Class 12 Maths Sample Paper 5 (CBSE 2026-27 pattern)

The number of equivalence relations on the set \(\{a, b, c\}\) is

1. (a) \(3\)
2. (b) \(4\)
3. (c) \(5\)
4. (d) \(6\)

The value of \(\sin^{-1}(\sin 10)\) (angle in radians) is

1. (a) \(10\)
2. (b) \(3\pi - 10\)
3. (c) \(10 - 3\pi\)
4. (d) \(\pi - 10\)

If \(A\) is a square matrix with \(A^3 = O\), then \((I - A)^{-1}\) equals

1. (a) \(I + A\)
2. (b) \(I - A\)
3. (c) \(I + A + A^2\)
4. (d) \(I - A + A^2\)

If \(A\) and \(B\) are square matrices of order \(3\) with \(|A| = 6\) and \(|B| = -2\), then \(\left|3A^{-1}B'\right|\) is

1. (a) \(-9\)
2. (b) \(9\)
3. (c) \(-1\)
4. (d) \(-3\)

The number of \(2 \times 2\) skew-symmetric matrices whose entries are all from \(\{-1, 0, 1\}\) is

1. (a) \(1\)
2. (b) \(2\)
3. (c) \(3\)
4. (d) \(9\)

If \(A\) is a non-singular matrix of order \(3\) and \(|\operatorname{adj} A| = 64\), then \(|A|\) is

1. (a) \(8\) only
2. (b) \(\pm 8\)
3. (c) \(\pm 4\)
4. (d) \(4\) only

The area of the triangle with vertices \((a, b + c)\), \((b, c + a)\) and \((c, a + b)\), where \(a, b, c\) are distinct, is

1. (a) \(0\)
2. (b) \(\dfrac12(a + b + c)\)
3. (c) \(\dfrac12|(a - b)(b - c)(c - a)|\)
4. (d) \(1\)

Let \(f(x) = x^3\) for \(x \le 1\) and \(f(x) = 3x - 2\) for \(x \gt 1\). Then at \(x = 1\), \(f\) is

1. (a) discontinuous
2. (b) continuous but not differentiable
3. (c) differentiable with \(f'(1) = 3\)
4. (d) differentiable with \(f'(1) = 1\)

If \(y = \dfrac{\sin^{-1}x}{\sqrt{1 - x^2}}\), \(|x| \lt 1\), then \((1 - x^2)\dfrac{dy}{dx} - xy\) equals

1. (a) \(0\)
2. (b) \(1\)
3. (c) \(-1\)
4. (d) \(y\)

The number of critical points of \(f(x) = |x|e^{-|x|}\) on \(\mathbb{R}\) (points where \(f' = 0\) or \(f'\) does not exist) is

1. (a) \(1\)
2. (b) \(2\)
3. (c) \(3\)
4. (d) \(4\)

For every \(n \gt 0\), \(\displaystyle\int_0^{\pi/2}\dfrac{\sin^n x}{\sin^n x + \cos^n x}\,dx\) equals

1. (a) \(\dfrac\pi2\)
2. (b) \(\dfrac\pi4\)
3. (c) \(\dfrac{n\pi}{4}\)
4. (d) \(1\)

The solution of \(\dfrac{dy}{dx} - y = e^{x}\) with \(y(0) = 1\) is

1. (a) \(y = e^{x}\)
2. (b) \(y = xe^{x}\)
3. (c) \(y = (x + 1)e^{x}\)
4. (d) \(y = (x - 1)e^{x}\)

If \(|\vec a| = 3\), \(|\vec b| = 4\) and \(\vec a \cdot \vec b = 6\), then \(|\vec a \times \vec b|\) is

1. (a) \(6\)
2. (b) \(6\sqrt3\)
3. (c) \(12\)
4. (d) \(6\sqrt2\)

The direction cosines of a line equally inclined to the three coordinate axes are

1. (a) \(\pm\left(\dfrac{1}{\sqrt3}, \dfrac{1}{\sqrt3}, \dfrac{1}{\sqrt3}\right)\)
2. (b) \((1, 1, 1)\)
3. (c) \(\left(\dfrac13, \dfrac13, \dfrac13\right)\)
4. (d) \(\pm\left(\dfrac{1}{\sqrt2}, \dfrac{1}{\sqrt2}, 0\right)\)

The angle which the line \(\dfrac{x - 2}{3} = \dfrac{y + 1}{-2} = \dfrac{z}{6}\) makes with the positive \(x\)-axis is

1. (a) \(\cos^{-1}\dfrac37\)
2. (b) \(\cos^{-1}\dfrac27\)
3. (c) \(\cos^{-1}\dfrac67\)
4. (d) \(\cos^{-1}\dfrac{3}{\sqrt{41}}\)

The system of constraints \(x + y \le 4\), \(x + y \ge 6\), \(x, y \ge 0\)

1. (a) has a bounded feasible region
2. (b) has an unbounded feasible region
3. (c) has no feasible solution
4. (d) has exactly one feasible point

On a bounded feasible region, \(Z = 3x + 2y\) attains its maximum value \(18\) at the corner points \((4, 3)\) and \((6, 0)\). Then

1. (a) the maximum occurs only at these two points
2. (b) the maximum occurs at every point of the segment joining \((4, 3)\) and \((6, 0)\)
3. (c) the problem has no maximum
4. (d) the minimum is also 18

An urn has \(5\) red and \(5\) black balls. A ball is drawn, its colour noted, and it is returned together with \(2\) more balls of the same colour. A second ball is then drawn. The probability that the second ball is red is

1. (a) \(\dfrac{5}{12}\)
2. (b) \(\dfrac12\)
3. (c) \(\dfrac{7}{12}\)
4. (d) \(\dfrac35\)

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The vectors \(\vec a = \hat i + 2\hat j - \hat k\) and \(\vec b = -2\hat i - 4\hat j - 2\hat k\) are collinear.

