Class 12 Maths Sample Paper 4 (CBSE 2026-27 pattern)

The function \(f : \mathbb{R} \to \mathbb{R}\), \(f(x) = x^3 - x\), is

1. (a) one-one and onto
2. (b) one-one but not onto
3. (c) onto but not one-one
4. (d) neither one-one nor onto

The value of \(\sin\left(2\tan^{-1}\dfrac13\right)\) is

1. (a) \(\dfrac35\)
2. (b) \(\dfrac45\)
3. (c) \(\dfrac23\)
4. (d) \(\dfrac34\)

If \(A\) and \(B\) are square matrices of the same order with \(AB = A\) and \(BA = B\), then \(A^2\) equals

1. (a) \(A\)
2. (b) \(B\)
3. (c) \(I\)
4. (d) \(O\)

If \(\begin{vmatrix} 2x & 3 \\ x & x \end{vmatrix} = 5\), then \(x\) equals

1. (a) \(\dfrac52\) or \(-1\)
2. (b) \(-\dfrac52\) or \(1\)
3. (c) \(\dfrac52\) only
4. (d) \(1\) or \(5\)

If \(A\) is an invertible matrix of order \(3\) with \(|A| = 5\), then \(|A^{-1}\operatorname{adj} A|\) equals

1. (a) \(\dfrac15\)
2. (b) \(5\)
3. (c) \(25\)
4. (d) \(125\)

The number of symmetric matrices of order \(3 \times 3\) whose entries are all \(0\) or \(1\) is

1. (a) \(8\)
2. (b) \(32\)
3. (c) \(64\)
4. (d) \(512\)

For \(\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{vmatrix}\), with \(A_{ij}\) the cofactor of \(a_{ij}\), the value of \(a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}\) is

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

The set of all points at which \(f(x) = |x - 1| + |x - 2|\) is not differentiable is

1. (a) \(\{1\}\)
2. (b) \(\{2\}\)
3. (c) \(\{1, 2\}\)
4. (d) \(\varnothing\)

For \(-1 \lt x \lt 1\), \(\dfrac{d}{dx}\left(\sin^{-1}x + \cos^{-1}x\right)\) equals

1. (a) \(0\)
2. (b) \(\dfrac{\pi}{2}\)
3. (c) \(\dfrac{2}{\sqrt{1 - x^2}}\)
4. (d) \(1\)

If \(y = x^{1/x}\), \(x \gt 0\), then \(\dfrac{dy}{dx} = 0\) at

1. (a) \(x = 1\)
2. (b) \(x = e\)
3. (c) \(x = \dfrac1e\)
4. (d) \(x = e^2\)

The function \(f(x) = xe^{-x^2}\) is strictly increasing on

1. (a) \(\left(-\dfrac{1}{\sqrt2}, \dfrac{1}{\sqrt2}\right)\)
2. (b) \((0, \infty)\)
3. (c) \(\mathbb{R}\)
4. (d) \(\left(\dfrac{1}{\sqrt2}, \infty\right)\)

The points on the curve \(y = x^2\) nearest to \(\left(0, \dfrac32\right)\) are

1. (a) \((0, 0)\)
2. (b) \((\pm1, 1)\)
3. (c) \((\pm\sqrt2, 2)\)
4. (d) \(\left(\pm\dfrac{1}{\sqrt2}, \dfrac12\right)\)

\(\displaystyle\int 2^x e^x\,dx\) equals

1. (a) \(2^x e^x + C\)
2. (b) \(\dfrac{(2e)^x}{\log 2} + C\)
3. (c) \(\dfrac{(2e)^x}{1 + \log 2} + C\)
4. (d) \(\dfrac{2^x e^x}{\log 2 - 1} + C\)

If \(\displaystyle\int_0^a (3x^2 + 2x + 1)\,dx = 14\), then \(a\) is

1. (a) \(1\)
2. (b) \(2\)
3. (c) \(3\)
4. (d) \(\sqrt{14}\)

The integrating factor of \(x\dfrac{dy}{dx} - y = x^2\log x\) \((x \gt 0)\) is

1. (a) \(x\)
2. (b) \(\dfrac1x\)
3. (c) \(\log x\)
4. (d) \(e^x\)

If \(\hat a\) and \(\hat b\) are unit vectors and \(\hat a + \hat b\) is also a unit vector, the angle between \(\hat a\) and \(\hat b\) is

1. (a) \(\dfrac{\pi}{3}\)
2. (b) \(\dfrac{\pi}{2}\)
3. (c) \(\dfrac{2\pi}{3}\)
4. (d) \(\dfrac{5\pi}{6}\)

The angle between the lines \(x = y = z\) and \(\dfrac{x - 1}{1} = \dfrac{y}{-1} = \dfrac{z}{0}\) is

1. (a) \(0\)
2. (b) \(\dfrac{\pi}{4}\)
3. (c) \(\dfrac{\pi}{3}\)
4. (d) \(\dfrac{\pi}{2}\)

The shortest distance between the lines \(\vec r = \lambda\hat i\) and \(\vec r = \hat j + \mu\hat k\) is

1. (a) \(0\)
2. (b) \(1\)
3. (c) \(\sqrt2\)
4. (d) \(\dfrac{1}{\sqrt2}\)

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

Assertion (A): If \(\vec a \times \vec b = \vec 0\) and \(\vec a \cdot \vec b = 0\), then \(\vec a = \vec 0\) or \(\vec b = \vec 0\).

Reason (R): For non-zero vectors, \(\vec a \times \vec b = \vec 0\) means \(\vec a \parallel \vec b\) and \(\vec a \cdot \vec b = 0\) means \(\vec a \perp \vec b\), and two non-zero vectors cannot be both parallel and perpendicular.

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 \(P(A) = 0.5\) and \(P(B) = 0.7\), then \(A\) and \(B\) cannot be mutually exclusive.

Reason (R): For any two events, \(P(A \cap B) \le P(A)\).

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.

Find the value of \(x\) for which \(\sin^{-1}x = \cos^{-1}x\).

OR

For the curve \(x = t^2 + t\), \(y = t^3 - 3t\), find \(\dfrac{dy}{dx}\) in terms of \(t\), and the points of the curve at which \(\dfrac{dy}{dx} = 0\).

The side of an equilateral triangle is increasing at \(2\) cm/s. Find the rate at which its area is increasing when the side is \(10\) cm.

Evaluate \(\displaystyle\int_{-1}^{1} e^{|x|}\,dx\).

OR

For \(\vec a = 2\hat i - \hat j + \hat k\) and \(\vec b = \hat i + \hat j - 2\hat k\), show that \(\vec a + \vec b\) is perpendicular to \(\vec a - \vec b\), and explain why this happens.

If \(y = \sqrt{x + \sqrt{x^2 - 1}}\), \(x \gt 1\), show that \((x^2 - 1)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} - \dfrac{y}{4} = 0\).

Find \(\displaystyle\int (2x + 3)\tan^{-1}x\,dx\).

OR

Solve \(x^2\dfrac{dy}{dx} = y^2 + 2xy\), given \(y = 1\) when \(x = 1\).

Find the intervals in which \(f(x) = \dfrac{\log x}{x}\), \(x \gt 0\), is strictly increasing or decreasing. Hence decide which is greater, \(e^\pi\) or \(\pi^e\).

Find the points on the line \(\dfrac{x - 1}{2} = \dfrac{y + 2}{-2} = \dfrac{z - 3}{1}\) at a distance of \(6\) units from the point \((1, -2, 3)\). Also find the angle the line makes with the \(z\)-axis.

Two fair dice are thrown. Let \(X\) be the absolute difference of the two numbers obtained. Find the probability distribution of \(X\) and its mean.

OR

An e-commerce hub in Nagpur loads a delivery van with small and large parcels. A small parcel weighs \(10\) kg and a large one \(30\) kg; the van can carry at most \(600\) kg. Each parcel, small or large, takes one slot and the van has \(30\) slots. The hub must send at least \(5\) large parcels on every trip. The delivery charge earned is ₹50 per small and ₹120 per large parcel. How many parcels of each kind should be loaded to maximise the charge earned? Formulate and solve graphically.

A small firm making solar lanterns in Jaipur finds that when it makes and sells \(x\) lanterns a week, the selling price per lantern is ₹\((600 - 3x)\) and the total weekly cost is ₹\((2x^2 + 60x + 1000)\). Find the weekly output that maximises profit, the maximum profit and the price per lantern at that output.

OR

For the lines \(L_1: \vec r = (3\hat i - \hat j + \hat k) + \lambda(2\hat i - \hat j)\) and \(L_2: \vec r = (2\hat j + 4\hat k) + \mu(2\hat i - \hat k)\): (a) find the shortest distance between them; (b) find the points \(P\) on \(L_1\) and \(Q\) on \(L_2\) such that \(PQ\) is perpendicular to both lines; (c) verify that \(PQ\) equals the shortest distance.

Meenakshi invests ₹60,000 in three schemes A, B and C that pay \(6\%\), \(7\%\) and \(8\%\) simple interest per year. Her total annual interest is ₹4,250, and the amount in scheme C is ₹20,000 less than the amounts in A and B together. Using matrices, find the amount invested in each scheme.

OR

Divisor chart. A Class XII group studies the relation \(R = \{(a, b) : a \text{ divides } b\}\) on the set \(S = \{1, 2, 3, 4, 6, 12\}\) of divisors of \(12\).

1. Is \(R\) reflexive? Justify. [1 mark]

2. Is \(R\) symmetric? Justify. [1 mark]

3. Show that \(R\) is transitive. Is \(R\) an equivalence relation? [2 marks]

   OR

   Find the number of ordered pairs in \(R\). [2 marks]

Reservoir inflow. During a heavy shower, the rate of inflow of water into a village reservoir \(t\) hours after it starts is \(r(t) = 40 + 6t - 0.3t^2\) kilolitres per hour \((0 \le t \le 20)\).

1. Find the rate of inflow at \(t = 5\). [1 mark]

2. At what time is the rate of inflow greatest, and what is it? [1 mark]

3. Find the total inflow in the first \(10\) hours. [2 marks]

   OR

   Find the total inflow between \(t = 10\) and \(t = 20\), and compare it with the first 10 hours. [2 marks]

Garment quality check. In a garment unit in Tiruppur, tailors A, B and C stitch \(50\%\), \(30\%\) and \(20\%\) of the shirts respectively. Of the shirts they stitch, \(2\%\), \(3\%\) and \(5\%\) respectively are defective. A shirt is picked at random.

1. Find the probability that the shirt was stitched by C and is defective. [1 mark]

2. Find the probability that the shirt is defective. [1 mark]

3. The shirt is found to be defective. Find the probability that it was stitched by A. [2 marks]

   OR

   The shirt is defective. Find the probability that it was not stitched by C. [2 marks]