Case Study Based Questions Class XII

Case study questions on Relations & Functions, Matrices, Determinants, Applications of Derivatives, Applications of Integral, Differential equations,

CASE STUDY-1 CHAPTER -1

RELATIONS & FUNCTIONS

A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever

Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:

R = {(𝑉1, 𝑉2) ∶ 𝑉1, 𝑉2 ∈ 𝐼 and both use their voting right in general election – 2019}

Q1) Two neighbors X and Y∈ I. X exercised his voting right while Y did not cast her vote in general election – 2019. Which of the following is true?

a) (X, Y) ∈ R

b) (Y, X) ∈ R

c) (X, X) ∉ R

d) (X, Y) ∉ R

Q2. Mr.’𝑋’ and his wife ‘𝑊’both exercised their voting right in general election -2019, Which of the following is true?

a) both (X, W) and (W, X) ∈ R

b) (X, W) ∈ R but (W, X) ∉ R

c) both (X, W) and (W, X) ∉ R

d) (W, X) ∈ R but (X, W) ∉ R

Q3) Three friends F1, F2 and F3 exercised their voting right in general election-2019, then which of the following is true?

a) (F1, F2 ) ∈R, (F2, F3) ∈ R and (F1,F3) ∈ R

b) (F1, F2 ) ∈ R, (F2,F3) ∈ R and (F1,F3) ∉ R

c) (F1, F2 ) ∈ R, (F2,F2) ∈R but (F3,F3) ∉ R

d) (F1, F2 ) ∉ R, (F2,F3) ∉ R and (F1,F3) ∉ R

Q4. The above defined relation R is ______

a) Symmetric and transitive but not reflexive

b) Universal relation

c) Equivalence relation

d) Reflexive but not symmetric and transitive

Q5. Mr. Shyam exercised his voting right in General Election – 2019, then Mr. Shyam is related to which of the following?

a) All those eligible voters who cast their votes

b) Family members of Mr. Shyam

c) All citizens of India

d) Eligible voters of India

Answer

Q No.

Option

Answer

1

d

(X, Y) ∉R

2

a

both (X, W) and (W, X) ∈ R

3

a

(F1, F2 ) ∈R, (F2, F3) ∈ R and (F1, F3) ∈ R

4

c

Equivalence relation

5

a

All those eligible voters who cast their votes

CASE STUDY-2 CHAPTER -1
RELATIONS & FUNCTIONS
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes

A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Q1. Let 𝑅 ∶ 𝐵 → 𝐵 be defined by R = {(𝑥, 𝑦): 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 } is
a) Reflexive and transitive but not symmetric
b) Reflexive and symmetric and not transitive
c) Not reflexive but symmetric and transitive
d) Equivalence
Q2. Raji wants to know the number of functions from A to B. How many number of functions are possible?
a) 6 ²
b) 2 ⁶
c) 6!
d) 2 ¹²
Q3. Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is
a) Symmetric
b) Reflexive
c) Transitive
d) None of these three
Q4. Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
a) 6²
b) 2 ⁶
c) 6!
d) 2 ¹²
Q5. Let 𝑅: 𝐵 → 𝐵 be defined by R={(1,1),(1,2), (2,2), (3,3), (4,4), (5,5),(6,6)}, then R is
a) Symmetric
b) Reflexive and Transitive
c) Transitive and symmetric
d) Equivalence
Answers

Q No.

Option

Answer

1

a

Reflexive and transitive but not symmetric

2

a

6²

3

d

None of these three

4

d

2¹²

5

b

Reflexive and Transitive

CASE STUDY-3 CHAPTER -1
RELATIONS & FUNCTIONS
Students of a school are taken to a railway museum to learn about railways heritage and its history.
An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by
R = {(l₁, l₂) : l₁ is parallel to l₂}
On the basis of the above information, answer the following questions :
(i) Find whether the relation R is symmetric or not.
(ii) Find whether the relation is transitive or not.
(iii) If one of the rail lines on the railway track is represented by the equation y = 3x + 2, then find the set of rail lines in R related to it.
Answer: (i) Yes R is symmetric   (ii) Yes R is transitive
              (iii) The set is {I : I is a line of type y = 3x + c, c ∈ R}
CASE STUDY-1 CHAPTER -3
MATRICES
A manufacture produces three stationery products Pencil, Eraser and Sharpener which he sells in two markets. Annual sales are indicated below

