List of top Mathematics Questions asked in CBSE CLASS XII

$ \begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix} $ is equal to:

A manufacturer produces three products $x,y,z$ which he sells in two markets. Annual sales are indicated below:

Market	Products
I	10000	2000	18000
II	6000	20000	8000

(a)If unit sale prices of $x,y$ and $z$ are Rs 2.50,Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

Find the inverse of each of the matrices,if it exists.$\begin{bmatrix} 1 & 3 & -2\\ -3 & 0 & -5\\ 2&5&0 \end{bmatrix}$

Find $ P(A \,|\, E_1) + P(A \,|\, E_2) $:

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L₁, L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

The vector with terminal point A (2, -3, 5) and initial point B (3, -4, 7) is:

Write two different vectors having same magnitude.

Write two different vectors having the same direction.

For the differential equations, find the general solution:$\frac {dy}{dx} =(1+x^2)(1+y^2)$

Find the number of all onto functions from the set {1,2,3,… ,n) to itself.

Show that \[ f(x) = e^x - e^{-x} + x - \tan^{-1} x \] is strictly increasing in its domain.

If $ M $ and $ m $ denote the local maximum and local minimum values of the function $ f(x) = x + \frac{1{x} $ ($ x \neq 0 $) respectively, find the value of $ M - m $.}

The number of points of discontinuity of \[ f(x) = \begin{cases} |x| + 3, & \text{if } x \leq -3, \\ -2x, & \text{if } -3 < x < 3, \\ 6x + 2, & \text{if } x \geq 3 \end{cases} \] is:

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights.
Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55\%, 37\% and 17\% due to severe, moderate and light turbulence respectively.
On the basis of the above information, answer the following questions:
Find the probability that an airplane reached its destination late. If the airplane reached its destination late, find the probability that it was due to moderate turbulence.

If $ A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix} $, find $ A^{-1} $ and use it to solve the following system of equations:

\[ x - 2y = 10, \quad 2x - y - z = 8, \quad -2y + z = 7. \]

Solve the following linear programming problem graphically:

Minimise $ z = 5x - 2y $

Subject to the constraints:

\[ x + 2y \leq 120, \\ x + y \geq 60, \\ x - 2y \geq 0, \\ x \geq 0, \\ y \geq 0. \]

A relation $ R $ defined on a set of human beings as $ R = \{(x, y) : x \text{ is 5 cm shorter than } y\} $ is:

If $ \vec{a} $ and $ \vec{b} $ are two non-zero vectors such that $ (\vec{a} + \vec{b}) \perp \vec{a} $ and $ (2\vec{a} + \vec{b}) \perp \vec{b} $, then prove that $ |\vec{b}| = \sqrt{2} |\vec{a}| $.

Find the local maximum value and local minimum value (whichever exists) for the function \[ f(x) = 4x^2 + \frac{1}{x}, \quad (x \neq 0). \]

Check the differentiability of the function $ f(x) = \lfloor x \rfloor $ at $ x = -3 $, where $ \lfloor \cdot \rfloor $ denotes the greatest integer function.

ABCD is a parallelogram. If $ \vec{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k} $ and $ \vec{DB} = 3\hat{i} - 6\hat{j} + 2\hat{k} $, then find $ \vec{AD} $ and hence find the area of parallelogram ABCD.

Assertion (A): A line in space cannot be drawn perpendicular to $ x $, $ y $, and $ z $ axes simultaneously.
Reason (R): For any line making angles $ \alpha, \beta, \gamma $ with the positive directions of $ x $, $ y $, and $ z $ axes respectively, \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \]

Assertion (A): For the matrix \[ A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}, \quad \text{where } \theta \in [0, 2\pi], \] $|A| \in [2, 4].$

Reason (R): $\cos \theta \in [-1, 1], \ \forall \ \theta \in [0, 2\pi].$

The integrating factor of the differential equation \[ (1 - x^2) \frac{dy}{dx} + xy = ax, \quad -1 < x < 1, \] is:

If

\[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \]

then the value of

\[ \left(\frac{24}{x} + \frac{24}{y}\right) \]

is: