List of top Mathematics Questions asked in CBSE CLASS XII

Evaluate the definite integral: \[ \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{1}{1 + \sqrt{\cot x}} \, dx \]

Mother, Father and Son line up at random for a family picture. Let events E : Son on one end and F : Father in the middle. Find \(P(E/F)\).

Find the indefinite integral: \[ \int \frac{2x + 1}{x^2 + 6x + 2} \, dx \]

Check whether the function \( f : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} \) defined as \( f(x, y) = (2y, 3x) \) is injective (one-to-one) or not.

The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that:
(i) target is hit
(ii) atleast one shot misses the target.

Find the general solution of the differential equation: \( y \log_e y \frac{dx}{dy} + x = \frac{y^2}{2} \).

Solve the following homogeneous differential equation: \( x \frac{dy}{dx} = y - x \sin^2\left(\frac{y}{x}\right) \), given the initial value condition that \( y = \frac{\pi}{6} \) when \( x = 1 \).

Solve the following linear programming problem graphically: Maximize \( Z = 8000x + 12000y \) subject to the boundary constraints: \( 3x + 4y \le 60 \), \( x + 3y \le 30 \), with non-negativity restrictions \( x \ge 0, y \ge 0 \).

If \( x = \sin t - \cos t \) and \( y = \sin t \cos t \), find the value of \(\frac{dy}{dx}\) evaluated at \( t = \frac{\pi}{4} \).

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

Find the coordinates of the foot of the perpendicular drawn from the origin \((0, 0, 0)\) to the straight line given by the equations \( \frac{x-2}{1} = \frac{y+1}{1} = \frac{z-3}{-1} \).

Find the absolute maximum value of \( f(x) = \cos x + \sin^2 x \) in the closed interval \( x \in [0, \pi] \).

If \(\vec{AB} = \hat{j} + \hat{k}\) and \(\vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k}\) represent two vectors along the sides \(AB\) and \(AC\) of \(\Delta ABC\), prove that the median vector satisfies \(\vec{AB} + \vec{AC} = 2\vec{AD}\), where \(D\) is the midpoint of \(BC\). Hence, find the length of the median \(AD\).

Assertion (A): A line can have direction cosines $\langle 1, 1, 1 \rangle$.
Reason (R): $\cos\theta = 1$ is possible for $\theta = 0$.

Assertion (A): $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.
Reason (R): $|\vec{a} \times \vec{b}| = (\vec{a} \cdot \vec{b})\tan\theta$, where $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ ($\theta \neq \frac{\pi}{2}$).

If $e^{x + y} = 3x$, then $\frac{dy}{dx}$ is equal to:

If $A$ and $B$ are square matrices of the same order, then which of the following statements is/are always true?
(i) $(A + B)(A - B) = A^2 - B^2$
(ii) $AB = BA$
(iii) $(A + B)^2 = A^2 + AB + BA + B^2$
(iv) $AB = 0 \implies A = 0$ or $B = 0$

A relation $R$ on the set $A = \{1, 2, 3\}$ defined as $R = \{(1, 2), (2, 1), (2, 2)\}$ is:

For a square matrix $A$ of order $n$, the inverse matrix $(3A)^{-1}$ is equal to:

For two events $A$ and $B$ such that $P(A) \neq 0$ and $P(B) \neq 1$, the conditional probability $P(A'/B')$ is equal to:

The direction cosines of the line given by $x = y = 1 - z$ are:

A vector of magnitude 3 making equal angles with the $x$ and $y$ axes and perpendicular to the $z$ axis is:

For a feasible region bounded by the corner points $(0,3)$, $(3,2)$, and $(5,0)$ in the first quadrant, the non-trivial constraints of the linear programming problem are:

In a linear programming problem, the linear function which has to be maximized or minimized is called:

If $|\vec{a}| = 5$ and $-2 \le \lambda \le 1$, then the sum of the greatest and the smallest value of $|\lambda \vec{a}|$ is: