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# CBSE Class 12 Maths Term 1 2021 Answer Key

The Central Board of Secondary Education (CBSE) conducted Term-1 Class 12 Board examination 2021-22 of Mathematics on Monday, December 6, 2021. The paper was rated lengthy and difficult as stated by students.

Lets discuss the structure of CBSE Class 12 Maths Term 1 Question Paper 2021.

The question paper consists of 50 questions divided into sections A, B, and C in which the Section:

• A consists of 20 questions of 1 mark each. Students needs to attempt any 16 questions,
• B consists of 20 questions of 1 mark each. Students needs to attempt any 16 questions,
• C consists of 10 questions of 1 mark each. Only eight questions are needed to be attempt.

Section A evaluated the knowledge and understanding of the students, where a few questions required little calculations. Section B was challenging, requiring students to have a thorough comprehension of the ideas in order to answer the problems. Linear Programming questions take a long time to answer. Section C assessed the students’ ability to apply their knowledge. This portion appears to be challenging for an average student to pass.

Now, lets analyze the CBSE Class 12 Maths Term 1 Paper Series SSJ/1 – Set 4 for 2021 with the answer keys and their solutions of the Class 12 Maths Paper.

The chapter ‘Applications of Derivatives‘ got the most marks and questions, followed by ‘Matrices‘ and ‘Relations and Functions.’ The least number of questions were asked from ‘Determinants‘ and ‘Inverse Trigonometric Functions.’ There is only one case study question in ‘Applications of Derivatives.

Section A was easy, section B was very complex, and section C was relatively easy but time-consuming, also the Question 42 from section C had symbols not present in the NCERT textbooks. Therefore, it is remarked that certain questions required students to think creatively and that the case study problems required good thinking skills. Hence, the CBSE Term 1 Maths paper 2021 was moderately difficult and time-consuming, with certain questions requiring additional time to answer.

Here, the Solved Questions Paper (along with the Answers and the explanation in-depth for each question) of the Maths CBSE Class 12th Term 1 Exam is provided as:

### Term- I

Time allowed- 90 minutes

Maximum Marks- 40

General Instructions

1. This question paper contains 50 questions out of which 40 questions are to be attempted. All questions carry equal marks.
2. The question paper consists of three Sections – Section A, B, and C.
3. Section – A contains 20 questions. Attempt any 16 questions from Q. No. 01 to 20.
4. Section – B also contains 20 questions. Attempt any 16 questions from Q. No. 21 to 40.
5. Section – C contains two Case Studies containing 5 questions in each case. Attempt any 4 questions from Q. No. 41 to 45 and another 4 from Q. No. 46 to 50.
6. There is only one correct option for every Multiple Choice Question (MCQ). Marks will not be awarded for answering more than one option.
7. There is no negative marking.

### SECTION – A

Question Number 1 to 20 are of 1 mark each. Any 16 Questions from Question 1 to 20 are needed to be attempted.

Question 1: Differential of log [log (log x5)] w.r.t. x is Question 2: The number of all possible matrices of order 2 × 3 with each entry 1 or 2 is

(a) 16

(b) 6

(c) 64

(d) 24

Question 3: A function f : R → R is defined as f(x) = x³ + 1. Then the function has

(a) no minimum value

(b) no maximum value

(c) both maximum and minimum values

(d) neither maximum value nor minimum value

Question 4: If sin y = x cos (a + y), then dx/dy is

(a) cos a/cos² (a+y)

(b) – cos a/cos² (a+y)

(c) cos a/sin² y

(d) – cos a/sin² y

Question 5: The points on the curve x²/9 + y²/25 + 1, where the tangent is parallel to the x-axis are

(a) (±5, 0)

(b) (0, ±5)

(c) (0, ±3)

(d) (±3, 0)

Question 6: Three points P(2x, x + 3), Q(0, x) and R(x + 3, x + 6) are collinear, then x is equal to