Market

Products in numbers

Pencil

Eraser

Sharpener

A

10000

2000

18000

B

6000

20000

8000

If the unit Sale price of Pencil, Eraser and Sharpener are Rs. 2.50, Rs. 1.50 and Rs. 1.00 respectively, and unit cost of the above three commodities are Rs. 2.00, Rs. 1.00 and Rs. 0.50 respectively, then,
Based on the above information answer the following:
1) Total revenue of market A
a) Rs. 64,000
b) Rs. 60,400
c) Rs. 46,000
d) Rs. 40600
2) Total revenue of market B
a) Rs. 35,000
b) Rs. 53,000
c) Rs. 50,300
d) Rs. 30,500
3) Cost incurred in market A
a) Rs. 13,000
b) Rs.30,100
c) Rs. 10,300
d) Rs. 31,000
4) Profit in market A and B respectively are
a) (Rs. 15,000, Rs. 17,000)
b) (Rs. 17,000, Rs. 15,000)
c) (Rs. 51,000, Rs. 71,000)
d) ( Rs. 10,000, Rs. 20,000)
5) Gross profit in both market
a) Rs.23,000
b) Rs. 20,300
c) Rs. 32,000
d) Rs. 30,200
Answers
Q No.

Option

Answer

1

c

Rs. 46,000

2

b

Rs. 53,000

3

d

Rs.31,000

4

a

Rs.15, 000, Rs.17, 000

5

c

Rs. 32,000

CASE STUDY-2 CHAPTER -3
MATRICES
CASE STUDY 2:
Amit, Biraj and Chirag were given the task of creating a square matrix of order 2. Below are the matrices created by them. A, B , C are the matrices created by Amit, Biraj and Chirag respectively.
A = $\left [ \begin{matrix}1 & 2 \\-1 & 3 \\\end{matrix} \right ]$ , B = $\left [ \begin{matrix}4 & 0 \\1 & 5 \\\end{matrix} \right ]$ , C = $\left [ \begin{matrix}2 & 0 \\1 & -2 \\\end{matrix} \right ]$
If a = 4 and b = −2, based on the above information answer the following:

1. Sum of the matrices A, B and C , A+(𝐵 + 𝐶) is
a) $\left [ \begin{matrix}1 & 6 \\2 & 7 \\\end{matrix} \right ]$
b) $\left [ \begin{matrix}6 & 1 \\7 & 2 \\\end{matrix} \right ]$
c) $\left [ \begin{matrix}7 & 2 \\1 & 6 \\\end{matrix} \right ]$
d) $\left [ \begin{matrix}2 & 1 \\7 & 6 \\\end{matrix} \right ]$
2. (𝐴 ^𝑇 ) ^𝑇 is equal to
a) $\left [ \begin{matrix}1 & 2 \\-1 & 3 \\\end{matrix} \right ]$
b) $\left [ \begin{matrix}2 & 1 \\3 & -1 \\\end{matrix} \right ]$
c) $\left [ \begin{matrix}1 & -1 \\2 & 3 \\\end{matrix} \right ]$
d) $\left [ \begin{matrix}2 & 3 \\-1 & 1 \\\end{matrix} \right ]$
3. (𝑏𝐴) ^𝑇 is equal to
a) $\left [ \begin{matrix}-2 & -4 \\2 & -6 \\\end{matrix} \right ]$
b) $\left [ \begin{matrix}-2 & 2 \\-4 & -6 \\\end{matrix} \right ]$
c) $\left [ \begin{matrix}-2 & 2 \\-6 & -4 \\\end{matrix} \right ]$
d) $\left [ \begin{matrix}-6 & -2 \\2 & 4 \\\end{matrix} \right ]$
4. AC−𝐵𝐶 is equal to
a) $\left [ \begin{matrix}-4 & -6 \\-4 & 4 \\\end{matrix} \right ]$
b) $\left [ \begin{matrix}-4 & -4 \\4 & -6 \\\end{matrix} \right ]$
c) $\left [ \begin{matrix}-4 & -4 \\-6 & 4 \\\end{matrix} \right ]$
d) $\left [ \begin{matrix}-6 & 4 \\-4 & -4 \\\end{matrix} \right ]$
5. (𝑎 + 𝑏)𝐵 is equal to
a) $\left [ \begin{matrix}0 & 8 \\10 & 2 \\\end{matrix} \right ]$
b) $\left [ \begin{matrix}2 & 10 \\8 & 0 \\\end{matrix} \right ]$
c) $\left [ \begin{matrix}8 & 0 \\2 & 10 \\\end{matrix} \right ]$
d) $\left [ \begin{matrix}2 & 0 \\8 & 10 \\\end{matrix} \right ]$
Answers

Q No.

