Cbse Class 12 math paper-2023 set-2 series-EF1GH/4

Here we provide CBSE Class 12 math paper-2023 set-2 series EF1GH/4 for practice and student get more marks after understanding the question type in board exam. Case study question are in the paper of 12 marks.

MATHEMATICS

Cbse Class 12 math paper-2023 set-1 series-EF1GH/C

CBSE Class 12 math paper-2023 set-2 series-EF1GH/4 Download

Time allowed : 3 hours Maximum Marks: 80

General instruction:

Read the following instructions very carefully and strictly follow them:

(i) This question paper cantains 38 questions. All are compulsory.

(ii) This question paper is divided into five section – A, B, C, D and E

(iii) In Section A, question no. 1 to 18 are multiple choice questions(MCQ) and question number 19 and 20 are Assertion-Reason based question of 1 mark each.

(iv) In Section B, Question no. 21 to 25 are very short answer (VSA) type question, carrying 2 marks each.

(v) In Section C, Question no. 26 to 31 are short answer (SA) type question, carrying 3 marks each.

(vi) In Section D, Question no. 32 to 35 are long answer (LA) type question carrying 5 marks each.

(vii) In Section E, Question no. 36 to 38 are case study based question carrying 4 marks each.

(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 question in Section C, 2 question in Section D and 2 question in Section E.

(ix) Use of calculators is not allowed.

SECTION A

This section comprises multiple choice question (MCQs) of 1 marks each.

1. If $\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\0 & 0 & 1\end{bmatrix}.\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 3 \\ 2 \end{bmatrix}$ , then the value of (2x + y – z) is :

(i) 1 (ii) 2

(iii) 3 (iv) 5

2. If a matrix $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ , then the matrix AA‘ (where A’ is the transpose of A) is :

(i) 14 (ii) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

(iii) $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{bmatrix}$ (iv) $\begin{bmatrix} 14 \end{bmatrix}$

3. If $x\begin{bmatrix} 1 & 2 \end{bmatrix} + y\begin{bmatrix} 2 & 5 \end{bmatrix} = \begin{bmatrix} 4 & 9 \end{bmatrix}$ , then :

(a) x = 1, y = 2 (b) x = 2, y = 1

4. If A is a square matrix and A² = A, then (I + A)² – 3A is equal to:

(a) I (b) A

5. The value of determinant $\begin{vmatrix} 2 & 7 & 1 \\ 1 & 1 & 1 \\ 10 & 8 & 1 \end{vmatrix}$ is :

(a) 47 (b) -79

6. The function f(x) = |x| is

(a) Continuous and differentiable everywhere;

(b) Continuous and differentiable nowhere.

(d) Continuous everywhere, but differentiable nowhere.

7. If $y = \log (\sin e^x) \dx$ , then $\frac{dy}{dx}$ is :

(a) $\cot e^x$ (b) $\operatorname{cosec} e^x$

8. $\displaystyle \int e^{5 \log x} \ dx$ is equal to :

(a) $\frac{x^5}{5} + C$ (b) $\frac{x^6}{6} + C$

9. $\displaystyle \int_0^4 (e^{2x} + x) \ dx$ is equal to :

(a) $\frac{15 + e^8}{2}$ (b) $\frac{16 - e^8}{2}$

10. A unit vector along the vector $4\hat{i} - 3\hat{k}$ is :

(a) $\frac{1}{7}(4\hat{i} - 3\hat{k})$

(b) $\frac{1}{5}(4\hat{i} - 3\hat{k})$

(d) $\frac{1}{\sqrt{5}}(4\hat{i} - 3\hat{k})$

11. If θ is the angle between two vectors $\vec{a}$ and $\vec{b}$ , then $\vec{a}.\vec{b} \geq 0$ only

(a) 0 < θ < π/2 (b) 0 ≤ θ ≤ π/2

12. The integrating factor for solving the differential equation $x.\frac{dy}{dx} - y = 2x^2$ is :

(a) $e^{-y}$ (b) $e^{-x}$

13. The number of solutions of the differential equation $\frac{dy}{dx} = \frac{y + 1}{x - 1}$ , when y(1) = 2, is :

(a) zero (b) one

14. Distance of the point (p, q, r) from y-axis is :

(a) q (b) |q|

15. If direction cosines of a line are $(\frac{1}{a}, \frac{1}{a}, \frac{1}{a})$ , then : (a) 0 < a < 1 (b) a > 2

16. For two events A and B, if P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6, then

(a) 0.24 (b) 0.3

17. Which of the following point satisfies both the inequations 2x + y ≤ 10 and x + 2y ≥ 8 ?

(a) (-2, 4) (b) (3, 2)

18. The solution set of the inequation 3x + 5y < 7 is :

(a) whole xy-plane except the points lying on the line 3x + 5y = 7

(b) whole xy-plane along with the points Iying on the line 3x +5y = 7.

(d) open half plane not containing the origin.

Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (¢) and (d as given below.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assértion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

19. Assertion (A) : All trigonometric functions have their inverses over their respective domains.

Reason (R): The inverse of $\tan^{-1} x$ exists for some x ∈ R.

20. Assertion (A): The lines $\vec{r} = \vec{a_1}+ \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2}+ \mu \vec{b_2}$ are
perpendicular, when $\vec{b_1}.\vec{b_2 }= 0$ .