(a) 0

(b) 2

(c) 3

(d) 1

Question 7: The principal value of is

(a) π/12

(b) π

(c) π/3

(d) π/6

Question 8: If (x²+ y²)² = xy, then dy/dx is Question 9: If a matrix A is both symmetric and skew-symmetric, then A is necessarily

(a) Diagonal matrix

(b) Zero square matrix

(c) Square matrix

(d) Identity matrix

Question 10: Let set X = {1, 2, 3} and a relation R is defined in X as : R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are

(a) {(1, 1), (2, 3), (1, 2)}

(b) {(3, 3), (3, 1), (1, 2)}

(c) {(1, 1), (3, 3), (3, 1), (2, 3)}

(d) {(1, 1), (3, 3), (3, 1), (1, 2)}

Question 11: A Linear Programming Problem is as follows:

Minimise                             z = 2x + y

subject to the constraints   x ≥3 , x ≤9, y ≥ 0

x – y ≥ 0, x + y≤14

The feasible region has

(a) 5 corner points including (0, 0) and (9, 5)

(b) 5 corner points including (7. 7) and (3, 3)

(c) 5 corner points including (14, 0) and (9, 0)

(d) 5 corner points including (3, 6) and (9, 5)

Question 12: The function is continuous at x = 0 for the value of k as

(a) 3

(b) 5

(c) 2

(d) 8

Question 13: If Cij denotes the cofactor of element pij of the matrix , then the value of C31 . C23 is

(a) 3

(b) 5

(c) 2

(d) 8

Question 14: The function of y = x2 e-x  is decreasing in the interval

(a) (0, 2)

(b) (2, ∞)

(c) (-∞, 0)

(d) (-∞, 0) ∪ (2, ∞)

Question 15: If R= {(x, y); x, y ∈ z, x² + y² ≤ 4} is a relation in set Z, then domain of R is

(a) {0, 1, 2}

(b) {-2, -1, 0, 1, 2}

(c) {0, -1, -2}

(d) {-1, 0, 1}

Question 16: The system of linear equations

5x + ky = 5,

3x + 3y = 5;

will be consistent if

(a) k ≠ -3

(b) k = -5

(c) k = 5

(d) k ≠ 5

Question 17: The equation of the tangent to the curve y (1 + x2) = 2 – x, where it crosses the x-axis is

(a) x – 5y = 2

(b) 5x – y = 2

(c) x + 5y = 2

(d) 5x + y = 2

Question 18: If are equal, then value of ab – cd is

(a) 4

(b) 16

(c) -4

(d) -16

Question 19: The principal value of is

(a) π/8

(b) 3π/8

(c) -π/8

(d) -3π/8

Question 20: For two matrices and P – Q is ### SECTION B

In this Section attempt any 16 questions out of the Questions 21-40. Each question is of one mark.

Question 21: The function f(x) = 2x3 – 15x2 + 36 x + 6 is increasing in the interval

(a) (∞, 2) ∪ (3, ∞)

(b) (-∞, 2)

(c) (-∞, 2) ∪ (3, ∞)

(d) (3, ∞)

Question 22: If x = 2 cosθ – cos 2θ and y = 2 sinθ – sin 2θ, then dy/dx is Question 23: What is the domain of the function cos-1 (2x – 3)?

(a) [-1, 1]

(b) (1, 2)

(c) (-1, 1)

(d) [1, 2]

Question 24: A matrix A = [aij]3 × 3 is defined by The number of elements in A which are more than 5, is

(a) 3

(b) 4

(c) 5

(d) 6

Question 25: If a function f defined by. is continuous at x = π/2. then the value of k is

(a) 2

(b)3

(c) 6

(d) – 6

Question 26: For the matrix is

(a) 2I

(b) 3I

(c) I

(d) 5I

Question 27: Let X = {x2 : x ∈ N} and the function f : N → X is defined by f(x) = x2, x ∈ N. Then this function is

(a) injective only

(b) not bijective

(c) surjective only

(d) bijective

Question 28: The corner points of the feasible region for a Linear Programming problem are P(0, 5). Q(1, 5), R(4. 2) and S(12, 0). The minimum value of the objective function Z = 2x + 5y is at the point

(a) P

(b) Q

(c) R

(d) S

Question 29: The equation of the normal to the curve ay2 = x3 at the point (am2, am3) is

(a) 2y – 3mx + am3 = 0

(b) 2x + 3my – 3am4 – am2 = 0

(c) 2x + 3my + 3am4 – 2am2 = 0

(d) 2x + 3my – 3am4 – 2am2 = 0

Question 30: If A is a square matrix of order 3 and | A | = -5, then | adj A | is

(a) 125

(b) – 25

(c) 25

(d) ± 25

Question 31: The simplest form of is Question 32: If for the matrix , | A3 | =125, then the value of α is

(a) ±3

(b) -3

(c) ±1

(d) 1

Question 33: If y = sin (m sin-1 x), then which one of the following equations is true? Question 34: The principal value of [tan-1 √3 – cot-1 (-√3)] is

(a) π

(b) -π/2

(c) 0

(d) 2√3

Question 35: The maximum value of is Question 36: Let matrix X = [xij] is given by , Then the matrix Y = [mij] = Minor of xij is Question 37: A function f : R ⇢ R defined by f(x) = 2 + x2 is

(a) not one-one

(b) one-one

(c) not onto

(d) neither one-one nor onto

Question 38: A Linear Programming Problem is as follows:

Maximise/Minimise objective function Z = 2x – y + 5

Subject to the constraints

3x + 4y ≤ 60

x + 3y ≤ 30

x ≥ 0, y ≥ 0

If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and O(0, 0), then which of the following is true?

(a) Maximum value of Z is 40

(b) Minimum value of Z is – 5

(c) Difference of maximum and minimum values of Z is 35

(d) At two corner points, the value of Z are equal

Question 39: If x = -4 is a root of , then the sum if te other two roots is

(a) 4

(b) -3

(c) 2

(d) 5

Question 40: The absolute maximum value of the function f(x) = 4x – 1/2 x2 in the interval [-2, 9/2] is

(a) 8

(b) 9

(c) 6

(d) 10

### SECTION – C

Attempt any 8 questions out of Questions 41-50. Each question is of one mark.

Question 41: In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of the curved surface of the cone is

(a) 2πr2rh (2rh + h2)

(b) π2hr (2rh + h2

(c) 2π2r (2rh2 – h3)

(d) 2π2r2 (2rh – h2)

Question 42: The corner points of the feasible region determined by a set of constraints (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and the objective function is Z = ax + 2by where a, b > 0. The condition on a and b such that the maximum Z occurs at Q and S is

(a) a – 5b = 0

(b) a – 3b = 0

(c) a – 2b = 0

(d) a – 8b = 0

Question 43: If curve y2 = 4x and xy = c cut at right angles, then the value of c is

(a) 4√2

(b) 8

(c) 2√2

(d) -4√2

Question 44: the inverse of the matrix is Question 45: For an L.P.P. the objective function is Z = 4x + 3y, and the feasible region determined by a set of constraints (linear inequations) is shown in the graph. Which one of the following statements is true?

(a) Maximum value of Z is at R.

(b) Maximum value of Z is at Q.

(c) Value of Z at R is less than the value at P.

(d) Value of Z at Q is less than the value at R.

Case Study

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this, they identified a square area in the local park. Local authorities charged an amount of ₹ 50 per square metre for space so that there is no misuse of the space and the Resident welfare association takes it seriously. The association hired a labourer for digging out 250 m3 and he charged ₹ 400 × (depth)2. Association will like to have a minimum cost.

Based on this information, answer any 4 of the following questions.

Question 46: Let side of square plot is x m and its depth is h metres, then cost c for the pit is Question 47: Value of h (in m) for which dc/dh = 0 is

(a) 1.5

(b) 2

(c) 2.5

(d) 3

Question 48: d2c/dh2 is given by Question 49: Value of c (in m) for minimum cost is Question 50: Total minimum cost of digging the pit (in ₹) is

(a) 4,100

(b) 7,500

(c) 7,850

(d) 3,220