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

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Here we provide CBSE Class 12 math paper-2023 set-1 series EF1GH/C 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 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 A is a square matrix of order 3 and |A| = 6, then the value of | adj A | is:

(a) 6 (b) 36

2. The value of $\displaystyle\int_0^{\pi/6} \sin 3x dx$ is :

(a) $-\frac{\sqrt{3}}{2}$ (b) $-\frac{1}{3}$

3. If $\overrightarrow{a}, \overrightarrow{b}$ and $(\overrightarrow{a} + \verrightarrow{b})$ are all unit vectors and θ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ , then the value of θ is :

(a) 2π/3 (b) 5π/6

4. The projection of vector $\hat{i}$ on the vector $\hat{i} + \hat{j} + 2\hat{k}$ is :

(a) 1/√6 (b) √6

5. A family has 2 children and the elder child is a girl. The probability that both children are girls is :

(a) 1/4 (b) 1/8

6. The vector equation of a line which passes through the point (2, -4, 5) and is parallel to the line $\dfrac{x + 3}{3} = \dfrac{4 - y}{2} = \dfrac{z + 8}{6}$ is :

(a) $\overrightarrow{r} = (-2\hat{i} + 4\hat{j} - 5\hat{k}) + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$

(b) $\overrightarrow{r} = (2\hat{i} - 4\hat{j} + 5\hat{k}) + \lambda(3\hat{i} - 2\hat{j} + 6\hat{k})$

(d) $\overrightarrow{r} = (-2\hat{i} + 4\hat{j} - 5\hat{k}) + \lambda(3\hat{i} - 2\hat{j} - 6\hat{k})$

7. For which value of x, are the determinants $\begin{vmatrix} 2x & -3 \\ 5 & x\end{vmatrix} = \begin{vmatrix} 10 & 1 \\ -3 & 2\end{vmatrix}$

(a) ±3 (b) -3

8. The value of the cofactor of the element of second row and third column in the matrix $\begin{bmatrix} 4 & 3 & 2 \\ 2 & -1 & 0 \\ 1 & 2 & 3\end{bmatrix}$ is :

(a) 5 (b) -5

9. The difference of the order and the degree of the differential equation $(\dfrac{d^2y}{dx^2}) + (\dfrac{dy}{dx})^3 + x^4 = 0$

(a) 1 (b) 2

10. If matrix $A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ and A² = kA, then the value of k is :

(a) 1 (b) -2

11. $\displaystyle \int \dfrac{\cos 2x}{\sin^2 x. \cos^2 x} dx$ is equal to:

(a) $\tan x - \cot x+ C$ (b) $-\cot x - \tan x + C$

12. The integrating factor of the differential equation $(3x^2 + y)\dfrac{dx}{dy} = x$ is :

(a) 1/x (b) 1/x²

13. The point which lies in the half-plane 2x + y – 4 ≤ 0

(a) (0, 8) (b) (1, 1)

14. If $(\cos x)^y = (\cos y)^x$ , then dy/dx is equal to:

(a) $\dfrac{y\tan x + \log (\cos y)}{x \tan y - \log (\cos x)}$

(b) $\dfrac{x\tan y + \log (\cos x)}{y \tan x + \log (\cos y)}$

(d) $\dfrac{y\tan x + \log (\cos y)}{x \tan y + \log (\cos x)}$

15. It is given that $X.\begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\2 & 3 \end{bmatrix}$

(a) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

(b) $\begin{bmatrix} 0 & -1 \\ 1 & 1 \end{bmatrix}$

(d) $\begin{bmatrix} 1 & -1 \\ 1 & -1 \end{bmatrix}$

16. If ABCD is a parallelogram and AC and BD are its diagonals, then $\overrightarrow{AC} + \overrightarrow{BD}$ is :

(a) $2\overrightarrow{DA}$ (b) $2\overrightarrow{AB}$

17. If $x = a\cos \theta + b \sin \theta, y =a \sin \theta - b \cos \theta$ , then which one of the following is true ?

(a) $y^2 \dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} + y = 0$

(b) $y^2 \dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} + y = 0$

(d) $y^2 \dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} - y = 0$

18. The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function Z = 2a x + by occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is :

(a) 2a = b (b) 2a = 3b

Question number 19 and 20 are Assertion and Reason based questions carrying 1 marks each. 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.

(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 and Reason (R) is not the correct explanation of the Assertion (A).

(c) Assertion (A) is true, but Reason (R) is false.

(d) Assertion (A) is flase, but Reason (R) is true.

19. Assertion (A) : The principal value of $\cot^{-1} (\sqrt{3})$ is π/6

Reason (R) : Domain of $\cot^{-1} x$ is R – {-1, 1}.

20. Assertion (A) : Quadrilateral formed by vertices A(0, 0, 0), B(3, 4, 5), C(8, 8, 8) and D(5, 4, 3) is rhombus.

Reason (R) : ABCD is a rhombus if AB = BC = CD = DA, AC ≠ BD.

SECTION B

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

21. If three non-zero vectors are $\vec{a},\vec{b}$ and $\vec{c}$ such that $\vec{a}.\vec{b} = \vec{a}.\vec{c}$

and $\vec{a} \times \vec{b} = \vec{a} \times {c}$ , then show that $\vec{b} = \vec{c}$ .

22. (a) Simplify :

$\tan^{-1}(\dfrac{\cos x}{1 - \sin x})$

(b) Prove that the greatest integer function f:R → R, given by f(x) = [x], is neither one-one nor onto.

