Subject: Mathematics
Class XII
Time: 3 hours
Max. Marks: 80
General Instructions:
1. This Question paper contains – five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.
2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.
3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with sub parts.
Section – A
(Each question carries 1 mark)
Q1. If A =[aij] is a skew-symmetric matrix of order n, then
a) aij = 1/aji ∀ i, j b) aij ≠ 0 ∀ i, j
c) aij = 0, where i = j d) aij ≠ 0 where i = j
Q2. If A is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| =
(a) 9 (b) -9
(c) 3 (d) -3
Q3. The area of a triangle with vertices A, B, C is given by
Ans. Option (b)
Q4. The value of ‘k’ for which the function
(a) 0 (b) -1
(c) 1 (d) 2
Q5. If 𝑓 ᇱ(𝑥) = 𝑥 + 1/ 𝑥, then 𝑓(𝑥) is
a) 𝑥2 + log |𝑥| + C b) 𝑥2/2 + log | 𝑥| + C
c) 𝑥/2 + log |𝑥| + C d) 𝑥/2 – log |𝑥| + C
Q6. If m and n, respectively, are the order and the degree of the differential equation
(a) 1 (b) 2
(c) 3 (d) 4
Q7. The solution set of the inequality 3x + 5y < 4 is
(a) an open half-plane not containing the origin.
(b) an open half-plane containing the origin.
(c) the whole XY-plane not containing the line 3x + 5y = 4.
(d) a closed half plane containing the origin.
Q8. The scalar projection of the vector
a) 7/√14 b) 7/14
b) 6/13 d) 7/2
9. The value of
(a) log4 (b) log 3/2
(c) ½ log2 (d) log 9/4
Q10. If A, B are non-singular square matrices of the same order, then (𝐴𝐵-1) -1 =
(a) A-1B (b) A-1B-1
(c) BA-1 d) AB
11. The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at
(a) (0.6, 1.6) 𝑜𝑛𝑙𝑦 (b) (3, 0) only (c) (0.6, 1.6) and (3, 0) only
(d) at every point of the line-segment joining the points (0.6, 1.6) and (3, 0)
Q12. The possible value(s) of ‘x’ is/are if
a) 3 (b) √3 (c) -√3 (d) √3, -√3
Q13. If A is a square matrix of order 3 and |A| = 5, then |𝑎𝑑𝑗𝐴| =
(a) 5 (b) 25 (c) 125 (d) 1/8
Q14. Given two independent events A and B such that P(A) =0.3, P(B) = 0.6 and P(𝐴’∩ 𝐵’) is
(a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1
Q15. The general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 is
(a) xy = C (b) x = Cy2 (c) y = Cx (d) y = Cx2
Q16. If 𝑦 = 𝑠𝑖𝑛-1𝑥, then (1 – 𝑥2) 𝑦2 𝑖𝑠 equal to
(a) 𝑥𝑦1 (b) 𝑥𝑦 (c) 𝑥𝑦2 (d) 𝑥2
a) √2 (b) 2√6 (c) 24 (d) 2√2
Q18. P is a point on the line joining the points 𝐴(0,5, −2) and 𝐵(3, −1,2). If the x-coordinate of P is 6, then its z-coordinate is
(a) 10 (b) 6 (c) -6 (d) -10
ASSERTION-REASON BASED QUESTIONS
In the following questions, a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices.
(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.
Ans. (c) A is true but R is false.
Ans. (a) Both A and R are true and R is the correct explanation of A.
SECTION B
This section comprises of very short answer type-questions (VSA) of 2 marks each
Q21. Find the value of 𝑠𝑖𝑛-1[𝑠𝑖𝑛 (13 π/7)]
Ans.
OR
Ans. Let 𝑦 ∈ 𝑁(codomain). Then ∃ 2𝑦 ∈ 𝑁(domain) such that
𝑓(2𝑦) = 2y/2 = 𝑦. Hence, f is surjective.
1, 2 ∈ 𝑁(domain) such that 𝑓(1) = 1 = 𝑓(2)
Hence, f is not injective.
Q22. A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
Ans. Let AB represent the height of the street light from the ground. At any time t seconds, let the man represented as ED of height 1.6 m be at a distance of x m from AB and the length of his shadow EC be y m. Using similarity of triangles, we have 4/1.6 = (x + y)/y = 3y = 2x
Ans.
OR
Find the direction ratio and direction cosines of a line parallel to the line whose equations are
6𝑥 − 12 = 3𝑦 + 9 = 2𝑧 – 2
Ans.
Q24. If 𝑦√(1 – 𝑥2) + 𝑥(√1 – 𝑦2) = 1, 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 dy/dx = -(√1 – 𝑦2)/ √(1 – 𝑥2)
Ans.
Ans.
SECTION C
Ans.
Q27. Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the “odd person” pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made?
Ans.
OR
Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size.
Ans.
Ans.
OR
Ans.
Q29. Solve the differential equation: 𝑦𝑑𝑥 + (𝑥 – 𝑦2) 𝑑𝑦 = 0
Ans.
OR
Solve the differential equation: 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = √(𝑥2 + 𝑦2) 𝑑𝑥
Ans.
Q30. Solve the following Linear Programming Problem graphically:
Maximize Z = 400x + 300y subject to 𝑥 + 𝑦 ≤ 200, 𝑥 ≤ 40, 𝑥 ≥ 20, 𝑦 ≥ 0
Ans.
Maximum profit occurs at x= 40, y=160
and the maximum profit =₹ 64,000
Ans.
SECTION D
(This section comprises of long answer-type questions (LA) of 5 marks each)
Q32. Make a rough sketch of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥2, 0 ≤ 𝑦 ≤ 𝑥, 0 ≤ 𝑥 ≤ 2} and find the area of the region using integration.
Ans.
Q33. Define the relation R in the set 𝑁 × 𝑁 as follows: For (a, b), (c, d) ∈ 𝑁 × 𝑁, (a, b) R (c, d) iff ad = bc. Prove that R is an equivalence relation in 𝑁 × 𝑁.
Ans. Let (𝑎, 𝑏) ∈ 𝑁 × 𝑁. Then we have
ab = ba (by commutative property of multiplication of natural numbers)
⟹ (𝑎, 𝑏)𝑅(𝑎, 𝑏)
Hence, R is reflexive.
Let (𝑎, 𝑏), (𝑐, 𝑑) ∈ 𝑁 × 𝑁 such that (a, b) R (c, d). Then
ad = bc
⟹ 𝑐𝑏 = 𝑑𝑎 (by commutative property of multiplication of natural numbers
⟹ (𝑐, 𝑑)𝑅(𝑎, 𝑏)
Hence, R is symmetric.
Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) ∈ 𝑁 × 𝑁 such that
(a, b) R (c, d) and (c, d) R (e, f).
Then ad = bc, cf = de
⟹ 𝑎𝑑𝑐𝑓 = 𝑏𝑐𝑑𝑒
⟹ 𝑎𝑓 = 𝑏𝑒
⟹ (𝑎, 𝑏)𝑅(𝑒, 𝑓)
Hence, R is transitive.
Since, R is reflexive, symmetric and transitive, R is an equivalence relation on 𝑁 × 𝑁.
OR
Given a non-empty set X, define the relation R in P(X) as follows: For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not symmetric.
Ans. Let 𝐴 ∈ 𝑃(𝑋). Then 𝐴 ⊂ 𝐴
⟹ (𝐴, 𝐴) ∈ 𝑅
Hence, R is reflexive.
Let 𝐴, 𝐵, 𝐶 ∈ 𝑃(𝑋) such that
(𝐴, 𝐵), (𝐵, 𝐶) ∈ 𝑅
⟹ 𝐴 ⊂ 𝐵, 𝐵 ⊂ 𝐶
⟹ 𝐴 ⊂ 𝐶
⟹ (𝐴, 𝐶) ∈ 𝑅
Hence, R is transitive.
∅, 𝑋 ∈ 𝑃(𝑋) such that ∅ ⊂ 𝑋. Hence, (∅, 𝑋) ∈ 𝑅. But, 𝑋 ⊄ ∅, which implies that (𝑋, ∅) ∉ 𝑅.
Thus, R is not symmetric.
Ans. The given lines are non-parallel lines. There is a unique line segment PQ (P lying on one and Q on the other, which is at right angles to both the lines. PQ is the shortest distance between the lines. Hence, the shortest possible distance between the insects = PQ
The position vector of P lying on the line
OR
Ans.
Ans.
SECTION E
(This section comprises of 3 case study/passage based questions of 4 marks each with two sub-parts. First two case study questions have three sub-parts (i), (ii), (iii) of marks 1, 1, 2 respectively. The third case study question has two sub-parts of 2 marks each.)
Q36. Case-Study 1: Read the following passage and answer the questions given below
The temperature of a person during an intestinal illness is given by
𝑓(𝑥) = −0.1𝑥2 + 𝑚𝑥 + 98.6, 0 ≤ 𝑥 ≤12, m being a constant, where 𝑓(𝑥) is the temperature in °F at 𝑥 days.
(i) Is the function differentiable in the interval (0, 12)? Justify your answer.
Ans. f(𝑥) = −0.1𝑥2 + 𝑚𝑥 + 98.6, being a polynomial function, is differentiable everywhere, hence, differentiable in (0, 12)
(ii) If 6 is the critical point of the function, then find the value of the constant m.
Ans. 𝑓’(𝑥) = − 0.2𝑥 + 𝑚
Since, 6 is the critical point,
𝑓’(6) = 0 ⟹ 𝑚 = 1.2
(iii) Find the intervals in which the function is strictly increasing/strictly decreasing.
Ans.
OR
(iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute maximum/absolute minimum values of the function.
Ans. (iii) 𝑓(𝑥) = −0.1𝑥2 + 1.2𝑥 + 98.6,
𝑓’(𝑥) = − 0.2𝑥 + 1.2, 𝑓’(6) = 0,
𝑓’’(𝑥) = − 0.2
𝑓’’(6) = − 0.2 < 0
Hence, by second derivative test 6 is a point of local maximum. The local maximum value
= 𝑓(6) = − 0.1 × 62 + 1.2 × 6 + 98.6 = 102.2
We have 𝑓(0) = 98.6, 𝑓(6) = 102.2, 𝑓(12) = 98.6
6 is the point of absolute maximum and the absolute maximum value of the function = 102.2.
0 and 12 both are the points of absolute minimum and the absolute minimum value of the function = 98.6.
Q37. Case-Study 2: Read the following passage and answer the questions given below.
In an elliptical sport field, the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of
(i) If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
Ans.
(ii) Find the critical point of the function.
Ans.
(iii) Use First Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Ans.
OR
(iii) Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Ans.
Q38. Case-Study 3: Read the following passage and answer the questions given below.
There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time.
(i) What is the probability that the shell fired from exactly one of them hit the plane?
Ans. (i) Let P be the event that the shell fired from A hits the plane and Q be the event that the shell fired from B hits the plane. The following four hypotheses are possible before the trial, with the guns operating independently:
(ii) If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?
Ans.
