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CBSE Class 12 Mathematics 2025 question paper (65/5)

Maximum marks 80 · Time 3 hours

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Section A

1 mark each

  1. Q.11 mark
    If A=[500050005]A = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}, then A3A^3 is :

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  2. Q.21 mark
    If P(A∪B)=0⋅9P(A \cup B) = 0{\cdot}9 and P(A∩B)=0⋅4P(A \cap B) = 0{\cdot}4, then P(A‾)+P(B‾)P(\overline{A}) + P(\overline{B}) is :

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  3. Q.31 mark
    If A=[123−437]A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & 3 & 7 \end{bmatrix} and B=[43−1205]B = \begin{bmatrix} 4 & 3 \\ -1 & 2 \\ 0 & 5 \end{bmatrix}, then the correct statement is :

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  4. Q.41 mark
    If ∣2x512x∣=∣6−543∣\begin{vmatrix} 2x & 5 \\ 12 & x \end{vmatrix} = \begin{vmatrix} 6 & -5 \\ 4 & 3 \end{vmatrix}, then the value of x is :

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  5. Q.51 mark
    If f(x)={sin⁡2axx2,x≠01,x=0f(x) = \begin{cases} \dfrac{\sin^2 ax}{x^2}, & x \neq 0 \\[4pt] 1, & x = 0 \end{cases} is continuous at x=0x = 0, then the value of a is :

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  6. Q.61 mark
    If A=[aij]A = [a_{ij}] is a 3×33 \times 3 diagonal matrix such that a11=1a_{11} = 1, a22=5a_{22} = 5 and a33=−2a_{33} = -2, then ∣A∣|A| is :

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  7. Q.71 mark
    The principal value of cot⁡−1(−13)\cot^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) is :

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  8. Q.81 mark
    If [4+xx−1−23]\begin{bmatrix} 4+x & x-1 \\ -2 & 3 \end{bmatrix} is a singular matrix, then the value of x is :

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  9. Q.91 mark
    If f(x)={[x], x∈R}f(x) = \{[x],\ x \in R\} is the greatest integer function, then the correct statement is :

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  10. Q.101 mark
    The slope of the curve y=−x3+3x2+8x−20y = -x^3 + 3x^2 + 8x - 20 is maximum at :

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  11. Q.111 mark
    ∫1+sin⁡x dx\displaystyle\int \sqrt{1 + \sin x}\ dx is equal to :

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  12. Q.121 mark
    ∫0π/2cos⁡x⋅esin⁡x dx\displaystyle\int_{0}^{\pi/2} \cos x \cdot e^{\sin x}\ dx is equal to :

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  13. Q.131 mark
    The area of the region enclosed between the curve y=x∣x∣y = x|x|, x-axis, x=−2x = -2 and x=2x = 2 is :

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  14. Q.141 mark
    The integrating factor of the differential equation (e−2xx−yx)dxdy=1\left(\dfrac{e^{-2\sqrt{x}}}{\sqrt{x}} - \dfrac{y}{\sqrt{x}}\right)\dfrac{dx}{dy} = 1 is :

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  15. Q.151 mark
    The sum of the order and degree of the differential equation [1+(dydx)2]3=d2ydx2\left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^3 = \dfrac{d^2y}{dx^2} is :

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  16. Q.161 mark
    For a Linear Programming Problem (LPP), the given objective function Z=3x+2yZ = 3x + 2y is subject to constraints : x+2y≤10x + 2y \le 10 3x+y≤153x + y \le 15 x, y≥0x,\ y \ge 0 The correct feasible region is :

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  17. Q.171 mark
    Let a⃗\vec{a} be a position vector whose tip is the point (2,−3)(2, -3). If AB→=a⃗\overrightarrow{AB} = \vec{a}, where coordinates of A are (−4,5)(-4, 5), then the coordinates of B are :

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  18. Q.181 mark
    The respective values of ∣a⃗∣|\vec{a}| and ∣b⃗∣|\vec{b}|, if given (a⃗−b⃗)⋅(a⃗+b⃗)=512(\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 and ∣a⃗∣=3∣b⃗∣|\vec{a}| = 3|\vec{b}|, are :

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  19. Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
    Q.191 mark
    Assertion (A) : The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP). Min Z=50x+70yZ = 50x + 70y subject to constraints 2x+y≥82x + y \ge 8, x+2y≥10x + 2y \ge 10, x, y≥0x,\ y \ge 0 Z=50x+70yZ = 50x + 70y has a minimum value =380= 380 at B(2,4)B(2, 4). Reason (R) : The region representing 50x+70y<38050x + 70y < 380 does not have any point common with the feasible region.

