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Class: XII Session: 2020-21

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.    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 SectionIV and 3 questions of Section-V. 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) = 𝑥3 is one-one 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 𝐴1,𝐴2, 𝐴3. What is the value of 𝐴1 ∪ 𝐴2 ∪ 𝐴3 and 𝐴1 ∩ 𝐴2 ∩ 𝐴3

 

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 𝐴2, 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 = [𝑎𝑖𝑗] be a square matrix of order 3×3 and |A|= -7. Find the value of 

𝑎11 𝐴21 + 𝑎12𝐴22 + 𝑎13 𝐴23

where 𝐴𝑖𝑗 is the cofactor of element 𝑎𝑖𝑗

 

1

7

    Find ∫ 𝑒𝑥(1 − cot 𝑥 + 𝑐𝑜𝑠𝑒𝑐2𝑥) 𝑑𝑥  

 

    OR

    Evaluate  

 

1

 

 

1

8

Find the area bounded by 𝑦 = 𝑥2,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 (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

 

 

 

(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 multi-national company is interested in maximizing the area 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

 

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

 

(

E2 and E3 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 E1, 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 𝐴2 = 2𝐴, then find the value of |A|.

 

 

OR

 

If , show that A2 − 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) = {1𝑥sin𝑥                 

             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 𝑦2 = 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 (̄ | 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 𝑥2 + 9 𝑦2 = 36 using integration

 

 

3

 

 

 

 

3

35

Find the general solution of the following differential equation:                            

𝑥 𝑑𝑦 − (𝑦 + 2𝑥2)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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At the helm of GMS Learning is Principal Balkishan Agrawal, a dedicated and experienced educationist. Under his able guidance, our school has flourished academically and has achieved remarkable milestones in various fields. Principal Agrawal’s visio…

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