Reason (R) : The angle θ between the lines $\vec{r} = \vec{a_1}+ \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2}+ \mu \vec{b_2}$ is given by $\cos \theta= \dfrac{\vec{b_1}.\vec{b_2}}{|b_1||b_2|}$

SECTION B

Case Study Question

This section comprises very short answer (VSA) type questions of 2 marks each.

21. Find the interval in which the function f(x) = 2x³ – 3x is strictly increasing.

22. (a) Find the vector equation of the line passing through the point (2, 1, 3) and perpendicular to both the lines

$\dfrac{x - 1}{1} = \dfrac{y - 2}{2} = \dfrac{z - 3}{3} ; \dfrac{x}{3} = \dfrac{y}{2} = \dfrac{z}{5}$

(b) The equations of a line are 5x – 3 = 15y + 7 = 3 – 10z. Write the
direction cosines of the line and find the coordinates of a point through which it passes.

23. (a) Find the domain of $y = sin^{-l} (x^2 - 4)$ .

(b) Evaluate:

$\cos^{-1}[\cos (-\dfrac{7\pi}{3})]$

24. If $(x^2+ y^3)^2 =xy$ , then find $\dfrac{dy}{dx}$ .

25. If $\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}$ , then find a unit vector along the vector $\vec{a } \times \vec{b}$ .

SECTION C

This section comprises short answer (SA) type questions of 3 marks eaeh.

26. Find :

$\displaystyle \int \dfrac{x^2}{x^2 + 6x + 12}\ dx$

27. Two fair dice are thrown simultaneously. If X denotes the number of sixes, find the mean of X.

28. (a) Find the particular solution of the differential equation

$\dfrac{dy}{dx} = \dfrac{x + y}{x}$ , y(1) = 0.

(b) Find the general solution of the differential equation

$e^x \tan y \ dx + (1 - e^x) \sec^2 y\ dy = 0$ .

29. (a) Evaluate :

$\displaystyle \int_{\pi/4}^{\pi/2} e^{2x} (\dfrac{1 - \sin 2x}{1 - \cos 2x})\ dx$

(b) Evaluate :

$\displaystyle \int_{-2}^{2} \dfrac{x^2}{1 + 5^x}\ dx$

30. (a) Find:

$\displaystyle \int \dfrac{e^x}{\sqrt{5 - 4e^x - e^{2x}}} \ dx$

(b) Evaluate:

$\displaystyle \int_0^{\pi/2} \sqrt{\sin x} \cos^5 x \ dx$

31. Solve the following linear programming problem graphically:
Maximise z = -3x – 5y
Subject to the constraints
-2x + y ≤ 4,
x + y ≥ 3,
x – 2y ≤ 2,
x ≥ 0, y ≥ 0.

SECTION D

This section comprises long answer (LA) type questions of 5 marks each.

32. Using integration, find the area of the region bounded by the circle x² + y² = 16, line y = x and y-axis, but lying in the 1st quadrant.

33. (a) Show that the following lines do not intersect each other :

$\dfrac{x - 1}{3} = \dfrac{y + 1}{2} = \dfrac{z - 1}{5} ; \dfrac{x + 2}{4} = \dfrac{y - 1}{3} = \dfrac{z + 1}{-2}$

(b) Find the angle between the lines

$2x = 3y = -z$ and $6x = -y = -4z$

34. (a) If N denotes the set of all natural numbers and R is the N x N relation on defined by (a, b) R (c, d), if ad(b + c) = bcla + d). Show that R is an equivalence relation.

(b) Let f: R-{-4/3} → R be a function defined as $fx)= \dfrac{4x}{3x+4}$ Show that f is a one-one function. Also, check whether f is an onto function or not.

35. Find the inverse of the matrix $A= \begin{bmatrix} 1 & - 1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$ . Using the inverse, $A^{-1}$ . solve the system of linear equations

x – y + 2z = 1; 2y – 3z = 1; 3x -2y + 4z = 3.

SECTION E

This section comprises 3 case study based questions of 4 marks each.

Case Study – 1

36. A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0.65. The probability that many are not present and still the work gets completed on time is 0.35. The probability that work will be completed on time when all workers are present is 0.80.

Let: $E_1$ : represent the event when many workers were not present for the job;
$E_2$ : represent the event when all workers were present;

and E : represent completing the construction work on time.

Based on the above information, answer the following questions :
(i) What is the probability that all the workers are present for the job? 1
(ii) What is the probability that construction will be completed on time? 1
(iii) (a) What is the probability that many workers are not present given that the construction work is completed on time ? 2

(iii) (b) What is the probability that all workers were present given that the construction job was completed on time ?

Case Study – 2

37. Let f(x) be a real valued function. Then its

• Left Hand Derivative (L.H.D.) : $\displaystyle Lf '(a) = lim_{h \to 0} \dfrac{f(a - h) - f(a)}{- h}$

• Right Hand Derivative (R.H.D.) : $\displaystyle Rf '(a) = lim_{h \to 0} \dfrac{f(a + h) - f(a)}{ h}$

Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and both are equal.

For the function $f(x) = \begin{cases} |x - 3|, & x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & x < 1 \end{cases}$

Answer the following question :

(i) What is R.H.D. of f(x) at x = 1 ? 1

(ii) What is L.H.D. of f{x) at x = 1 ? 1

(iii) (a) Check if the function f(x) is differentiable at x = 1. 2

(iii) (b) Find f ‘(2) and f ‘(-1). 2

Case Study – 3

38. Sooraj’s father wants to construct a rectangular garden using brick wall on one side of the garden and wire fencing for the other three sides as shown in the figure. He has 200 metres of fencing wire .