CBSE Class 12 Mathematics Case Studies

CBSE Class 12 Mathematics Case Study 1

1. DETERMINANTS: A determinant is a square array of numbers (written within a pair of vertical lines) which represents a certain sum of We can solve a system of equations using determinants, but it becomes very tedious for large systems. We will only do 2 × 2 and 3 × 3 systems using determinants. Using the properties of determinants solve the problem given below and answer the questions that follow:

Three shopkeepers Ram Lal, Shyam Lal, and Ghansham are using polythene bags, handmade bags (prepared by prisoners), and newspaper's envelope as carry bags. It is found that the shopkeepers Ram Lal, Shyam Lal, and Ghansham are using (20,30,40), (30,40,20), and (40,20,30) polythene bags, handmade bags, and newspapers envelopes respectively. The shopkeepers Ram Lal, Shyam Lal, and Ghansham spent ₹250, ₹270, and ₹200 on these carry bags respectively.

i. What is the cost of one polythene bag?

a. ₹ 1

b. ₹ 2

c. ₹ 3

d. ₹ 5

ii. What is the cost of one handmade bag?

a. ₹ 1

b. ₹ 2

c. ₹ 3

d. ₹ 5

iii. What is the cost of one newspaper bag?

1. ₹ 1

2. ₹ 2

3. ₹ 3

4. ₹ 5

iv. Keeping in mind the social conditions, which shopkeeper is better?

a. Ram Lal

b. Shyam Lal

c. Ghansham

d. None of these

v. Keeping in mind the environmental conditions, which shopkeeper is better?

a. Ram Lal

b. Shyam Lal

c. Ghansham

d. None of these

CBSE Class 12 Mathematics Case Study 2

2. A manufacturer produces three products, x, y, z which he sells in two mark Annual sales are indicated below:

If unit sales prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00 respectively, find the following:

i. the total revenue in the market-I with the help of matrix algebr

a. 44000

b. 48000

c. 46000

d. 53000

ii. the total revenue in the market-II with the help of matrix algebr

a. 51000

b. 53000

c. 46000

d. 49000

iii. If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit from both the markets

a. 53000

b. 46000

c. 34000

d. 32000

iv. If matrix A = [ aij ]2 x² , where aij = 1, if i ≠ j = 0 and if i = j, then A² is equal to

a. I

b. A 

c. 0

d. none of these

v. If A and B are matrices of same order then (AB'- BA') is a

a. skew-symmetric matrix

b. null matrix

c. symmetric matrix

d. unit matrix

CBSE Class 12 Mathematics Case Study 3

3. Matrices/Determinant: In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the The determinant of a matrix A is denoted det(A) or |A|. Using determinants/ Matrices calculate the following:

Ram purchases 3 pens, 2 bags, and 1 instrument box and pays ₹ 41. From the same shop, Dheeraj purchases 2 pens, 1 bag, and 2 instrument boxes and pays ₹29, while Ankur purchases 2 pens, 2 bags, and 2 instrument boxes and pays ₹44.

Read the above information and answer the following questions:

i. Find the cost of one

a. ₹ 2

b. ₹ 5

c. ₹ 10

d. ₹15

ii. What are the cost of one pen and one bag?

a. ₹ 12

b. ₹ 15

c. ₹ 17

d. ₹25

iii. What is the cost of one pen & one instrument box?

a. ₹ 7

b. ₹ 12

c. ₹ 17

d. ₹25

iv. What is the cost of one bag & one instrument box?

a. ₹ 20

b. ₹ 25

c. ₹ 10

d. ₹15

v. Find the cost of one pen, one bag, and one instrument

a. ₹ 22

b. ₹ 25

c. ₹ 20

d. ₹24

CBSE Class 12 Mathematics Case Study 4

4. Two farmers Ramkishan and Gurcharan Singh cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.

i. Find the combined sales in September and October for farmer Gurcharan Singh for each variety.

a. 80000

b. 90000

c. 130000

d. 135000

ii. Find the combined sales in September and October for farmer Ramkishan in each variety.

a. 90000

b. 80000

c. 86000

d. 81000

iii. Which variety of Rice has the lowest selling value in the month of September for the farmer Ramkisan?

a. Basmati

b. Peermal

c. Naura

d. All of these have the same price

iv. If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October.

v. Which variety of Rice has the highest selling value in the month of September for the farmer Gurcharan Singh?

a. Basmati

b. Peermal

c. Naura

d. All of these have the same price

CBSE Class 12 Mathematics Case Study 5

5. Two schools P and Q want to award their selected students on the values of Tolerance, Kindness, and The school P wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students respectively with total award money of Rs. 2200.
School Q wants to spend Rs 3100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each value is Rs1200, using matrices, find the following:

i. What is award money for Tolerance?

