Class: XII Session: 202021
Subject: Mathematics
Sample Question Paper (Theory)
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. PartA 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 casebased 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
SectionIV and 3 questions of SectionV. You have to attempt only one of the
alternatives in all such questions.
Sr. No. 
Part – A 
Mark
s 


Section I All questions are compulsory. In case of internal choices attempt any
one. 


1 
Check whether the function 𝑓:
𝑅 → 𝑅 defined as 𝑓(𝑥)
= 𝑥^{3} is oneone or not. OR 
1 


How many reflexive relations are possible in a set A
whose 𝑛(𝐴) = 3.

1 

2 
A
relation R in 𝑆 = {1,2,3}
is defined as 𝑅 = {(1,1), (1,
2), (2, 2), (3, 3)}. Which element(s) of relation R be removed to make
R an equivalence relation? 
1 

3 
A relation R in the set of real numbers R defined as 𝑅 = {(𝑎, 𝑏): √𝑎 = 𝑏}
is a function or not. Justify OR An equivalence relation R in A divides it into equivalence
classes 𝐴_{1},𝐴_{2},
𝐴_{3}. What is the value of 𝐴_{1} ∪ 𝐴_{2} ∪ 𝐴_{3}
and 𝐴_{1} ∩ 𝐴_{2}
∩ 𝐴_{3} 
1 1 

4 
If A and B are matrices of order 3 × 𝑛 and 𝑚
× 5 respectively, then find the order of matrix 5A – 3B, given that it
is defined. 
1 

5 
Find the value of 𝐴^{2}, where A is
a 2×2 matrix whose elements are given by
OR Given that A is a square matrix of order 3×3 and A = 
4. Find adj A 
1 1 

6 
Let A = [𝑎_{𝑖𝑗}]
be a square matrix of order 3×3 and A= 7. Find the value of 𝑎11 𝐴21
+ 𝑎12𝐴22
+ 𝑎13 𝐴23
where 𝐴_{𝑖𝑗}
is the cofactor of element 𝑎_{𝑖𝑗}

1 

7 
Find ∫
𝑒^{𝑥}(1 − cot 𝑥 + 𝑐𝑜𝑠𝑒𝑐^{2}𝑥)
𝑑𝑥
OR Evaluate 
1 1 

8 
Find the area bounded by 𝑦 = 𝑥^{2},𝑡ℎ𝑒
𝑥 − axis and the lines 𝑥
= −1 and 𝑥 = 1. 
1 

9 
How many arbitrary
constants are there in the particular solution of the differential equation ; y (0) = 1 OR For what value of n is the following a homogeneous
differential equation: 
1 1 

10 
Find a unit vector
in the direction opposite to 
1 

11 
Find the area of the
triangle whose two sides are represented by the vectors 2𝑖
̂ 𝑎𝑛𝑑 − 3𝑗̂. 
1 

12 
Find the angle between
the unit vectors 𝑎̂ 𝑎𝑛𝑑 𝑏̂,
given that  𝑎̂ + 𝑏̂
= 1 
1 

13 
Find the direction cosines of the normal to YZ plane? 
1 

14 
Find the coordinates of the point where the line cuts the XY plane. 
1 

15 
The probabilities of A and B solving a problem
independently are respectively. If both of them try to solve
the problem independently, what is the probability that the problem is
solved? 
1 

16 
The probability that it will rain on any particular day is
50%. Find the probability that it rains only on first 4 days of the
week. 
1 


Section II Both the Case
study based questions are compulsory. Attempt any 4 sub parts from each
question (1721) and (2226). Each question carries 1 mark 


17 
An architect designs a building for a multinational
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 



(ii)The area of the rectangular region A expressed as a
function of x is a) b) c) d) 
1 


(iii) The maximum value
of area A is a)
b)
c)
d)

1 


(iv) The CEO of the multinational company is
interested in maximizing the area of the whole floor including the
semicircular ends. For this to happen the valve of x should be a)
0 m b)
30 m c)
50 m d)
80 m 
1 


(v) The extra area
generated if the area of the whole floor is
maximized is : a) b)
c)
d)
No change Both areas are equal 
1 

18

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 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
Sonia processed the form is : a)
0.0210 b)
0.04 c)
0.47 d) 0.06

1 


(ii)The probability that Sonia processed the form and
committed an error is : a)
0.005 b)
0.006 c)
0.008 d)
0.68 
1 


(iii)The total
probability of committing an error in processing the form is a)
0 b)
0.047 c) 0.234

1 


d) 1 



(iv)The manager of the company wants to do a quality
check. During inspection 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

1 


( E_{2} and E_{3}
value of 
a) 0
b) 0.03
c) 0.06
d) 1

v)Let A be the
event of committing an error in processing the form and let E_{1}, be
the events that Vinay, Sonia and Iqbal processed the form. The is

1 



Part – B 



Section III 

19 
Express 

in the simplest form. 
2 
20 
If A is a square matrix of order 3 such that 𝐴^{2 }= 2𝐴,
then find the value of A. OR If , show that A^{2 }− 5A + 7I = O. Hence find A^{−1}.

2 2 

21 
Find the value(s) of k so that the following function is
continuous at 𝑥 = 0 
2 


1−cos𝑘𝑥 𝑖𝑓 𝑥 ≠ 0 𝑓(𝑥)
= ^{{}_{1}𝑥sin𝑥 𝑖𝑓 𝑥 = 0 2 


22 
Find
the equation of the normal to the curve
y = perpendicular
to the line 3𝑥 − 4𝑦 = 7. 
2 

23 
Find
OR Evaluate 
2 2 

24 
Find the area of the
region bounded by the parabola 𝑦^{2
}= 8𝑥 and the line 𝑥
= . 
2 

25 
Solve the following
differential equation:
. 
2 

26 
Find the area of the
parallelogram whose one side and a diagonal are represented by coinitial
vectors 𝑖 ̂ 
+ 𝑘̂ and 4𝑖 ̂ + 5𝑘̂
respectively 
2 

27 
Find the vector equation
of the plane that passes through the point (1,0,0) and contains the line 𝑟⃗ = λ 𝑗.̂ 
2 

28 
A refrigerator box contains
2 milk chocolates and 4 dark chocolates. Two chocolates are drawn at random.
Find the probability distribution of the number of milk chocolates. What is
the most likely outcome?
OR Given that E and F are
events such that P(E) = 0.8, P(F) = 0.7, P (EF) = 0.6. Find P (Ē̄  F̄) 
2 2 


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 = {(𝑎, 𝑏) ∶ 𝑎 + 𝑏
is "divisible by 2"} is reflexive, symmetric or transitive.
Write the equivalence class containing 0 i.e. [0].

3 

30 
If
y = 𝑒
𝑥
𝑠𝑖𝑛2 𝑥 + (sin𝑥)𝑥,
find . 
3 

31 
Prove
that the greatest integer function defined by 𝑓(𝑥)
= [𝑥], 0 < 𝑥 < 2 is not differentiable at 𝑥 = 1 
3 


OR If 
3 

32 
Find the intervals in which the function given by
is a) strictly increasing b) strictly
decreasing 
3 

33 
Find . 
3 

34 
Find the area of the region bounded by the curves
OR Find the area of the ellipse 𝑥^{2 }+ 9 𝑦^{2 }= 36
using integration 
3 3 

35 
Find the general solution
of the following differential equation: 𝑥
𝑑𝑦 − (𝑦 + 2𝑥^{2})𝑑𝑥 =
0 
3 


Section V All questions are compulsory. In case of internal choices attempt any
one. 


36 
If , find 𝐴^{−1}. Hence Solve the system of equations; 𝑥 − 2𝑦 = 10 2𝑥 − 𝑦 − 𝑧 = −2𝑦 + 𝑧 = 7
OR Evaluate the
product AB, where 1 −1 0 2 2 −4 𝐴 = [2 3 4] 𝑎𝑛𝑑 𝐵
= [−4 2 −4] 0 1 2 2 −1 5 𝐻𝑒𝑛𝑐𝑒 𝑠𝑜𝑙𝑣𝑒
𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓
𝑙𝑖𝑛𝑒𝑎𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠
𝑥 − 𝑦 = 3 
5 5 


2𝑥 + 3𝑦 + 4𝑧 = 17 𝑦 + 2𝑧 = 7 


37 
Find the shortest
distance between the lines 𝑟⃗
= 3𝑖̂ + 2𝑗̂ − 4𝑘̂ + 𝜆(𝑖̂ + 2𝑗̂
+ 2𝑘̂) 𝑎𝑛𝑑 𝑟⃗ = 5𝑖̂ − 2𝑗̂
+ 𝜇 (3𝑖̂ + 2𝑗̂ + 6𝑘̂) If the lines intersect find their point of
intersection OR Find
the foot of the perpendicular drawn from the point (1, 3, 6) to the plane 2𝑥 + 𝑦 − 2𝑧 + 5
= 0. Also find the equation and length of the perpendicular. 
5 5 

38 
Solve the following linear
programming problem (L.P.P) graphically. Maximize 𝑍 = 𝑥 + 2𝑦 subject to constraints ; 𝑥 + 2𝑦 ≥ 100 2𝑥 − 𝑦 ≤ 0 2𝑥 + 𝑦 ≤ 200 𝑥, 𝑦 ≥ 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 𝑍 = 3𝑥 − 4𝑦
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. (ii) Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where 𝑝, 𝑞 > 𝑜 be the objective function. Find the condition on and so that the maximum value of occurs at B(4,10)𝑎𝑛𝑑 C(6,8). Also mention the number of optimal solutions in this case.

5 5 
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