Class: XII Session: 2020-21
Subject: Mathematics
Time Allowed: 3 Hours Maximum Marks: 80
General Instructions:
1. This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks
2. Part-A has Objective Type Questions and Part -B has Descriptive Type Questions
3. Both Part A and Part B have choices.
Part – A:
1. It consists of two sections- I and II.
2. Section I comprises of 16 very short answer type questions.
3. Section II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs.
Part – B:
1. It consists of three sections- III, IV and V.
2. Section III comprises of 10 questions of 2 marks each.
3. Section IV comprises of 7 questions of 3 marks each.
4. Section V comprises of 3 questions of 5 marks each.
5. Internal choice is provided in 3 questions of Section –III, 2 questions of Section IV and 3 questions of Section-V. You have to attempt only one of the alternatives in all such questions.
Sr.
Part – A Mark
No.
s
Section I
All questions are compulsory. In case of internal choices attempt any one.
1 Check whether the function ��: �� → �� defined as ��(��) = ��3is one-one or not. 1
OR
Page 1 of 10
How many reflexive relations are possible in a set A whose ��(��) = 3. 1
2 A relation R in �� = {1,2,3} is defined as �� = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which
1
element(s) of relation R be removed to make R an equivalence relation? 3 A relation R in the set of real numbers R defined as �� = {(��, ��): √�� = ��} is a
1
function or not. Justify
OR
1
An equivalence relation R in A divides it into equivalence classes ��1,��2, ��3. What is the value of ��1 ∪ ��2 ∪ ��3 and ��1 ∩ ��2 ∩ ��3
4 If A and B are matrices of order 3 × �� and �� × 5 respectively, then find the
1
order of matrix 5A – 3B, given that it is defined.
5 Find the value of ��2, where A is a 2×2 matrix whose elements are given by
1
������ = {10���������� ≠ ��
�� = ��
OR
1
Given that A is a square matrix of order 3×3 and |A| = - 4. Find |adj A| 6 Let A = [������] be a square matrix of order 3×3 and |A|= -7. Find the value of
1
��11 ��21 + ��12��22 + ��13 ��23
where ������ is the cofactor of element ������
7 Find ∫ ����(1 − cot �� + ����������2��) ����
1
OR
��
1
Evaluate ∫ ��2sin �� ����
2
−��2
8 Find the area bounded by �� = ��2,��ℎ�� �� − axis and the lines
1
�� = −1 and �� = 1.
9 How many arbitrary constants are there in the particular solution of the
1
differential equation ����
����= −4����2; y (0) = 1
OR
1
For what value of n is the following a homogeneous differential equation: ���� =��3 − ����
����
��2�� + ����2
10 Find a unit vector in the direction opposite to −34��̂ 1
11 Find the area of the triangle whose two sides are represented by the vectors
1
2��̂������ − 3��̂.
Page 2 of 10
12 Find the angle between the unit vectors ��̂ ������ ��̂, given that | ��̂ + ��̂| = 1 1 13 Find the direction cosines of the normal to YZ plane? 1 14 Find the coordinates of the point where the line ��+3
3=��−1
plane.
−1=��−5
1
−5cuts the XY
15 The probabilities of A and B solving a problem independently are 13������ 14
1
respectively. If both of them try to solve the problem independently, what is the probability that the problem is solved?
16 The probability that it will rain on any particular day is 50%. Find the probability
1
that it rains only on first 4 days of the week.
Section II
Both the Case study based questions are compulsory. Attempt any 4 sub parts from each question (17-21) and (22-26). Each question carries 1 mark
17 An architect designs a building for a multi-national company. The floor consists of a rectangular region with semicircular ends having a perimeter of 200m as shown below:
Design of Floor
Building
Based on the above information answer the following:
(i) If x and y represents the length and breadth of the rectangular region, then the relation between the variables is
a) x + π y = 100
b) 2x + π y = 200
c) π x + y = 50
d) x + y = 100
Page 3 of 10
(ii)The area of the rectangular region A expressed as a function of x is 1
a)2��(100 �� − ��2)
b)1��(100 �� − ��2)
c)����(100 − ��)
d) ����2+ 2��(100 �� − ��2)
(iii) The maximum value of area A is
1
a)��
3200��2
b)3200
����2
c)5000
����2
d)1000
����2
(iv) The CEO of the multi-national company is interested in maximizing the area
1
of the whole floor including the semi-circular ends. For this to happen the valve of x should be
a) 0 m
b) 30 m
c) 50 m
d) 80 m
(v) The extra area generated if the area of the whole floor is maximized is : 1
a)3000
����2
b)5000
����2
c)7000
����2
d) No change Both areas are equal
Page 4 of 10
In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal 18
the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03
Based on the above information answer the following:
(i) The conditional probability that an error is committed in processing given that
1
Sonia processed the form is :
a) 0.0210
b) 0.04
c) 0.47
d) 0.06
(ii)The probability that Sonia processed the form and committed an error is : 1
a) 0.005
b) 0.006
c) 0.008
d) 0.68
(iii)The total probability of committing an error in processing the form is 1
a) 0
b) 0.047
c) 0.234
Page 5 of 10
d) 1
(iv)The manager of the company wants to do a quality check. During inspection
1
he selects a form at random from the days output of processed forms. If the form selected at random has an error, the probability that the form is NOT processed by Vinay is :
a) 1
b) 30/47
c) 20/47
d) 17/47
(v)Let A be the event of committing an error in processing the form and let E1,
1
E2 and E3 be the events that Vinay, Sonia and Iqbal processed the form. The 3��=1is
value of ∑ �� (����| A)
a) 0
b) 0.03
c) 0.06
d) 1
Part – B
Section III
19
2< �� <��2in the simplest form.2
1−��������), −3��
Express ������−1(��������
20 If A is a square matrix of order 3 such that ��2 = 2��, then find the value of |A|. 2
OR
If �� = [3 1
2
−1 2], show that A2 − 5A + 7I = O.
