Class 12 Maths Sample Paper 1 (CBSE 2026-27 pattern)

The relation \(R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}\) on \(A = \{1, 2, 3\}\) is

1. (a) reflexive, but neither symmetric nor transitive
2. (b) an equivalence relation
3. (c) symmetric and transitive but not reflexive
4. (d) transitive but not reflexive

The value of \(\cos^{-1}\left(\cos\dfrac{4\pi}{3}\right)\) is

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

If \(A\) is a square matrix of order \(3\) with \(|A| = 4\), then \(|\operatorname{adj} A|\) is

1. (a) \(4\)
2. (b) \(8\)
3. (c) \(16\)
4. (d) \(64\)

The matrix \(\begin{bmatrix} 2 & 3 \\ -1 & x \end{bmatrix}\) is singular when \(x\) equals

1. (a) \(\dfrac32\)
2. (b) \(-\dfrac32\)
3. (c) \(-\dfrac23\)
4. (d) \(3\)

If \(A\) and \(B\) are symmetric matrices of the same order, then \(AB - BA\) is

1. (a) symmetric
2. (b) skew-symmetric
3. (c) the zero matrix
4. (d) the identity matrix

For the \(2 \times 3\) matrix \(A = [a_{ij}]\) with \(a_{ij} = i - 2j\), the element \(a_{23}\) is

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

The system \(x + 2y = 3\), \(2x + ky = 6\) has infinitely many solutions when \(k\) equals

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

The function \(f(x) = \dfrac{1 - \cos 4x}{8x^2}\) for \(x \ne 0\), \(f(0) = k\), is continuous at \(x = 0\) when \(k\) equals

1. (a) \(\dfrac12\)
2. (b) \(1\)
3. (c) \(2\)
4. (d) \(\dfrac14\)

\(\dfrac{d}{dx}\left[\log(\sec x + \tan x)\right]\) equals

1. (a) \(\tan x\)
2. (b) \(\sec x\)
3. (c) \(\sec^2 x\)
4. (d) \(\sec x \tan x\)

If \(y = Ae^{3x} + Be^{-3x}\), then \(\dfrac{d^2y}{dx^2}\) equals

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

The function \(f(x) = x^3 - 3x^2 + 3x + 7\) is

1. (a) increasing on \(\mathbb{R}\)
2. (b) decreasing on \(\mathbb{R}\)
3. (c) decreasing on \((0, 1)\) only
4. (d) neither increasing nor decreasing on \(\mathbb{R}\)

\(\displaystyle\int e^x\left(x^2 + 2x\right)dx\) equals

1. (a) \(2xe^x + C\)
2. (b) \((x^2 + 2x)e^x + C\)
3. (c) \(x^2e^x + C\)
4. (d) \((x^2 - 2x)e^x + C\)

If \(|\vec a| = 3\), \(|\vec b| = 4\) and \(\vec a \cdot \vec b = 6\), the angle between \(\vec a\) and \(\vec b\) is

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

The direction cosines of the line joining \((1, -2, 3)\) and \((3, 0, 4)\), directed from the first point to the second, are

1. (a) \(\dfrac23, \dfrac23, \dfrac13\)
2. (b) \(2, 2, 1\)
3. (c) \(\dfrac25, \dfrac25, \dfrac15\)
4. (d) \(-\dfrac23, -\dfrac23, -\dfrac13\)

If \(|\vec a| = 2\) and \(|\vec b| = 5\), then \(|\vec a \times \vec b|^2 + (\vec a \cdot \vec b)^2\) equals

1. (a) \(49\)
2. (b) \(100\)
3. (c) \(10\)
4. (d) \(29\)

The corner points of the feasible region of an LPP are \((0, 4)\), \((2, 1)\) and \((5, 0)\). The objective function \(Z = px + qy\) \((p, q \gt 0)\) takes its minimum value at both \((0, 4)\) and \((2, 1)\) if

1. (a) \(p = \dfrac{3q}{2}\)
2. (b) \(q = \dfrac{3p}{2}\)
3. (c) \(p = q\)
4. (d) \(p = 2q\)

If the feasible region of a linear programming problem is bounded (and non-empty), then the objective function \(Z = ax + by\)

1. (a) has a maximum but may not have a minimum
2. (b) has both a maximum and a minimum, each at a corner point
3. (c) has neither a maximum nor a minimum
4. (d) has its optimum only at an interior point

If \(A\) and \(B\) are independent events with \(P(A) = 0.4\) and \(P(B) = 0.5\), then \(P(A \cup B)\) is

1. (a) \(0.9\)
2. (b) \(0.2\)
3. (c) \(0.7\)
4. (d) \(0.3\)

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

Assertion (A): The lines \(\vec r = (\hat i + \hat j) + \lambda(2\hat i - \hat j + \hat k)\) and \(\vec r = (2\hat i + \hat k) + \mu(-4\hat i + 2\hat j - 2\hat k)\) are parallel.

Reason (R): Two lines are parallel if their direction vectors are collinear.

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 \(P(A) = 0.3\) and \(P(B \mid A) = 0.4\), then \(P(A \cap B) = 0.12\).

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

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 \(\cot^{-1}\left(-\dfrac{1}{\sqrt3}\right) + \operatorname{cosec}^{-1}(2)\).

OR

If \(x^3 + xy^2 = 10\), find \(\dfrac{dy}{dx}\) at the point \((1, 3)\).

Find the intervals in which \(f(x) = 2x^3 - 15x^2 + 36x + 1\) is strictly increasing and strictly decreasing.

OR

Find a unit vector perpendicular to both \(\vec a = \hat i + 2\hat j - \hat k\) and \(\vec b = 3\hat i - \hat j + 2\hat k\).

Two cards are drawn one after the other, without replacement, from a well-shuffled pack of \(52\) cards. Find the probability that both are kings.

Find the values of \(a\) and \(b\) for which the function \(f(x) = \begin{cases} ax^2 + b, & x \lt 2 \\ 3x - 1, & x \ge 2 \end{cases}\) is differentiable at \(x = 2\).

Find \(\displaystyle\int \dfrac{3x + 1}{x^2 + 4x + 8}\,dx\).

Solve the differential equation \(x\dfrac{dy}{dx} - 3y = x^5e^x\), \(x \gt 0\), given that \(y = 2\) when \(x = 1\).

OR

Evaluate \(\displaystyle\int_0^3 x^2\sqrt{3 - x}\,dx\).

Find the coordinates of the foot of the perpendicular drawn from the point \(P(2, 3, 5)\) to the line \(\dfrac{x - 1}{1} = \dfrac{y + 1}{2} = \dfrac{z - 2}{-1}\). Hence find the perpendicular distance of \(P\) from the line.

OR

Solve the following linear programming problem graphically: maximise \(Z = 4x + 5y\) subject to \(2x + y \le 14\), \(x + 3y \le 12\), \(x \ge 0\), \(y \ge 0\).

Using matrices, solve the system of equations: \(2x - y + z = 3\), \(x + 3y - 2z = 1\), \(3x + y + 2z = 11\).

OR

Find the area of the region bounded by the parabola \(y^2 = 4x\) and the line \(y = x - 3\).

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

OR

Consider the lines \(L_1: \dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z - 2}{k}\) and \(L_2: \dfrac{x + 2}{1} = \dfrac{y - 3}{-2} = \dfrac{z - 1}{2}\). (a) Find \(k\) if \(L_1 \perp L_2\). (b) For this \(k\), find the shortest distance between \(L_1\) and \(L_2\). (c) Do the lines intersect? Justify.

Screening test. A diagnostic lab in Pune uses a quick screening test for a viral infection. In the population tested, \(2\%\) have the infection. The test is positive for \(95\%\) of infected people and also (falsely) for \(10\%\) of uninfected people. A person is tested at random.

1. Find the probability that the person is infected and tests positive. [1 mark]

2. Find the probability that the test is positive. [1 mark]

3. The test is positive. Find the probability that the person is actually infected, and comment on the result. [2 marks]

   OR

   The test is negative. Find the probability that the person is not infected. [2 marks]

Mall parking. A mall in Noida finds that if it charges ₹\(p\) per hour for parking, about \(200 - 8p\) cars use the parking each hour \((0 \le p \le 25)\).

1. Write the hourly revenue \(R(p)\). [1 mark]

2. Find the interval of \(p\) on which \(R\) is increasing. [1 mark]

3. Find the charge that maximises the revenue, and the maximum revenue. Justify that it is a maximum. [2 marks]

   OR

   The city levies a tax of ₹2.50 on every car parked. Find the charge \(p\) that maximises the mall's hourly profit \((p - 2.5)(200 - 8p)\), and the maximum profit. [2 marks]

Smart-card batches. A metro rail office issues smart cards numbered \(1, 2, \ldots, 12\). For audit, cards are grouped by the relation \(R = \{(a, b) : a + 2b \text{ is divisible by } 3\}\) on \(S = \{1, 2, \ldots, 12\}\).

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

2. How many distinct equivalence classes are there? [1 mark]

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

   OR

   Define \(f : S \to \{0, 1, 2\}\) by \(f(a) =\) remainder when \(a\) is divided by \(3\). Is \(f\) one-one? Is it onto? Justify. [2 marks]