List of top Questions asked in CBSE CLASS XII- 2026

There is a triangular park in the society. The park is divided into two sections as shown in the figure. The vertices of the triangular park ABC are A(0, 4), B(\(-2\), 0) and C(3, 0). Based on the above information, answer the following questions :

Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length \( l_1 \) and \( l_2 \) are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :

Opposite sides of a square are along the lines : \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \quad \text{and} \quad \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \] Find the area of the square if direction ratios of other pair of opposite sides of the square are given by $\langle -3, 6, p \rangle$. Also, find the value of $p$.

Show that $f : \mathbf{R}_+ \to [-5, \infty)$ given by $f(x) = 4x^2 + 4x - 5$ is both one-one and onto where $\mathbf{R}_+ = [0, \infty)$. Also, find $p \in \mathbf{R}_+$ such that $f(p) = 3$.

Solve the following Linear Programming Problem graphically :
Maximise $Z = 12x + 18y$
subject to the constraints \[ x + y \le 1200, \quad x - 2y \ge 0, \quad x + 3y \ge 600, \quad x \ge 0, \, y \ge 0 \]

Find : \( \int \frac{dx}{x^{1/2} + x^{1/3}} \)

Find : \( \int \tan^{-1}\left(\frac{1 - x}{1 + x}\right) \, dx \)

Evaluate : \( \int_{-\pi/2}^{\pi/2} \frac{\cos^2 x}{2^x + 1} \, dx \)

Represent the equations of lines $l_1$ and $l_2$ in vector form and check whether they are intersecting or not. \[ l_1 : \frac{x + 3}{-3} = \frac{y - 1}{1} = \frac{z - 5}{5} \quad \text{and} \quad l_2 : \frac{x + 1}{-1} = \frac{2 - y}{-2} = \frac{z - 5}{5} \]

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that :
(i) exactly one surgery is successful,
(ii) at most two surgeries are successful.

Find the general solution of the differential equation \( (x^2 + y^2) \, dy = xy \, dx \).

Find the particular solution of the differential equation \( \frac{dy}{dx} - 3y \cot x = \sin 2x \), given that \( y = 2 \) when \( x = \frac{\pi}{2} \).

Find : \( \int \frac{x + 3}{x^2 + 4x + 5} \, dx \)

If \( x = e^{t + \frac{1}{t}} \) and \( y = e^{t - \frac{1}{t}} \), then find \( \frac{dy}{dx} \) at \( t = -2 \).

Find the vector equation of a line passing through the origin and perpendicular to both the lines \( \vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k}) \) and \( \vec{r} = \mu(\hat{i} - \hat{j} + \hat{k}) \).

Find the value of \( \sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))] \).

If the lines \( \frac{x - 3}{1} = \frac{1 - y}{1} = \frac{z + 2}{p} \) and \( \frac{2 - x}{3} = \frac{y + 1}{5} = \frac{z + 56}{2p} \) are perpendicular to each other, then find the value(s) of p.

If for two unit vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}| \), then find the angle between \( \vec{a} \) and \( \vec{b} \).

Assertion (A) : One of the particular solutions of the differential equation \( \frac{dy}{dx} = e^{x+y} \) can be \( e^x + e^{-y} = -2 \).
Reason (R) : \( e^x + e^{-y} = C \) is the general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \).

Assertion (A) : The vectors \( \vec{a} \) and \( (-2\vec{a}) \), where \( \vec{a} \neq \vec{0} \), are collinear vectors.
Reason (R) : \( \vec{a} \cdot (-2\vec{a}) = 0 \).

If A and B are skew-symmetric matrices of same order, then \( AB' + BA' \) is a/an :

If \( A^2 = 4A + 3I \) and \( A^{-1} = xA + yI \), then the value of \( (x + y) \) is :

If a square matrix A is such that \( A^2 = A \) and \( (I - A)^3 = xA + I \), then value of x must be :

If a matrix B is such that \( B \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 0 & 1 & 1 2 & 0 & 1 \end{bmatrix} \), then the order of matrix B is :