Class 12th Maths Mid- Term Previous year Question Paper (2023-24)

CLASS → 12^th [MATHS]

MID-TERM EXAMINATION (2023–24)

CLASS: XII
SUBJECT: MATHEMATICS (041)

Time Allowed: 3 hours
Maximum Marks: 80
समय: 3 घंटे
अधिकतम अंक: 80

सामान्य निर्देश:

• 1. इस प्रश्न पत्र में 38 प्रश्न हैं। सभी प्रश्न अनिवार्य हैं।
• 2. एक कुल 5 खंडों में विभाजित किया गया है जैसे – क, ख, ग, घ और ड।
• 3. खंड क में प्रश्न आंक 1 से 18 तक एक विकल्पीय और प्रश्न आंक 19 और 20 आधारण-कारण चार के हैं।
• 4. खंड ख में प्रश्न आंक 21 से 25 तक अति लघु उत्तरीय प्रश्न हैं जिने 2 अंक मिले हैं।
• 5. खंड ग में प्रश्न आंक 26 से 31 तक लघु उत्तरीय प्रश्न हैं जिने 3 अंक मिले हैं।
• 6. खंड घ में प्रश्न आंक 32 से 35 तक दीर्घ उत्तरीय प्रश्न हैं जिने 5 अंक मिले हैं।
• 7. खंड ड में प्रश्न आंक 36 से 38 तक स्थिति आधारित प्रश्न हैं, जो 4 अंक के हैं।
• 8. कुल पत्र में कौन्छ विकल्प नहीं दी गई है। तथापि, खंड ख के 2, खंड ग के 3, खंड घ के 2 और खंड ड के 2 प्रश्नों में आंतरिक विकल्प का प्रावधान किया गया है।
• 9. कैलक्युलेटर का उपयोग वर्जित है।

GENERAL INSTRUCTIONS:

• 1. This question paper contains 38 questions. All questions are compulsory.
• 2. This question paper is divided into five sections – A, B, C, D and E.
• 3. Section A: Q1 to Q18 are MCQs, Q19 & Q20 are Assertion-Reason based (1 mark each).
• 4. Section B: Q21 to Q25 are Very Short Answer (VSA) type questions (2 marks each).
• 5. Section C: Q26 to Q31 are Short Answer (SA) type questions (3 marks each).
• 6. Section D: Q32 to Q35 are Long Answer (LA) type questions (5 marks each).
• 7. Section E: Q36 to Q38 are Case Study based questions (4 marks each).
• 8. No overall choice. Internal choice in 2 Qs in B, 3 Qs in C, 2 Qs in D, 2 Qs in E.
• 9. Use of calculator is not allowed.

SECTION-A

This section comprises multiple choice questions (MCQs) of 1 mark each.

1. If y = sin²(x²), then dy/dx is:

(a) cos²(x²)

(b) cos²(2x)

(c) 2·sin(2x)

(d) 2x·sin(2x²)

2. The side of a square increases at the rate of 4 cm per second. Find the rate at which the area of the square increases when the side is 5 cm.

(a) 20 sq.cm/sec

(b) 40 sq.cm/sec

(c) 15 sq.cm/sec

(d) 25 sq.cm/sec

3. Evaluate: ∫ x·sin(2x) dx

(a) (x·cos(2x))/2 + (sin(2x))/4 + c

(b) (x·cos(2x))/2 − (sin(2x))/4 + c

(c) −(x·cos(2x))/2 + (sin(2x))/4 + c

(d) (x·cos(2x))/2 + (sin(2x))/2 + c

4. Let A = {1, 2, 3} and R = {(1,1), (2,3), (1,2)}; which ordered pairs among the following can be added in relation R to make it reflexive and transitive?

(a) (2, 2), (3, 3)

(b) (2, 2), (1, 3)

(c) (1, 3), (3, 3)

(d) (1, 3), (2, 2), (3, 3)

5. Minimum value of f(x) = |x + 2| − 1, x ∈ ℝ, is:

(a) –1

(b) 1

(c) –2

(d) 2

6. ∫ (2 − 3·sin(x)) / cos²(x) dx is equal to:

(a) 2·sec(x) − 3·tan(x) + c

(b) 3·sec(x) + 2·tan(x) + c

(c) 2·tan(x) − 3·sec(x) + c

(d) 3·tan(x) − 2·sec(x) + c

7. cos⁻¹(½) + 2·sin⁻¹(½) + 4·tan⁻¹(1/√3) is equal to:

(a) π/6

(b) π/3

(c) 4π/3

(d) 3π/4

8. Order and degree of the differential equation (d²y/dx²)³ = 1 + (dy/dx)⁴:

(a) 2, 3

(b) 2, 6

(c) 3, ½

(d) 3, 4

9. If 3x² + 8xy + 5y² = 1, then dy/dx equals:

(a) −(6x + 8y)/(16x + 40y)

(b) (16x + 8y)/(18x + 10y)

(c) −(6x + 8y)/(8x + 10y)

(d) (18x + 16y)/(12x + 20y)

10. Value of cot[cos⁻¹(7/25)] is:

(a) 24/25

(b) 24/7

(c) 25/24

(d) 7/24

11. ∫ dx / √(16 − 9x²) is equal to:

(a) (1/3)·sin⁻¹(3x/4) + c

(b) (1/4)·sin⁻¹(4/3x) + c

(c) log|x + √(16 − 9x²)| + c

(d) (1/8)·log| (4 − 3x)/(4 + 3x) | + c

12. If A = [ [2, −7], [9, −5] ], then (adj A)′ is:

(a) [ [5, 9], [7, 2] ]

(b) [ [−5, 7], [−9, 2] ]

(c) [ [5, 9], [−7, −2] ]

(d) [ [−5, −9], [7, 2] ]

13. If f(x) = { 3x − 8, if x ≤ 5 2k,   if x > 5 } is continuous, then the value of k is:

(a) 1/2

(b) 3/2

(c) 5/2

(d) 7/2

14. ∫ (x⁴ + 1)/(x² + 1) dx is equal to:

(a) (x³)/3 − x + 2·tan⁻¹(x) + c

(b) (x⁵)/5 + tan⁻¹(x) + c

(c) (x³)/3 + x² + tan⁻¹(x) + c

(d) (2x³)/3 + x + 2·tan⁻¹(x) + c

15. If matrix A =

[ 2 0 -3 ]
[ 4 3 1 ]
[ -5 7 2 ]

is expressed as the sum of a symmetric matrix B and a skew-symmetric matrix C, then matrix C is equal to:

(a) [ 2 2 -4 ] [ 2 3 4 ] [ -4 4 2 ]

(b) [ 2 4 -5 ] [ 0 3 7 ] [ -3 1 2 ]

(c) [ 0 -2 -1 ] [ 2 0 -3 ] [ -1 3 0 ]

(d) [ 0 2 -1 ] [ 2 0 3 ] [ -1 3 0 ]

16. Derivative of sin(x) with respect to cos(x) is:

(a) −cot(x)

(b) 1

(c) tan(x)

(d) −tan(x)

17. If

Matrix A = [ 1 2 -1 ] [ 3 0 2 ] [ 4 5 0 ] Matrix B = [ 1 0 0 ] [ 2 1 0 ] [ 0 1 3 ]

then AB is equal to:

(a) [ 5 1 -3 ] [ 3 2 6 ] [ 14 5 0 ]

(b) [ 11 4 3 ] [ 1 2 3 ] [ 0 3 3 ]

(c) [ 1 8 4 ] [ 2 9 6 ] [ 0 2 0 ]

(d) [ 0 1 2 ] [ 5 4 3 ] [ 1 8 2 ]

18. If y = log [ tan(π/4 + x/2) ], then dy/dx is:

(a) sec(x)

(b) csc(x)

(c) tan(x)

(d) sec(x)·tan(x)

Questions 19 and 20 (1 mark each)

In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer from the following:

(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

19.
Assertion (A): The principal value of cot⁻¹(√3) is π/6 — True
Reason (R): Domain of cot⁻¹x is ℝ = (−1, 1) — False
Correct option: (c) A is true but R is false ✔️

20.
Assertion (A): Let A = {−1, 1, 2, 3} and B = {1, 4, 9}, where f: A → B given by f(x) = x². Then f is a many-one function. — False
Reason (R): If x₁ ≠ x₂ ⇒ f(x₁) ≠ f(x₂), for every x₁, x₂ ∈ domain, then f is one-one, else many-one. — True
Correct option: (d) A is false but R is true ✔️

SECTION-B

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

21. Using integration, find the area of the region enclosed by the parabola y² = 4x, and vertical lines x = 2 and x = 4 in the first quadrant.

22. If x = a·cos(θ) + b·sin(θ) and y = a·sin(θ) − b·cos(θ), then prove:
y²·(d²y/dx²) − x·(dy/dx) + y = 0

OR

Differentiate sin(xˣ) with respect to x.

23. Show that the modulus function f: ℝ → ℝ given by f(x) = |x| is neither one-one nor onto.

24. Using integration, find the area enclosed by the x-axis, y-axis, and the line x + y + 1 = 0 in quadrant III.
OR
Find the area of the region bounded by the curve y = cos(x) between x = 0 and x = π using integration.