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  20. Q.201 mark
    Assertion (A) : Let A={x∈R:−1≤x≤1}A = \{x \in R : -1 \le x \le 1\}. If f:A→Af : A \to A be defined as f(x)=x2f(x) = x^2, then f is not an onto function. Reason (R) : If y=−1∈Ay = -1 \in A, then x=±−1∉Ax = \pm\sqrt{-1} \notin A.

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Section B

2 marks each

  1. Q.212 marks
    Find the domain of the function f(x)=cos⁡−1(x2−4)f(x) = \cos^{-1}(x^2 - 4).
  2. Q.222 marks
    Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm2^2/s. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
  3. Q.23 (a)2 marks
    Differentiate sin⁡xcos⁡x\dfrac{\sin x}{\sqrt{\cos x}} with respect to x.

  4. OR

    Q.23 (b)2 marks
    If y=5cos⁡x−3sin⁡xy = 5\cos x - 3\sin x, prove that d2ydx2+y=0\dfrac{d^2y}{dx^2} + y = 0.
  5. Q.24 (a)2 marks
    Find a vector of magnitude 5 which is perpendicular to both the vectors 3i^−2j^+k^3\hat{i} - 2\hat{j} + \hat{k} and 4i^+3j^−2k^4\hat{i} + 3\hat{j} - 2\hat{k}.

  6. OR

    Q.24 (b)2 marks
    Let a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} be three vectors such that a⃗⋅b⃗=a⃗⋅c⃗\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} and a⃗×b⃗=a⃗×c⃗\vec{a} \times \vec{b} = \vec{a} \times \vec{c}, a⃗≠0⃗\vec{a} \neq \vec{0}. Show that b⃗=c⃗\vec{b} = \vec{c}.
  7. Q.252 marks
    A man needs to hang two lanterns on a straight wire whose end points have coordinates A(4,1,−2)A(4, 1, -2) and B(6,2,−3)B(6, 2, -3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.

Section C

3 marks each

  1. Q.263 marks
    Find the value of 'a' for which f(x)=3sin⁡x−cos⁡x−2ax+6f(x) = \sqrt{3}\sin x - \cos x - 2ax + 6 is decreasing in R.
  2. Q.27 (a)3 marks
    Find : ∫2x(x2+3)(x2−5) dx\displaystyle\int \dfrac{2x}{(x^2 + 3)(x^2 - 5)}\ dx

  3. OR

    Q.27 (b)3 marks
    Evaluate : ∫14(∣x−2∣+∣x−4∣) dx\displaystyle\int_{1}^{4} \left(|x - 2| + |x - 4|\right)\ dx
  4. Q.283 marks
    Find the particular solution of the differential equation [xsin⁡2(yx)−y]dx+x dy=0\left[x\sin^2\left(\dfrac{y}{x}\right) - y\right]dx + x\,dy = 0 given that y=π4y = \dfrac{\pi}{4}, when x=1x = 1.
  5. Q.293 marks
    In the Linear Programming Problem (LPP), find the point/points giving maximum value for Z=5x+10yZ = 5x + 10y subject to constraints x+2y≤120x + 2y \le 120 x+y≥60x + y \ge 60 x−2y≥0x - 2y \ge 0 x, y≥0x,\ y \ge 0
  6. Q.30 (a)3 marks
    If a⃗+b⃗+c⃗=0⃗\vec{a} + \vec{b} + \vec{c} = \vec{0} such that ∣a⃗∣=3|\vec{a}| = 3, ∣b⃗∣=5|\vec{b}| = 5, ∣c⃗∣=7|\vec{c}| = 7, then find the angle between a⃗\vec{a} and b⃗\vec{b}.