Reason (R): Two non-zero vectors are collinear if and only if one is a scalar multiple of the other.

1. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (c) Assertion (A) is true but Reason (R) is false.
4. (d) Assertion (A) is false but Reason (R) is true.

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): If \(A\) and \(B\) are independent events, then \(P(A \cup B) = 1 - P(A')P(B')\).

Reason (R): If \(A\) and \(B\) are independent, then \(A'\) and \(B'\) are also independent.

1. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (c) Assertion (A) is true but Reason (R) is false.
4. (d) Assertion (A) is false but Reason (R) is true.

Show that \(f : \mathbb{R} \to \mathbb{R}\), \(f(x) = 2x + |x|\), is both one-one and onto.

OR

Let \(f(x) = |x|^3\). Show that \(f''(0)\) exists and find it.

Find \(\displaystyle\int \sqrt{e^x - 1}\,dx\).

OR

Using vectors, find the acute angle between two diagonals of a cube.

Given \(P(A) = \dfrac25\), \(P(B \mid A) = \dfrac12\) and \(P(A \mid B) = \dfrac13\), find \(P(B)\) and \(P(A' \cap B')\).

Show that \(f(x) = e^{-|x|}\) is continuous at \(x = 0\) but not differentiable there. Write \(f'(x)\) for \(x \ne 0\).

If \(y = (\sin x)^{\tan x}\), \(0 \lt x \lt \dfrac\pi2\), find \(\dfrac{dy}{dx}\), and its value at \(x = \dfrac\pi4\).

OR

Find \(\displaystyle\int \dfrac{x^2 + 1}{(x^2 + 4)(x^2 + 9)}\,dx\).

Evaluate \(\displaystyle\int_0^{\pi/2}\dfrac{\sin x\cos x}{\sin x + \cos x}\,dx\).

Find the vector equation of the line through \(P(2, 1, -1)\) that meets the line \(L: \dfrac{x - 1}{1} = \dfrac{y + 2}{2} = \dfrac{z - 3}{-2}\) at right angles. Also find the point where they meet.

OR

Maximise \(Z = 2x + 2y\) subject to \(x + y \le 5\), \(x \le 4\), \(y \le 3\), \(x, y \ge 0\). Show that the problem has infinitely many optimal solutions and describe them.

Find \(A^{-1}\) for \(A = \begin{bmatrix} 2 & 1 & 0 \\ 1 & -1 & 3 \\ 0 & 2 & 1 \end{bmatrix}\). Hence solve the system \(2x + y = 4\), \(x - y + 2z = 5\), \(3y + z = 9\).

OR

A farmer in Punjab wants to fence a rectangular field along a straight canal; no fence is needed along the canal. He also puts one fence across the field, perpendicular to the canal, to split it into two plots. Fencing parallel to the canal costs ₹30 per metre and fencing perpendicular to it costs ₹20 per metre, and his budget is ₹6000. Find the dimensions that maximise the total area, and the maximum area.

OR

A cup of tea at \(90^\circ\)C is left in a room at \(30^\circ\)C. Its temperature \(T\) falls at a rate proportional to \(T - 30\). After \(5\) minutes it is \(70^\circ\)C. (a) Form and solve the differential equation. (b) Find how long after the start the tea reaches \(50^\circ\)C. (Answer in logarithmic form and to one decimal place, using \(\log 3 = 1.0986\), \(\log 1.5 = 0.4055\).)

The vertices of \(\triangle ABC\) are \(A(1, 2, 3)\), \(B(3, 4, 1)\) and \(C(-1, 6, 5)\). (a) Write the vector equation of the line \(BC\). (b) Find the foot \(D\) of the perpendicular from \(A\) to \(BC\), and the length \(AD\). (c) Verify that \(\dfrac12|\vec{AB} \times \vec{AC}| = \dfrac12 \cdot BC \cdot AD\).

Motor insurance. A motor insurer classifies its policy-holders in Chennai as careful \((60\%)\), average \((30\%)\) and risky \((10\%)\). The probabilities that a driver of each type has an accident in a year are \(0.01\), \(0.03\) and \(0.15\) respectively.

1. Find the probability that a randomly chosen policy-holder has an accident in the year. [1 mark]

2. A policy-holder had an accident. Find the probability that the driver is of the risky type. [1 mark]

3. A policy-holder had an accident. Find the probability that the driver is careful, and comment on the insurer's classification. [2 marks]

   OR

   Two policy-holders are chosen at random, independently. Find the probability that exactly one of them has an accident in the year. [2 marks]

Giant wheel. At a mela in Lucknow, the height (in metres) of a rider on a giant wheel above the ground \(t\) minutes after the ride starts is \(h(t) = 12 - 10\cos\dfrac{\pi t}{2}\), \(0 \le t \le 4\).

1. Find the height of the rider at \(t = 1\). [1 mark]

2. Find the rate of change of height at \(t = 1\). [1 mark]

3. Find the times in \([0, 2]\) at which the rider is rising at \(2.5\pi\) m/min. [2 marks]

   OR

   Find \(\dfrac{d^2h}{dt^2}\) at \(t = \dfrac13\) and state whether the rider's rate of rising is increasing or decreasing then. [2 marks]

Square residues. A number-theory club studies the relation \(R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} : 5 \text{ divides } a^2 - b^2\}\).

1. Is \((2, 3) \in R\)? Justify. [1 mark]

2. Write the equivalence class of \(0\). [1 mark]

3. Show that \(R\) is an equivalence relation, and find the number of distinct equivalence classes. [2 marks]

   OR

   Describe the equivalence class of \(1\). [2 marks]