CBSE Class 12 2025 Mathematics 65-5-1 Question Paper Set-1: Download Solutions with Answer Key

If \[ A = \begin{bmatrix} 5 & 0\\0 & 5 \end{bmatrix}, \]
then \( A^3 \) is:

• (A) \( \begin{bmatrix} 5^3 & 0\\0 & 5^3 \end{bmatrix} = \begin{bmatrix} 125 & 0\\0 & 125 \end{bmatrix} \)
• (B) \( \begin{bmatrix} 0 & 125\\0 & 125 \end{bmatrix} \)
• (C) \( \begin{bmatrix} 15 & 0\\0 & 15 \end{bmatrix} \)
• (D) \( \begin{bmatrix} 5^3 & 0\\0 & 5^3 \end{bmatrix} \)

If \( P(A \cup B) = 0.9 \) and \( P(A \cap B) = 0.4 \), then \( P(A) + P(B) \) is:

• (A) \( 0.3 \)
• (B) \( 1 \)
• (C) \( 1.3 \)
• (D) \( 0.7 \)

If \[ A = \begin{bmatrix} 1 & 2 & 3\\-4 & 3 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 3\\-1 & 2\\0 & 5 \end{bmatrix}, \]
then the correct statement is:

If \[ \left| \frac{2x}{5} \right| = \left| \frac{6 - 5}{4} \right|, \quad then the value of x is: \]

If \[ f(x) = \begin{cases} \frac{\sin^2(ax)}{x^2}, & x \neq 0\\1, & x = 0 \end{cases} \]
is continuous at \( x = 0 \), then the value of \( a \) is:

If \[ A = \begin{bmatrix} 1 & 0 & 0\\0 & 5 & 0\\0 & 0 & -2 \end{bmatrix}, \]
then \( |A| \) is:

The principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \) is:

If \[ \begin{bmatrix} 4 + x & x - 1\\-2 & 3 \end{bmatrix} \]
is a singular matrix, then the value of \( x \) is:

If \( f(x) = \lfloor x \rfloor \) is the greatest integer function, then the correct statement is:

The slope of the curve \( y = -x^3 + 3x^2 + 8x - 20 \) is maximum at:

The integral \[ \int \sqrt{1 + \sin x} \, dx \]
is equal to:

The integral \[ \int_0^{\frac{\pi}{2}} \cos x \cdot e^{\sin x} \, dx \]
is equal to:

The area of the region enclosed between the curve \( y = |x| \), x-axis, \( x = -2 \) and \( x = 2 \) is:

The integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \]
is:

The sum of the order and degree of the differential equation \[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right) \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^3 \]
is:

For a Linear Programming Problem (LPP), the given objective function \( Z = 3x + 2y \) is subject to constraints:
\[ x + 2y \leq 10, \] \[ 3x + y \leq 15, \] \[ x, y \geq 0. \]

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The correct feasible region is:

Let \( \vec{a} \) be a position vector whose tip is the point (2, -3). If \( \overrightarrow{AB} = \vec{a} \), where coordinates of A are (–4, 5), then the coordinates of B are:

The respective values of \( |\vec{a}| \) and \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad and \quad |\vec{a}| = 3 |\vec{b}|, \]
are:

Assertion (A): The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP).

Reason (R): The region representing \( Z = 50x + 70y \) such that \( Z < 380 \) does not have any point common with the feasible region.


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Assertion (A): Let \( A = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \). If \( f : A \to A \) be defined as \( f(x) = x^2 \), then \( f \) is not an onto function.

Reason (R): If \( y = -1 \in A \), then \( x = \pm \sqrt{-1} \notin A \).

Find the domain of the function \( f(x) = \cos^{-1}(x^2 - 4) \).

Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm²/s. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.

(a) Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) \text{ with respect to x.

(b) If y = 5 \cos x - 3 \sin x, \text{ prove that \frac{d^2y{dx^2 + y = 0.