Hence find A−1.
21 Find the value(s) of k so that the following function is continuous at �� = 0 2
Page 6 of 10
��(��) = {
1−cos ����
�� sin ������ �� ≠ 0 1
2���� �� = 0
22 Find the equation of the normal to the curve
2
y = �� + 1��, �� > 0 perpendicular to the line 3�� − 4�� = 7.
23 Find ∫1
2
������2��(1−��������)2����
OR
Evaluate ∫ ��(1 − ��)�� ���� 10
2
24 Find the area of the region bounded by the parabola ��2 = 8�� and the line �� =
2
2.
25 Solve the following differential equation:
2
����
����= ��3���������� ��, ���������� ��ℎ���� ��(0) = 0.
26 Find the area of the parallelogram whose one side and a diagonal are
2
represented by coinitial vectors ��̂- ��̂+ ��̂ and 4��̂+ 5��̂ respectively
27 Find the vector equation of the plane that passes through the point (1,0,0) and
2
contains the line ��⃗ = λ ��̂.
28 A refrigerator box contains 2 milk chocolates and 4 dark chocolates. Two
2
chocolates are drawn at random. Find the probability distribution of the number of milk chocolates. What is the most likely outcome?
OR
2
Given that E and F are events such that P(E) = 0.8, P(F) = 0.7, P (E∩F) = 0.6. Find P (̄Ē | F̄)
Section IV
All questions are compulsory. In case of internal choices attempt any one.
29 Check whether the relation R in the set Z of integers defined as R =
3
{(��, ��) ∶ �� + �� is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].
30If y = ���� ������2 ��+ (sin ��)��, find ����
����.3
31 Prove that the greatest integer function defined by ��(��) = [��], 0 < �� < 2 is not
3
differentiable at �� = 1
Page 7 of 10
OR
If �� = �� ������ �� , �� = �� �������� �������� ��2��
����2���� �� =��63
32 Find the intervals in which the function �� given by
3
��(��) = tan �� − 4��, �� ∈ (0,��2) is
a) strictly increasing b) strictly decreasing
Find ∫��2+ 1
33
(��2+2)(��2+3)����.3
34 Find the area of the region bounded by the curves
3
��2 + ��2 = 4, y = √3�� ������ �� − �������� ���� ��ℎ�� ���������� ����������������
OR
3
Find the area of the ellipse ��2 + 9 ��2 = 36 using integration
35 Find the general solution of the following differential equation:
3
�� ���� − (�� + 2��2)���� = 0
Section V
All questions are compulsory. In case of internal choices attempt any one.
36 If �� = [1 2 0
−2 −1 −2
0 −1 1
], find ��−1. Hence
5
Solve the system of equations;
�� − 2�� = 10
2�� − �� − �� = 8
−2�� + �� = 7
OR
5
Evaluate the product AB, where
�� = [
1 −1 0 2 3 4 0 1 2
] ������ �� = [
2 2 −4 −4 2 −4 2 −1 5
]
���������� ���������� ��ℎ�� ������������ ���� ������������ ������������������ �� − �� = 3
Page 8 of 10
2�� + 3�� + 4�� = 17
�� + 2�� = 7
37 Find the shortest distance between the lines
5
��⃗ = 3��̂+ 2��̂− 4��̂ + ��(��̂+ 2��̂+ 2��̂)
������ ��⃗ = 5��̂− 2��̂+ �� (3��̂+ 2��̂+ 6��̂)
If the lines intersect find their point of intersection
OR
5
Find the foot of the perpendicular drawn from the point (-1, 3, -6) to the plane 2�� + �� − 2�� + 5 = 0. Also find the equation and length of the
perpendicular.
38 Solve the following linear programming problem (L.P.P) graphically.
5
Maximize �� = �� + 2��
subject to constraints ;
�� + 2�� ≥ 100
2�� − �� ≤ 0
2�� + �� ≤ 200
��, �� ≥ 0
OR
The corner points of the feasible region determined by the system of linear constraints are as shown below:
Answer each of the following:
(i) Let �� = 3�� − 4�� be the objective function. Find the maximum and minimum value of Z and also the corresponding points at which the
maximum and minimum value occurs.
Page 9 of 10
(ii) Let �� = ���� + ����, where ��, �� > �� be the objective function. Find the condition on �� and �� so that the maximum value of �� occurs at B(4,10)������ C(6,8). Also mention the number of optimal solutions in this case.
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