23. Function f is defined as

$f(x) = \begin{cases} 2x + 2, & \text{ if }x < 2 \\ k, &\tex{ if } x = 2 \\ 3x, & \text{ if } x > 2 \end{cases}$

Find the value of k for which the funtion f is continuous at x = 2

24. Find the intervals in which the function $f(x) = x^4 - 4x^3 + 4x^2 + 15$ , is strictly increasing.

25. (a) If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $|\vec{a}| = 7, |\vec{b}| = 24, |\vec{c}| = 25$ and $\vec{a} + \vec{b} + \vec{c} = 0$ , then find the value of $\vec{a}.\vec{b} + \vec{b}.\vec{c} + \vec{c}.\vec{a}$ .

(b) If a line makes angles α, β and γ with x-axis, y-axis and z-axis respectively, then prove that $\sin^2 \alpha + \sin^2 \beta +\sin^2 \gamma = 2$

SECTION C.

This section comprises short answe (SA) type questions of 3 marks each.

26. (a) Evaluate:

$\displaystyle \int^{\pi/2}_0 \dfrac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx$

(b) Evaluate:

$\displaystyle \int^3_1 (|x - 1| + |x - 2|) dx$

27. (a) Find the particular solution of the differential equation $\dfrac{dy}{dx} = \dfrac{xy}{x^2 + y^2}$ , given that y = 1 when x = 0.

(b) Find the particular solution of the differential equation $(1 + x^2)\dfrac{dy}{dx} + 2xy = \dfrac{1}{1 + x^2}$ , given that y = 0 when x = 0.

28. (a) Out of two bags, bag A contains 2 white and 3 red balls and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag B.

(b) Out of a group of 50 people, 20 always speak the truth. Two persons are selected at random from the group (without replacement). Find the probability distribution of number of selected person who always speak th truth.

29. Find:

$\displaystyle \int \dfrac{\cos \theta}{\sqrt{3 - 3 \sin \theta - \cos^2 \theta}} d\theta$

30. Solve the following Linear Programming Problem graphically :

Minimise Z = 3 x + 8 y

subject to the constraints

3x + 4y ≥ 8

5x + 2y ≥ 11

x ≥ 0, y ≥ 0

31. Find:

$\displaystyle \int \dfrac{2x^2 + 1}{x^2(x^2 + 4)}dx$

SECTION D

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

32. If matrix $A = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\1 & 1 & 1 \end{bmatrix}$ find $A^{-1}$ and hence solve the following system of linear equation:

3x + 2y + z = 2000

4x + y + 3z = 2500

x + y + z = 900,

33. (a) Show that the lines $\dfrac{x + 1}{3} = \dfrac{y + 3}{5} = \dfrac{z + 5}{7}$ and $\dfrac{x - 2}{1} = \dfrac{y - 4}{3} = \dfrac{z - 6}{5}$ intersect. Also find their point of intersection.

(b) Find the shortest distance between the pair of lines $\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = z$ and $\dfrac{x + 1}{5} = \dfrac{y - 2}{1} = \dfrac{z + 5}{7}; z = 2$ .

34. Find the area of triangle ABC bounded by the lines represented by the equations 5x – 2y – 10 =0, x – y – 9 = 0 and 3x – 4y – 6 = 0, using integration method.

35. (a) Show that the relation S in the set R of real numbers defined by

S = {(a, b) : a ≤ b³, a ∈ R, b ∈ R}

is neither reflexive, nor symmetric, nor transitive.

(b) Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by

R = {(a, b) : both a and b are either odd or even}.

Show that R is an equivalence relation. Hence, find the elements of equivalence class [1].

SECTION E

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

Case study – 1

36. In a group activity class, there are 10 students whose age are 16, 17, 15, 14, 19, 17, 16, 19, 16 and 15 years. One student is selected at random such that each has equal chance of being chosen and age of the student is recorded.

On the basis of the above information, answer the following question:

(i) Find the probability that the age of the selected student is a composite number. 1

(ii) Let X be the age of the selected student. What can be the value of X ? 1

(iii) (a) Find the probability distributionof random variable X and hence find the mean age. 2

(b) A student was selected at random and his age was found to be greater than 15 years. Find the probability that his age is a prime number. 2

See solution

Case study – 2

37. A housing society wants to commission a swimming pool for its residents. For this, they have to purchase a square piece of land and dig this to such a depth that its capacity is 250 cubic metres. Cost of land is Rs 500 per square metre. The cost of digging increases with the depth and cost for the whole pool is Rs 4000 (depth)²

Suppose the side of the square plot is x metres and depth is h metres. On the basis of the above information, answer the following question:

(i) Write cost C(h) as function in terms of h. 1

(ii) Find critical point. 1

(iii) (a) Use second derivative test to find the value of h for which cost of constructing the pool is minimum. What is the minimum cost of construction of the pool ? 2

(iii) (b) Use first derivative test to find the depth of the pool so that cost of construction is minimum. Also, find relation between x and h for minimum cost. 2

See solution

Case study – 3

38. In an agriculture institute, scientists do experiments with varieties of seeds to grow them in different environments to produce healthy plants and get more yield.

A scientist observed that a particular seeds grew very fast after germination. He had recorded growth of plant since germination and he said that its growth can be defined by the function

f(x) = 1/3 x³ – 4 x² + 15 x + 2, 0 ≤ x ≤ 10

where x is the number of days the plant is exposed to sunlight.

On the basis of the above information, answer the following questions :

(i) What are the critical points of the function f(x) ? 2

(ii) Using second derivative test, find the minimum value of the function. 2

A housing society wants to commission a swimming pool

CBSE