List of top Questions asked in CBSE CLASS XII- 2026

Find the foot of the perpendicular from the point \( (0, 2, 3) \) on the line \( \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \) and hence find the length of the perpendicular.

Find the differential of \( x^{\cot x} + \frac{2x^2-3}{2x^2-x+2} \) with respect to \( x \).

Find the value of \( p \) if the shortest distance between the lines \( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \) is \( \frac{3}{\sqrt{2}} \) units.

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost Rs.60, 2 apples, 4 oranges and 6 bananas cost Rs.90, whereas 6 apples, 2 oranges and 3 bananas cost Rs.70. Using matrix method, find the cost of one fruit of each kind.

Sketch the curve \( \{(x, y) : 100x^2 + 25y^2 = 2500\ \) and find the area of the region enclosed by it, using integration.}

A die is rolled. Consider events : \( A = \{1, 2, 5\}, B = \{3, 5\}, C = \{2, 3, 4, 5\} \) and hence find : (i) \( P(A|C) \) and \( P(C|A) \) (ii) \( P(A \cap B | C) \) and \( P(A \cup B | C) \)

Find : \( \int \frac{x - \sin x}{1 - \cos x} \, dx \)

Evaluate : \( \int_0^2 \frac{1}{\sqrt{x^2 + 2x + 3}} \, dx \)

Solve the following Linear Programming Problem graphically :
Maximize \( Z = \frac{2x}{5} + \frac{3y}{10} \)
subject to constraints
\( 2x + y \leq 1000 \)
\( x + y \leq 800 \)
\( x, y \geq 0 \).

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are unit vectors, then prove that \( |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 \leq 9 \).

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are \( 55\hat{i} - 2\hat{j} \), \( 5\hat{i} + 8\hat{j} \) and \( a\hat{i} - 52\hat{j} \) respectively, find the value of 'a'.

A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected.
Find P(A and B) and find whether the events A and B are independent events or not.

Find the angle between the following pair of lines :
\(\frac{x - 2}{3} = \frac{y + 5}{2} = \frac{1 - z}{-6}\) and \(\frac{x - 7}{1} = \frac{y}{2} = \frac{6 - z}{-2}\).

Three honey bees were found flying along the vectors \(\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}\), \(\vec{b} = 4\hat{j} - 2\hat{k}\) and \(\vec{c} = 3\hat{i} + 2\hat{k}\) respectively.
Find the value of \(\lambda\) such that the path for \(\vec{a} + \lambda\vec{b}\) is perpendicular to \(\vec{c}\).

Differentiate \(x^x\) with respect to \(x \log x\).

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

If \(A, B\) and \(C\) be three non-collinear points such that \(\vec{AB} = \hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{AC} = 2\hat{i} - 3\hat{j}\), then find the area of \(\Delta ABC\).

If \(y = P \cos ux + Q \sin ux\), show that \(\frac{d^2y}{dx^2} + u^2y = 0\).

Evaluate \(\sin \left[\tan^{-1} \tan\left(\frac{3\pi}{4}\right)\right]\).

Assertion (A) : Consider a Linear Programming Problem with minimise Z = x + 2y subject to constraints 2x + y \( \geq \) 3, x + 2y \( \geq \) 6, x, y \( \geq \) 0 which gives minimum Z at infinitely many points. The corner points of feasible region are (0, 3) and (6, 0).
Reason (R) : If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.

If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

The value of \( \int_{-5}^{-1} \frac{1}{x} \, dx \) is equal to :

An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :

\( \int \frac{1}{\sqrt{1 + \cos 2x}} \, dx \) is equal to :