a. 350

b. 300

c. 500

d. 400

ii. What is the award money for Leadership?

a. 300

b. 280

c. 450

d. 500

iii. What is the award money for Kindness?

a. 500

b. 400

c. 300

d. 550

iv. If a matrix A is both symmetric and skew-symmetric, then

a. A is a diagonal matrix

b. A is a scalar matrix

c. A is a zero matrix

d. A is a square matrix

v. If A and B are two matrices such that AB = B and BA = A, then B2 is equal to

a. B

b. A

c. 1

d. 0

CBSE Class 12 Mathematics Case Study 6

6. A man has an expensive square shape piece of golden board of size 24 cm is to be made into a box without top by cutting from each corner and folding the flaps to form a box.

i. Volume of open box formed by folding up the flap:

a. 4(x³ - 24x² + 144x)

b. 4(x³- 34x² + 244x)

c. x³ - 24x² + 144x

d. 4x³ - 24x² + 144x

ii. In the first derivative test, if dy/dx changes its sign from positive to negative as x increases through c₁, then function attains a:

a. Local maxima at x = c₁

b. Local minima at x = c₁

c. Neither maxima nor minima at x = c₁

d. None of these

iii. What should be the side of the square piece to be cut from each corner of the board to be hold the maximum volume?

a. 14 cm

b. 12 cm

c. 4 cm

d. 5 cm

iv. What should be the maximum volume of open box?

a. 1034 cm³

b. 1024 cm³

c. 1204 cm³

d. 4021 cm³

v. The smallest value of the polynomial x³ – 18x² + 96x in [0, 9] is:

a. 126

b. 0

c. 135

d. 160

CBSE Class 12 Mathematics Case Study 7

7. As we know good planning can save energy, time, and money. A farmer wants to construct a circular well and square garden in his He wants to keep the sum of their perimeters 600 m.

i. If the radius of the circular garden be r m and the side of the square garden be x m then sum of area S is

a. S = πr²+ ( 600+2πr/4 )²

b. S = πr² + ( 300+πr)²

c. S = 2πr²( 600+2πr/4 )²

d. S = πr+ ( 600+πr/4)²

ii. Radius of circular well is

a. 600/π+4

b. r = 300/π+4

c. r = 300/π+4

d. r = 150/π+4

iii. For the given condition

a. d²s /dr² =0

b. d²s /dr² <0

c. d²s /dr² >0

d. None of these

iv. The relationship between the side of the square garden and the radius of the circular garden.

a. a = r

b. 2a = r²

c. a = r

d. a = 2r

v. Find the number which exceeds its square by the greatest possible

a. 1/2

b. 2

c. 1

d. 0

CBSE Class 12 Mathematics Case Study 8

8. A telephone company in a town has 500 subscribers on its list and collects fixed charges of 300 per subscriber per The company proposes to increases the annual subscription and it is believed that for every increase of 1 one subscriber will discontinue the service.

i. If x be the annual subscription then the total revenue of the company after increment will be:

a. R(x) = -x² + 200x + 150000

b. R(x) = x² - 200x - 140000

c. R(x) = 200x² + x + 150000

d. R(x) = -x² + 100 x + 100000

ii. To find maximum profit we put

a. R'(x) = 0

b. R'(x) > 0

c. R'(x) < 0

d. R''(x) = 0

iii. How much fee the company should increase to have maximum profit?

a. 150

b. Rs. 100

c. Rs. 200

d. Rs. 250

iv. Find the maximum profit that the company can make if the profit function is given by P(x)= 41 + 24x - 18x².

a. 25

b. 44

c. 45

d. 49

v. Find both the maximum and minimum value respectively of 3x⁴ - 8x³ + 48x + 1 on the interval [1, 4].

a. -63, 257

b. 258, -63

c. 257, -63

d. -63, -257

CBSE Class 12 Mathematics Case Study 9

9. A gardener wants to construct a rectangular bed of garden in a circular patch of He takes the maximum perimeter of the rectangular region as possible. (Refer to the images given below for calculations)

i. The perimeter of rectangle P is:

a. 4x + 4√(a² −x²)

b. x + √(a² −x²)

c. 4x + √(a² −x²)

d. x + 4√(a² −x²)

ii. To find critical points put

a. dp/dx > 0

b. dp/dx< 0

c. dp/dx= 0

d. none of these

iii. Value of y is

a. a/2

b. a/√2

c. 2a

d. √2a

iv. P is maximum when the rectangle is

a. Square

b. Parallelogram

c. Rectangle

d. Trapezium

v. If a rectangle of the maximum perimeter which can be inscribed in a circle of radius 10 cm is square then the sides of the region

a. 10√8 cm

b. 2√10cm

c. 20√2 cm

d. 10√2 cm

CBSE Class 12 Mathematics Case Study 10

10. An Apache helicopter of the enemy is flying along the curve given by y = x² + A soldier, placed at (3, 7) want to shoot down the helicopter when it is nearest to him