  7. OR

    Q.30 (b)3 marks
    If a⃗\vec{a} and b⃗\vec{b} are unit vectors inclined with each other at an angle θ\theta, then prove that 12∣a⃗−b⃗∣=sin⁡θ2\dfrac{1}{2}|\vec{a} - \vec{b}| = \sin\dfrac{\theta}{2}.
  8. Q.31 (a)3 marks
    The probability that a student buys a colouring book is 0⋅70{\cdot}7 and that she buys a box of colours is 0⋅20{\cdot}2. The probability that she buys a colouring book, given that she buys a box of colours, is 0⋅30{\cdot}3. Find the probability that the student : (i) Buys both the colouring book and the box of colours. (ii) Buys a box of colours given that she buys the colouring book.

  9. OR

    Q.31 (b)3 marks
    A person has a fruit box that contains 6 apples and 4 oranges. He picks out a fruit three times, one after the other, after replacing the previous one in the box. Find : (i) The probability distribution of the number of oranges he draws. (ii) The expectation of the random variable (number of oranges).

Section D

5 marks each

  1. Q.325 marks
    Sketch a graph of y=x2y = x^2. Using integration, find the area of the region bounded by y=9y = 9, x=0x = 0 and y=x2y = x^2.
  2. Q.335 marks
    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 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 matrix method.
  3. Q.34 (a)5 marks
    For a positive constant 'a', differentiate at+1ta^{t + \frac{1}{t}} with respect to (t+1t)a\left(t + \dfrac{1}{t}\right)^a, where t is a non-zero real number.

  4. OR

    Q.34 (b)5 marks
    Find dydx\dfrac{dy}{dx} if yx+xy+xx=aby^x + x^y + x^x = a^b, where a and b are constants.
  5. Q.35 (a)5 marks
    Find the foot of the perpendicular drawn from the point (1,1,4)(1, 1, 4) on the line x+25=y+12=−z+4−3\dfrac{x+2}{5} = \dfrac{y+1}{2} = \dfrac{-z+4}{-3}.

  6. OR

    Q.35 (b)5 marks
    Find the point on the line x−13=y+12=z−43\dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-4}{3} at a distance of 222\sqrt{2} units from the point (−1,−1,2)(-1, -1, 2).

Section E

  1. A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum. On the basis of the above information, answer the following questions :
    Q.36 (i)1 mark
    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.
  2. Q.36 (ii)1 mark
    Find dSdx\dfrac{dS}{dx}.
  3. Q.36 (iii) (a)2 marks
    Find a relation between x and y such that the surface area (S) is minimum.

  4. OR

    Q.36 (iii) (b)2 marks
    If surface area (S) is constant, the volume (V)=14(Sx−2x3)(V) = \dfrac{1}{4}(Sx - 2x^3), x being the edge of base. Show that volume (V) is maximum for x=S6x = \sqrt{\dfrac{S}{6}}.
  5. Let A be the set of 30 students of class XII in a school. Let f:A→Nf : A \to N, N is a set of natural numbers such that function f(x)=f(x) = Roll Number of student x. On the basis of the given information, answer the following :
    Q.37 (i)1 mark
    Is f a bijective function ?
  6. Q.37 (ii)1 mark
    Give reasons to support your answer to (i).
  7. Q.37 (iii) (a)2 marks
    Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where R={(x,y):x, yR = \{(x, y) : x,\ y are Roll Numbers of students such that y=3x}y = 3x\}. List the elements of R. Is the relation R reflexive, symmetric and transitive ? Justify your answer.

  8. OR

    Q.37 (iii) (b)2 marks
    Let R be a relation defined by R={(x,y):x, yR = \{(x, y) : x,\ y are Roll Numbers of students such that y=x3}y = x^3\}. List the elements of R. Is R a function ? Justify your answer.
  9. A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage and radish to be 25%, 35% and 40% respectively. But before he could plant the seeds, they got mixed up in the bag and he had to sow them randomly. Based upon the above information, answer the following questions :
    Q.38 (i)2 marks
    Calculate the probability of a randomly chosen seed to germinate.
  10. Q.38 (ii)2 marks
    What is the probability that it is a cabbage seed, given that the chosen seed germinates ?