(a) Find a vector of magnitude 5 which is perpendicular to both the vectors \( 3\hat{i} - 2\hat{j} + \hat{k} and 4\hat{i} + 3\hat{j} - 2\hat{k} \).

(b) Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c\) \text{ be three vectors such that \mathbf{a \times \mathbf{b = \mathbf{a \times \mathbf{c \text{ and \mathbf{a \times \mathbf{b \neq 0. \text{ Show that \mathbf{b = \mathbf{c.

A man needs to hang two lanterns on a straight wire whose end points have coordinates A (4, 1, -2) and B (6, 2, -3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.

Find the value of \( a \) for which \( f(x) = \sqrt{3} \sin x - \cos x - 2ax + 6 \) \text{ is decreasing in \mathbb{R.

27. (a) Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]

27. (b) Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]

Find the particular solution of the differential equation: \[ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 \quad given that \quad y = \frac{\pi}{4}, when x = 1. \]

In the Linear Programming Problem (LPP), find the point/points giving the maximum value for \( Z = 5x + 10y \text{ subject to the constraints:

x + 2y \leq 120

x + y \geq 60

x - 2y \geq 0

x \geq 0, y \geq 0
\

If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\) such that \(|\vec{a}| = 3\), \(|\vec{b}| = 5\), \(|\vec{c}| = 7\), then find the angle between \(\vec{a}\) and \(\vec{b}\).

The probability that a student buys a colouring book is 0.7, and a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find:


(i) The probability that she buys both the colouring book and the box of colours.

(ii) The probability that she buys a box of colours given she buys the colouring book.

A fruit box contains 6 apples and 4 oranges. A person picks out a fruit three times with replacement. Find:


(i) The probability distribution of the number of oranges he draws.

(ii) The expectation of the number of oranges.

Sketch a graph of \( y = x^2 \). Using integration, find the area of the region bounded by \( y = 9 \), \( x = 0 \), and \( y = x^2 \).

A furniture workshop produces three types of furniture: chairs, tables, and beds each day. On a particular day, the total number of furniture pieces produced is 45. It was also found that the production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using the matrix method.

(a) For a positive constant \( a \), differentiate \( \left( t + \frac{1}{t} \right)^a \) with respect to \( t \), where \( t \) is a non-zero real number.

(b) Find \( \frac{dy}{dx} \) if \( x^3 + y^3 + x^2 = a^b \), where \( a \) and \( b \) are constants.

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

(b) Find the point on the line \( \frac{x-1}{3} = \frac{y+1}{2} = \frac{z-4}{3} \) at a distance of \( \sqrt{2} \) units from the point \( (-1, -1, 2) \).

(i) Taking length = breadth = \( x \) m and height = \( y \) m, express the surface area \( S \) of the box in terms of \( x \) and its volume \( V \), which is constant.

(ii) Find \( \frac{dS}{dx} \).

(iii) (a) Find a relation between \( x \) and \( y \) such that the surface area \( S \) is minimum.

(iii) (b) If surface area \( S \) is constant, the volume \( V = \frac{1}{4}(Sx - 2x^3) \), \( x \) being the edge of the base. Show that the volume \( V \) is maximum for \( x = \frac{\sqrt{6}}{6} \).

(i) Is \( f \) a bijective function?

(ii) Give reasons to support your answer to (i).

iii)(a) Let \( R \) be a relation defined by the teacher to plan the seating arrangement of students in pairs, where \( R = \{(x, y) : x, y are Roll Numbers of students such that y = 3x \} \).
List the elements of \( R \). Is the relation \( R \) reflexive, symmetric, and transitive? Justify your answer.

iii)(b) Let \( R \) be a relation defined by \( R = \{(x, y) : x, y are Roll Numbers of students such that y = x^3 \} \).
List the elements of \( R \). Is \( R \) a function? Justify your answer.

(i) Calculate the probability of a randomly chosen seed to germinate.

(ii) What is the probability that it is a cabbage seed, given that the chosen seed germinates?