Option

Answer

1

c

$\left [ \begin{matrix}7 & 2 \\1 & 6 \\\end{matrix} \right ]$

2

a

$\left [ \begin{matrix}1 & 2 \\-1 & 3 \\\end{matrix} \right ]$

3

b

$\left [ \begin{matrix}-2 & 2 \\-4 & -6 \\\end{matrix} \right ]$

4

c

$\left [ \begin{matrix}-4 & -4 \\-6 & 4 \\\end{matrix} \right ]$

5

c

$\left [ \begin{matrix}8 & 0 \\2 & 10 \\\end{matrix} \right ]$

CASE STUDY-1 CHAPTER -4
DETERMINANTS
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2023 – 24, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports. In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000.

Based on the above information, answer the following questions :
i) Express the given information algebraically using matrices.
ii) Check whether the system of matrix equations so obtained is consistent or not.
iii) (a) Find the number of scholarships of each kind given by the school, using matrices.
OR
iii) (b) Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school ?

Answer (i)
Let No. of girl child scholarships = x
No. of meritorious achievers = y
x + y = 50
3000x + 4000y = 180000 ⇒ 3x + 4y = 180
$\mathbf{\left[\begin{matrix}1&1\\3&4\\\end{matrix}\right]\left[\begin{matrix}x\\y\end{matrix}\right]=\left[\begin{matrix}50\\180\end{matrix}\right]}$
Answer (ii)
$\mathbf{\left|\begin{matrix}1&1\\3&4\\\end{matrix}\right|=1\neq 0}$
Therefore the system of equations are consistent
Answer (iii) a
X  = A^-1B
$\mathbf{\left[\begin{matrix}x\\y\end{matrix}\right]=\left[\begin{matrix}4&-1\\-3&1\\\end{matrix}\right]\left[\begin{matrix}50\\180\end{matrix}\right]=\left[\begin{matrix}20\\30\end{matrix}\right]}$
⇒ x = 20, y = 30
Answer (iii) b
Required expenditure = ₹ [30(3000) + 20(4000)] = ₹ 1,70,000
Case study based questions

Chapter - 6 Class XII

Application of Derivatives

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 ^oF at x days.

(i) Is the function differentiable in the interval (0, 12) ? Justify your answer.

(ii) If 6 is the critical point of the function, then find the value of the constant m

(iii) Find the intervals in which the function is strictly increasing/strictly decreasing.

Answer

(i) Yes it is differentiable in (0, 12) because f(x) is a polynomial function which is differentiable at everywhere.

(ii) m = 1.2

(iii) f is strictly increasing in [0, 6]

f is strictly decreasing in [6, 12]

In the PDF given below there are six case study based questions with detailed explanation of the solutions.

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Case study 1 Chapter -13 Class XII
Probability

Case Study 1

The reliability of a COVID PCR test is specified as follows:

Of people having COVID, 90% of the test detects the disease but 10% goes undetected.

Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/her as COVID positive.

Based on the above information, answer the following

1) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually having COVID?

a) 0.001

b) 0.1

c) 0.8

d) 0.9

2) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually not having COVID’?

a) 0.01

b) 0.99

c) 0.1

d) 0.001

3) What is the probability that the ‘person is actually not having COVID?

a) 0.998

b) 0.999

c) 0.001

d) 0.111

4) What is the probability that the ‘person is actually having COVID given that ‘he is tested as COVID positive’?

a) 0.83

b) 0.0803

c) 0.083

d) 0.089

5) What is the probability that the ‘person selected will be diagnosed as COVID positive’?

a) 0.1089

b) 0.01089

c) 0.0189

d) 0.189

Q No.	Option	Answer
1	d	0.9
2	a	0.01
3	b	0.999
4	c	0.083
5	b	0.01089

Case study 2 Chapter -13 Class XII
Probability

Read the following passage and answer the questions given below:

In an Office three employees James, Sophia and Oliver process incoming copies of a certain form. James processes 50% of the forms, Sophia processes 20% and Oliver the remaining 30% of the forms. James has an error rate of 0.06 , Sophia has an error rate of 0.04 and Oliver has an error rate of 0.03 .

Based on the above information, answer the following questions.

(i) Find the probability that Sophia processed the form and committed an error.

(ii) Find the total probability of committing an error in processing the form.

(iii) 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 form. If the form selected at random has an error, find the probability that the form is not processed by James.

(iii) Let E be the event of committing an error in processing the form and let E₁, E₂ and E₃

be the events that James, Sophia and Oliver processed the form. Find the value of

$\sum_{i=1}^{3}P\left(E_{i}/E\right)$

Solution

(i)

(ii)

(iii)

(iii) (OR)

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