CBSE Class 12 2025 Mathematics 65-4-2 Question Paper Set-2: Download Solutions with Answer Key

The principal branch of \(\cos^{-1} x\) is:

• (1) \(\left[ \frac{\pi}{2}, \frac{3\pi}{2} \right]\)
• (2) \([\pi, 2\pi]\)
• (3) \([0, \pi]\)
• (4) \([2\pi, 3\pi]\)

The values of \(\lambda\) so that \(f(x) = \sin x - \cos x - \lambda x + C\) decreases for all real values of \(x\) are :

• (A) \(1 < \lambda < \sqrt{2}\)
• (B) \(\lambda \geq 1\)
• (C) \(\lambda \geq \sqrt{2}\)
• (D) \(\lambda < 1\)

If A and B are square matrices of same order such that AB = BA, then \(A^2 + B^2\) is equal to :

• (A) \(A + B\)
• (B) \(BA\)
• (C) \(2(A + B)\)
• (D) \(2BA\)

If \(f(x) = \left\{ \begin{array}{ll} \frac{1 - \sin^3 x}{3 \cos^2 x} & for \, x \neq \frac{\pi}{2},
k & for \, x = \frac{\pi}{2}, \end{array} \right. \) is continuous at \(x = \frac{\pi}{2}\), then the value of \(k\) is:

• (A) \(\frac{3}{2}\)
• (B) \(\frac{1}{6}\)
• (C) \(\frac{1}{2}\)
• (D) \(1\)

For real \(x\), let \(f(x) = x^3 + 5x + 1\). Then :

• (A) \(f\) is one-one but not onto on \(\mathbb{R}\)
• (B) \(f\) is onto on \(\mathbb{R}\) but not one-one
• (C) \(f\) is one-one and onto on \(\mathbb{R}\)
• (D) \(f\) is neither one-one nor onto on \(\mathbb{R}\)

If the direction cosines of a line are \(\lambda, \lambda, \lambda\), then \(\lambda\) is equal to:

• (A) \(\frac{-1}{\sqrt{3}}\)
• (B) \(1\)
• (C) \(\frac{1}{\sqrt{3}}\)
• (D) \(\pm \frac{1}{\sqrt{3}}\)

If \(\left| \begin{array}{ccc} -1 & 2 & 4
1 & x & 1
0 & 3 & 3x \end{array} \right| = -57\), the product of the possible values of \(x\) is:

• (A) \(-24\)
• (B) \(-16\)
• (C) \(16\)
• (D) \(24\)

The matrix \[ \begin{pmatrix} 0 & 1 & -2
-1 & 0 & -7
2 & 7 & 0 \end{pmatrix} \]
is a :

• (A) diagonal matrix
• (B) symmetric matrix
• (C) skew symmetric matrix
• (D) scalar matrix

If \(f(x) = \begin{cases} 3x - 2, & 0 \leq x \leq 1
2x^2 + ax, & 1 < x < 2 \end{cases}\) is continuous for \(x \in (0, 2)\), then \(a\) is equal to :

• (A) -4
• (B) -7
• (C) -2
• (D) -1

If \(f : \mathbb{N} \rightarrow \mathbb{W}\) is defined as \[ f(n) = \begin{cases} \frac{n}{2}, & if n is even
0, & if n is odd \end{cases} \]
then \(f\) is :

• (A) injective only
• (B) surjective only
• (C) a bijection
• (D) neither surjective nor injective

If \(f(x) = 2x + \cos x\), then \(f(x)\) :

• (A) has a maxima at \(x = \pi\)
• (B) has a minima at \(x = \pi\)
• (C) is an increasing function
• (D) is a decreasing function

If the sides \(AB\) and \(AC\) of \(\triangle ABC\) are represented by vectors \(\hat{i} + \hat{j} + 4 \hat{k}\) and \(3 \hat{i} - \hat{j} + 4 \hat{k}\) respectively, then the length of the median through A on BC is :

• (A) \(2 \sqrt{2}\) units
• (B) \(\sqrt{18}\) units
• (C) \(\frac{\sqrt{34}}{2}\) units
• (D) \(\frac{\sqrt{48}}{2}\) units

The function \(f\) defined by \[ f(x) = \begin{cases} x, & if x \leq 1
5, & if x > 1 \end{cases} \]
is not continuous at :

• (A) \(x = 0\)
• (B) \(x = 1\)
• (C) \(x = 2\)
• (D) \(x = 5\)

\(\int e^x (\cos x - \sin x) \, dx\) is equal to:

• (A) \(e^x \sin x + C\)
• (B) \(-e^x \sin x + C\)
• (C) \(-e^x \cos x + C\)
• (D) \(e^x \cos x + C\)

The area of the region enclosed by the curve \(y = \sqrt{x}\) and the lines \(x = 0\) and \(x = 4\) and the x-axis is :

• (A) \(\frac{16}{9}\) sq. units
• (B) \(\frac{32}{9}\) sq. units
• (C) \(\frac{16}{3}\) sq. units
• (D) \(32\) sq. units

