List of top Mathematics Questions asked in CBSE CLASS XII

Find the shortest distance between the lines: \[ \mathbf{r}_1 = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda (\hat{i} - 2\hat{j} + 3\hat{k}) \] \[ \mathbf{r}_2 = (\hat{i} + 4\hat{k}) + \mu (3\hat{i} - 6\hat{j} + 9\hat{k}) \]
If $P(A) = \frac{1}{7}$, $P(B) = \frac{5}{7}$ and $P(A \cap B) = \frac{4}{7}$, then $P(A | B)$ 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 $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \lambda \hat{k}$, then the value of $\lambda$ is:
If \( \int \frac{1}{2x^2} \, dx = k \cdot 2x + C \), then \( k \) is equal to:
Find the angle at which the given lines are inclined to each other: \[ l_1: \frac{x - 5}{2} = \frac{y + 3}{1} = \frac{z - 1}{-3} \] \[ l_2: \frac{x}{3} = \frac{y - 1}{2} = \frac{z + 5}{-1} \]
If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:
Evaluate: \[ \int_0^\pi \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]
\[ \int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha} \, dx \] is equal to :
Find the distance of the point $P(2, 4, -1)$ from the line \[ \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}. \]
(a) Find the point Q on the line \( \frac{2x + 4}{6} = \frac{y + 1}{2} = \frac{-2z + 6}{-4} \) at a distance of \( \frac{\sqrt{5}}{2} \) from the point \( P(1, 2, 3) \).
Solve for \( x \), \[ 2 \tan^{-1} x + \sin^{-1} \left( \frac{2x}{1 + x^2} \right) = 4\sqrt{3} \]
Let \( A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} -5 \\ -2 \end{bmatrix}, C = [9 \ \ 7], \) which of the following is defined?
Solve the differential equation \( 2(y + 3) - xy \frac{dy}{dx} = 0 \); given \( y(1) = -2 \).
Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).
The integrating factor of the differential equation \( \frac{dx}{dy} = \frac{x \log x}{2 \log x - y} \) is:
Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]
The corner points of the feasible region in graphical representation of a L.P.P. are \( (2, 72), (15, 20) \) and \( (40, 15) \). If \( Z = 1x + 9y \) be the objective function, then:
If a line makes angles of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) with the positive directions of \( x \), \( y \), and \( z \)-axes respectively, then \( \theta \) is:
If \( x = e^{\frac{x}{y}} \), then prove that \( \frac{dy}{dx} = \frac{x - y}{x \log x} \).
A man needs to hang two lanterns on a straight wire whose end points have coordinates A (4, 1, -2) and B (6, 2, -3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.
Let the volume of a metallic hollow sphere be constant. If the inner radius increases at the rate of 2 cm/s, find the rate of increase of the outer radius when the radii are 2 cm and 4 cm respectively.
Determine the values of $x$ for which the function $f(x) = \frac{x-4}{x+1}$ is an increasing or a decreasing function.
Sum of two skew-symmetric matrices of the same order is always a/an:
If $y = \sin^{-1} x$, then $(1 - x^2) \frac{d^2y}{dx^2}$ is equal to :
Evaluate \( \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \)