A | B) = \frac{2}{3}$ and $P(A \cup B) = \frac{3}{5}$, then $P(A)$ is :
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \hat{k}$, then the value of $\lambda$ is :
$ \int \frac{e^{10 \log x} - e^{8 \log x}}{e^{6 \log x} - e^{5 \log x}} \, dx$ is equal to :
Edge of a variable cube increases at the rate of 5 cm/s. The rate at which the surface area of the cube increases when the edge is 2 cm long is :
If $f(x) = 3x - b$, $x>1$ ; $f(x) = 11$, $x = 1$ ; $f(x) = -3x - 2b$, $x<1$ is continuous at $x = 1$, then the values of $a$ and $b$ are :
If $f(x) = 2x^8$, then the correct statement is :
The matrix $A = \begin{pmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{pmatrix}$ is a/an :
Solve the differential equation \((x^2 + y^2) \, dx + xy \, dy = 0\), \(y(1) = 1\).
(a) Find \[ \int \frac{x}{(x - 1)(x^2 + 4)} \, dx \]
Show that the derivative of \( \tan^{-1} (\sec x + \tan x) \) with respect to \( x \) is equal to \( \frac{1}{2} \) for \( \left( -\frac{\pi}{2}<x<\frac{\pi}{2} \right) \).
In a Linear Programming Program (LPP) for objective function \( Z = 14x - 10y \) subject to the constraints: \[ x + y \leq 8, \quad 3x - 2y \geq -6, \quad x, y \geq 0 \] shade the feasible region and mark the corner points in a neatly drawn graph.
If \[ \begin{bmatrix} 3 & -1 \\ 0 & 1 \\ 2 & -3 \end{bmatrix} A = \begin{bmatrix} 2 & -5 \\ -17 \end{bmatrix} \] then find matrix \( A \).
If \( P(A) = \frac{1}{5}, P(B) = \frac{3}{5} \) and \( P(A \cap B) = \frac{2}{5} \), then \( P(A' \cup B') \) is :
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \( \hat{i} + 15 \hat{j} + 6 \hat{k} \) and the other is along the vector \( 2 \hat{i} + 10 \hat{j} + 6 \hat{k} \), then the value of \( \lambda \) is :
\[ \int \frac{a^x}{\sqrt{1 - a^2 x}} \, dx \] is equal to :