The integrating factor of the differential equation \[ \frac{dy}{dx} + y \tan x - \sec x = 0 \quad is: \]

• (A) \(-\cos x\)
• (B) \(\sec x\)
• (C) \(\log \sec x\)
• (D) \(e^{\sec x}\)

The corner points of the feasible region of a Linear Programming Problem are \((0, 2)\), \((3, 0)\), \((6, 0)\), \((6, 8)\), and \((0, 5)\). If \(Z = ax + by; \, (a, b > 0)\) be the objective function, and maximum value of \(Z\) is obtained at \((0, 2)\) and \((3, 0)\), then the relation between \(a\) and \(b\) is :

• (A) \(a = b\)
• (B) \(a = 3b\)
• (C) \(b = 6a\)
• (D) \(a = 3b\)

The value of \[ \int_0^1 \frac{dx}{e^x + e^{-x}} \]
is :

• (A) \(-\frac{\pi}{4}\)
• (B) \(\frac{\pi}{4}\)
• (C) \(\tan^{-1} e - \frac{\pi}{4}\)
• (D) \(\tan^{-1} e\)

Assertion (A): If A and B are two events such that \(P(A \cap B) = 0\), then A and B are independent events.

Reason (R): Two events are independent if the occurrence of one does not affect the occurrence of the other.

Assertion (A): In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution.

Reason (R): A feasible region is defined as the region that satisfies all the constraints.

Using matrices and determinants, find the value(s) of \(k\) for which the pair of equations \[ 5x - ky = 2; \quad 7x - 5y = 3 \]
has a unique solution.

(a) Simplify \(\sin^{-1} \left( \frac{x}{\sqrt{1 + x^2}} \right)\).

(b) Find the domain of \(\sin^{-1} \sqrt{x - 1}\).

Calculate the area of the region bounded by the curve \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
and the x-axis using integration.

(a) Find the least value of ‘a’ so that \(f(x) = 2x^2 - ax + 3\) is an increasing function on \([2, 4]\).

(b) If \(f(x) = x + \frac{1}{x}, \, x \geq 1\), show that \(f\) is an increasing function.

Find the local maxima and local minima of the function \[ f(x) = \frac{8}{3} x^3 - 12x^2 + 18x + 5. \]

Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

A coin is tossed twice. Let \(X\) be a random variable defined as the number of heads minus the number of tails. Obtain the probability distribution of \(X\) and also find its mean.

If \(f : \mathbb{R}^+ \to \mathbb{R}\) is defined as \(f(x) = \log_a x\) where \(a > 0\) and \(a \neq 1\), prove that \(f\) is a bijection.
(R\(^+\) is the set of all positive real numbers.)

Let \(A = \{1, 2, 3\}\) and \(B = \{4, 5, 6\}\). A relation \(R\) from \(A\) to \(B\) is defined as \(R = \{(x, y) : x + y = 6, x \in A, y \in B \}\).

(i) Write all elements of \(R\).

(ii) Is \(R\) a function? Justify.

(iii) Determine domain and range of \(R\).

Find: \[ \int \frac{\cos x \, dx}{1 + \cos x + \sin x}. \]

(a) Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a die. Find the conditional probability of the event A: `the die shows a number greater than 3' given that B: `there is at least one tail'.

Find the distance of the point \((-1, -5, -10)\) from the point of intersection of the lines \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}, \quad \frac{x - 4}{5} = \frac{y - 1}{2} = z. \]

Solve the following Linear Programming Problem graphically:

Minimise \( Z = 3x + 5y \)

subject to the constraints: \[ x + 2y \geq 10, \quad x + y \geq 6, \quad 3x + y \geq 8, \quad x, y \geq 0. \]

The relation between the height of the plant (\(y\) cm) with respect to exposure to sunlight is governed by the equation \[ y = 4x - \frac{1}{2} x^2, \]
where \(x\) is the number of days exposed to sunlight.


(i) Find the rate of growth of the plant with respect to sunlight.

(ii) In how many days will the plant attain its maximum height? What is the maximum height?

Evaluate: \[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \, dx \]

Find: \[ J = \int \frac{\sqrt{x^2 + 1} \left[ \log(x^2 + 1) - 2 \log x \right]}{x^2} \, dx \]

Show that the area of a parallelogram whose diagonals are represented by \( \vec{a} \) and \( \vec{b} \) is given by \[ Area = \frac{1}{2} | \vec{a} \times \vec{b} |. \]
Also, find the area of a parallelogram whose diagonals are \( 2\hat{i} - \hat{j} + \hat{k} \) and \( \hat{i} + 3\hat{j} - \hat{k} \).

Find the equation of a line in vector and Cartesian form which passes through the point \( (1, 2, -4) \) and is perpendicular to the lines \[ \frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7}. \]
and \[ \vec{r} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu (3\hat{i} + 8\hat{j} - 5\hat{k}). \]