25. Let ℕ be the set of all natural numbers. Define relation R on ℕ × ℕ as:

(a, b) R (c, d) ⇔ 1/a + 1/d = 1/b + 1/c

Show that R is transitive.

SECTION–C (3 Marks Each)

26. If matrix A is:

[ 3 6 ]
[ -4 -11 ]

Show that:

A² + 8A − 9I = 0

Hence, find A⁻¹.
OR
If

[ 0 −tan(α/2) ]
[ tan(α/2) 0 ]

and I is the identity matrix of order 2, show that:

I + A = (I − A) × [ cos(α) −sin(α) ]
                     [ sin(α) cos(α) ]

SECTION–C (3 Marks Each)

27. Draw the graph of $y={\tan }^{-1}\left(x\right)$ in its principal value branch.
Write its domain and range.

28. Solve the differential equation:
$\left(\mathrm{(y-}-{\sin }^2\mathrm{x)}\right)\mathrm{dx}+\tan \left(x\right)\mathrm{dy}=0$
OR
Solve: $\left(1,+,e^{\frac{x}{y}}\right)\mathrm{dx}+e^{\frac{x}{y}}\left(1,-,\frac{x}{y}\right)\mathrm{dy}=0$

29. If $f\left(x\right)=\left\{\begin{array}{c}\frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}\text{if}-1\le x<0\\ \frac{2x+1}{x-1}\text{if}0\le x<1\end{array}\right.$ is continuous at x = 0, find the value of k.
OR
Find a and b such that $f\left(x\right)=\left\{\begin{array}{c}\frac{x-4}{|x-4|}+a\text{if}x<4\\ a+b\text{if}x=4\\ \frac{x-4}{|x-4|}+b\text{if}x>4\end{array}\right.$ is continuous at x = 4.

30. Evaluate the integral: $\int x^2·{\tan }^{-1}\left(x\right)\mathrm{dx}$

31. Differentiate $x^{{\sin }^{-1}}$ with respect to $\cos \left(x^2\right)$.

SECTION–D (5 Marks Each)

32. Evaluate:
$${\int }_0^{\frac{\pi }{4}}\sqrt{\tan \left(x\right)}+\sqrt{\cot \left(x\right)} \mathrm{dx}$$ OR
$${\int }_{-a}^a\frac{x^2}{1+e^x} \mathrm{dx}$$

33. Solve the differential equation:
$$\left({\tan }^{-1},x,-,y\right)\mathrm{dx}=\left(1,+,x^2\right)\mathrm{dy}$$

34. If X is a symmetric matrix and Y is a skew-symmetric matrix such that:
$$3X+2Y=\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right]$$ then find the product XY.

35. Find intervals in which the function $f\left(x\right)=\frac{3}{2}{x}^4-4x^3-45x^2+51$
is:
(a) strictly increasing
(b) strictly decreasing

OR
Find intervals in which the function $f\left(x\right)=\sin \left(3x\right)$, for $x\in \left[0,\frac{\pi }{2}\right]$
is:
(a) increasing
(b) decreasing

SECTION–E: Case Study Based Questions (4 Marks Each)

36. A shopkeeper has 3 varieties of pens: A, B, and C. Lakshay purchased 1 of each variety for a total of ₹21.
Aman purchased 4 pens of variety A, 3 pens of variety B and 2 pens of variety C for ₹60.
Sunil purchased 6 pens of variety A, 2 pens of variety B and 3 pens of variety C for ₹70.

Based on the above information, answer the following:

1. Represent the above situation as algebraic equations. [1 mark]
2. Using matrices, express the equations in the form AX = B. [1 mark]
3. Using matrix method, find:
   • Sum of the costs of pens of variety A and B. OR
   • Difference of the costs of pens of variety B and C. [2 marks]

37. Navita cuts her pizza into two pieces along a straight line. The circular pizza is represented by: $$x2+\; y2=\; 16$$ and the cutting line is: $y\; =\; x$

Based on this information:

1. Draw a suitable figure. Find the point of intersection of the circle and line in the first quadrant. [2 marks]
2. Using integration, find the area enclosed by the circle, line, and x-axis in the first quadrant. [2 marks]

38. Sahil wants to gift a handmade pen stand to his teacher. He took a square cardboard of side 20 cm.
He cut squares of side x cm from each corner and folded up the flaps to form an open box.

Application-Based Questions: Pen-Stand Problem

Answer the following:

(i) Express the volume V of the pen-stand in terms of x.
[1 mark]

(ii) Find the value of x, when the rate of change of volume with respect to height is zero.
[1 mark]

(iii) Find the volume of the largest such pen-stand.
[2 marks]

OR
Find the total surface area of the outer surface of the largest such pen-stand.
[2 marks]

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