List of top Mathematics Questions asked in CBSE CLASS XII

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: \[ \int x^2 \sin^{-1}(x^{3/2}) \, dx. \]
(a) Check the differentiability of \( f(x) = | \cos x | \) at \( x = \frac{\pi}{2} \).
Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]
Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
Evaluate: \[ \int_0^a \frac{x^2}{x^6 + a^6} \, dx \]
Assertion (A): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
Assertion (A): For any symmetric matrix \( A \), \( B'A B \) is a skew-symmetric matrix.
Reason (R): A square matrix \( P \) is skew-symmetric if \( P' = -P \).
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
If \[ \int_0^2 2e^{2x} \, dx = \int_0^a e^x \, dx, \text{ the value of 'a' is:} \]
If a line makes an angle of \( 30^\circ \) with the positive direction of \( x \)-axis, \( 120^\circ \) with the positive direction of \( y \)-axis, then the angle which it makes with the positive direction of \( z \)-axis is:
If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements is:
The unit vector perpendicular to both vectors \( \hat{i} + \hat{k} \) and \( \hat{i} - \hat{k} \) is:
Direction ratios of a vector parallel to the line \( \frac{x - 1}{2} = -y = \frac{2z + 1}{6} \) are:
The degree of the differential equation \( (y'')^2 + (y')^3 = x \sin(y') \) is:
The derivative \( \frac{d}{dx} [\cos(\log x + e^x)] \) at \( x = 1 \) is:
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
The number of all scalar matrices of order 3, with each entry \( -1, 0, 1 \), is:
The common region determined by all the constraints of a linear programming problem is called:
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The number of solutions of the differential equation \( \frac{dy}{dx} - y = 1 \), given that \( y(0) = 1 \), is:
The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy.
\includegraphics[width=\linewidth]{latex1.png} A dietician wishes to minimize the cost of a diet involving two types of foods, food \( X \) (in kg) and food \( Y \) (in kg), which are available at the rate of \( \text{₹} 16/\text{kg} \) and \( \text{₹} 20/\text{kg} \), respectively. The feasible region satisfying the constraints is shown in Figure-2. On the basis of the above information, answer the following questions: [(i)] Identify and write all the constraints which determine the given feasible region in Figure-2. [(ii)] 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.
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.
\includegraphics[width=\linewidth]{latex.png} 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.
\hspace*{0.8cm} OR
(b) Find the quantity of fuel required to travel \( 600 \, \text{km} \) at the speed \( V \) at which \( \frac{dF}{dV} = -0.01 \).
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.}