List of top Mathematics Questions asked in CBSE CLASS XII

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 \( P' = -P \).
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 \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
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 the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements 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:
Direction ratios of a vector parallel to the line \( \frac{x - 1}{2} = -y = \frac{2z + 1}{6} \) are:
The unit vector perpendicular to both vectors \( \hat{i} + \hat{k} \) and \( \hat{i} - \hat{k} \) 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 degree of the differential equation \( (y'')^2 + (y')^3 = x \sin(y') \) is:
The number of all scalar matrices of order 3, with each entry \( -1, 0, 1 \), is:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
The common region determined by all the constraints of a linear programming problem is called:
The number of solutions of the differential equation \( \frac{dy}{dx} - y = 1 \), given that \( y(0) = 1 \), is:
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.
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(b) Find the quantity of fuel required to travel \( 600 \, \text{km} \) at the speed \( V \) at which \( \frac{dF}{dV} = -0.01 \).
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.
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.}
Find: \[ \int x^2 \sin^{-1}(x^{3/2}) \, dx. \]
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.
Find the equation of the line which bisects the line segment joining points \( A(2, 3, 4) \) and \( B(4, 5, 8) \) and is perpendicular to the lines: \[ \frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7} \quad \text{and} \quad \frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{-5}. \]
The position vectors of vertices of \( \Delta ABC \) are \( A(2\hat{i} - \hat{j} + \hat{k}) \), \( B(\hat{i} - 3\hat{j} - 5\hat{k}) \), and \( C(3\hat{i} - 4\hat{j} - 4\hat{k}) \). Find all the angles of \( \Delta ABC \).
Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
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 \( X \).