List of top Mathematics Questions asked in CBSE CLASS XII

In an agricultural institute, scientists conduct experiments with varieties of seeds to grow them in different environments for producing healthy plants and obtaining higher yields.
A scientist observed that a particular seed grew very fast after germination. He recorded the growth of the plant from the time of germination and modeled its growth with the function:
Given:
\( f(x) = \frac{1}{3}x^3 - 4x^2 + 15x + 2 \),   \( 0 \leq x \leq 10 \)
where \( x \) is the number of days the plant is exposed to sunlight.
On the basis of the above information, answer the following questions: 

Relation \( R \) on \( A = \{1,2,3,4,5,6,7\} \):
\[ R = \{(a,b) : a \text{ and } b \text{ are both odd or both even} \} \] Show that \( R \) is an equivalence relation and find the equivalence class \([1]\).
Show that the relation \( S = \{ (a, b) : a \leq b^3,\ a, b \in \mathbb{R} \} \) is neither reflexive, nor symmetric, nor transitive.
Find the shortest distance between the pair of lines:
Line \( L_1 \) : \[ \frac{x - 1}{2} = \frac{y + 1}{3} = z, \] Line \( L_2 \) : \[ \frac{x + 1}{5} = \frac{y - 2}{1} = z = 2. \]
Show that the lines
\[ \frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7} \quad \text{and} \quad \frac{x - 2}{1} = \frac{y - 4}{3} = \frac{z - 6}{5} \] intersect and find their point of intersection.
Find the area of triangle ABC bounded by:
5x −2y−10 =0, x−y−9=0, 3x−4y−6=0, usingintegrationmethod.
Find the particular solution of: \[ (1 + x^2) \frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}, y = 0 \text{ when } x = 1. \]
Minimise $z = 3x + 8y$ subject to: \[ 3x + 4y \geq 8, 5x + 2y \geq 11, x \geq 0,\; y \geq 0. \]
Two bags: A (2W, 3R), B (4W, 5R). One red drawn. Find probability it came from B.
Given: \[ A = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix} \]
Find \( A^{-1} \) and hence solve the following system of equations:
\[ \begin{cases} 3x + 2y + z = 2000 \\ 4x + y + 3z = 2500 \\ x + y + z = 900 \end{cases} \]
Find: \[ I = \int \frac{2x^2 + 1}{x^2(x^2 + 4)} \, dx \]
Find the particular solution of the differential equation:
\[ \frac{dy}{dx} = \frac{xy}{x^2 + y^2} \] given that \( y = 1 \) when \( x = 0 \).
50 people, 20 always tell truth.Two selected. Find probability distribution of number of truth-tellers.
Find the integral:
\[ I = \int \frac{\cos \theta \, d\theta}{\sqrt{3 - 3\sin \theta - \cos^2 \theta}} \]
Evaluate: \[ \int_{1}^{3} (|x - 1| + |x - 2|) dx. \]
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto
Find the intervals in which the function
\( f(x) = x^4 - 4x^3 + 4x^2 + 15 \) is increasing .
If a line makes angles \( \alpha, \beta, \gamma \) with the x-axis, y-axis, and z-axis respectively,
then prove that: \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]
Evaluate: \[ I = \int_{0}^{\pi/2} \frac{ x \sin x \cos x }{ \sin^4 x + \cos^4 x } \, dx. \]
If \( |\vec{a}| = 7 \), \( |\vec{b}| = 24 \), \( |\vec{c}| = 25 \) and
\( \vec{a} + \vec{b} + \vec{c} = \vec{0} \),
find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

Simplify:
\( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \)

If $x = a \cos \theta + b \sin \theta$, $y = a \sin \theta - b \cos \theta$, then which one is true?
The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function $Z = 2ax + by$ occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is:
corner points of the bounded
It is given that \[ X \begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}. \text{ Then matrix X is:} \]
Assertion (A): Quadrilateral formed by vertices A(0, 0, 0), B(3, 4, 5), C(8, 8, 8) and D(5, 4, 3) is a rhombus.
Reason (R): ABCD is a rhombus if $AB = BC = CD = DA$, and $AC \ne BD$.