List of top Mathematics Questions asked in CBSE CLASS XII

If \( F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \) and \( [F(x)]^2 = F(kx) \), then the value of \( k \) is:
Find the value of \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\frac{\pi}{2}\right)\right]. \)
Find the domain of the function \( f(x) = \sin^{-1}(x^2 - 4) \). Also, find its range.

If \( y = A \sin 2x + B \cos 2x \) and \[ \frac{d^2y}{dx^2} - k y = 0, \text{ find the value of } k. \]

Show that a function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{2x}{1 + x^2} \) is neither one-one nor onto. Further, find set \( A \) so that the given function \( f : \mathbb{R} \to A \) becomes an onto function.
(a) If \[ A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] find \( (AB)^{-1} \). Also, find \( |AB|^{-1} \).
(b) Given \[ A = \begin{bmatrix} 2 & 3 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2 \end{bmatrix}, \text{ find } A^{-1}. \text{ Use it to solve the following system of equations:} \] \[ x + y + z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ x + y + 2z = 4. \]
A relation \( R \) is defined on \( \mathbb{N} \times \mathbb{N} \) (where \( \mathbb{N} \) is the set of natural numbers) as: \[ (a, b) \, R \, (c, d) \iff a - c = b - d. \] Show that \( R \) is an equivalence relation.
Identify and write all the constraints which determine the given feasible region in Figure-2.
If the objective is to minimize cost \( Z = 16x + 20y \), find the values of \( x \) and \( y \) at which cost is minimum. Also, find the minimum cost assuming that minimum cost is possible for the given unbounded region.

If \( \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is: 

Which of the following statements is \textit{not true about equivalence classes \( A_i \) (\( i = 1, 2, \ldots, n \)) formed by an equivalence relation \( R \) defined on a set \( A \)?}

Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h.

The relation between fuel consumption \( F \) (liters per 100 km) and speed \( V \) (km/h) under some constraints is given as:

\[ F = \frac{V^2}{500} - \frac{V}{4} + 14. \]

 

On the basis of the above information, answer the following questions:

(i) Find \( F \), when \( V = 40 \, \text{km/h} \).
(ii) Find \( \frac{dF}{dV} \).
(iii)(a) Find the speed \( V \) for which fuel consumption \( F \) is minimum.
OR
(b) Find the quantity of fuel required to travel \( 600 \, \text{km} \) at the speed \( V \) at which \( \frac{dF}{dV} = -0.01 \).

Find the probability that the airplane will not crash.
Find the value of \( p \) for which the lines \[ \vec{r_1} = \hat{i} + (2\lambda + 1) \hat{j} + (3\lambda + 2) \hat{k} \] and \[ \vec{r_2} = \hat{i} - 3 \hat{j} + (p\mu + 7) \hat{k} \] are perpendicular to each other and also intersect. Also, find the point of intersection of the given lines.
Evaluate: \[ \int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} \, dx. \]
Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
(b) Find \( \frac{dy}{dx} \), \text{ if } \[ 5^x + 5^y = 5^{x + y}. \]
A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).
If vectors \( \vec{a}, \vec{b} \) and \( 2\vec{a} + 3\vec{b} \) are unit vectors, then find the angle between \( \vec{a} \) and \( \vec{b} \).
Find: \[ \int x^2 \sin^{-1}(x^{3/2}) \, dx. \]
Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]
Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
(a) Check the differentiability of \( f(x) = | \cos x | \) at \( x = \frac{\pi}{2} \).
Evaluate: \[ \int_0^a \frac{x^2}{x^6 + a^6} \, dx \]