i. If (x1, y1) be the position of a helicopter on curve y = x² + 7, then distance D from soldier place at (3, 7) can be represented by

a. D² = x₁²- 6x₁ + 9 + x₁⁴

b. D² = x₁²- 6x₁ - 9 + x₁⁴

c. D² = 6x₁²- x₁ + 6 + x₁⁴

d. D² = 6x₁²- 6x₁ + 9 + x₁⁴

ii. Distance D² is minimum at

a. x₁ = 0

b. x₁ = 1

c. x₁ = 2

d. x₁ = -1

iii. The point on the curve y = x² +7 nearest to (3,7) is

a. (1, 8)

b. (8, 1)

c. (1, 0)

d. (0, 1)

iv. Condition for minima by the second derivative test is

a. dy/dx > 0

b. d²y/dx² = 0

c. d²y/dx² < 0

d. d²y/dx² > 0

v. Minimum distance between the helicopter and the soldier is

a. √2 units

b. √ 2 units

c. √5 units

d. 3 units

CBSE Class 12 Mathematics Case Study 11

11. Three friends Ganesh, Dinesh and Ramesh went for playing a Tug of war game. Team A, B, and C belong to Ganesh, Dinesh and Ramesh respectively.

Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below).

Team A pulls with F₁ = 4î + 0ĵKN

Team B F₂ = -2î + 4ĵKN

Team C F₃ = -3î - 3ĵ KN

i. Which team will win the game?

a. Team B

b. Team A

c. Team C

d. No one

ii. What is the magnitude of the teams combine Force?

a. 7 KN

b. 4 KN

c. 1.5 KN

d. 2 KN

iii. In with direction approx the ring getting pulls:

a. 0 radian

b. 2.5 radian

c. 2.4 radian

d. 3 radian

iv. What is the magnitude of the force of Team B?

a. 2√5 KN

b. 6 KN

c. 2 KN

d. √6 KN

v. How many KN Force is applied by Team A?

a. 5 KN

b. 4 KN

c. 2 KN

d. 16 KN

CBSE Class 12 Mathematics Case Study 12

12. Ram's house is situated at Gandhi Nagar at Point O, for going to school he first travels by bus in the east.

Here at Point A, a hospital is situated. From Hospital Ram takes an auto and goes 3 km in the north direction, here at point B school is situated. Suresh's house is at 30o east, 6 km from point B. (Refer image for information)

i. What is vector distance between Ram's house and school?

a. 4 km

b. 5 km

c. 7 km

d. 8 km

ii. How many km Ram travels to reach school?

a. 7 km

b. 5 km

c. 4 km

d. 8 km

iii. What is the vector distance from school to Suresh's home?

a. √3î + ĵ

b. 3√3î + 3ĵ

c. 6î

d. 6ĵ

iv. What is the displacement from Ram's house to Suresh house?

a. (4 + 3√3)î + 6ĵ

b. 4î + 6ĵ

c. 13î

d. 13ĵ

v. What is the total distance from Ram's house to Suresh's home?

a. 11 km

b. 13 km

c. 9 km

d. 5 km

CBSE Class 12 Mathematics Case Study 13

13. Once Ramesh was going to his native place at a village near Agra. From Delhi and Agra he went by flight, In the way, there was a river. Ramesh reached the river by taxi.

Then Ramesh used a boat for crossing the river. The boat heads directly across the river 40 m wide at 4 m/s. The current was flowing downstream at 3 m/s.

i. What is the resultant velocity of the boat?

a. 5 m/s

b. 4 m/s

c. 3 m/s

d. 6 m/s

ii. How much time does it take the boat to cross the river?

a. 5 sec

b. 8 sec

c. 9 sec

d. 10 sec

iii. How far downstream is the boat when it reaches the other side?

a. 24 m

b. 28 m

c. 18 m

d. 20 m

iv. What is the angle between the resultant velocity and bank of the river?

a. 36.8^o

b. 45^o

c. 40^o

d. 50^o

v. If speeds of boat and current were 5 m/s and 2.0 m/s then what will be resultant velocity?

a. 5 m/sec

b. 5 m/sec

c. 6 m/sec

d. 5 m/s

CBSE Class 12 Mathematics Case Study 14

14. Deepal left from his village on weekend. First, he travelled d1 displacement up to a temple. After this, he left for the zoo and travelled d2 displacement. After this he left for shopping in a mall - Total driving time of Deepal from village to Mall was 1.5 hr. If d₁ = (6, 8) d₂ = (3, 4) and d₃ = (7, 12) km

i. What is the total displacement from village to Mall?

a. 30 km

b. 20 km

c. 25 km

d. 40 km

ii. What is the speed of Deepal from Village to Mall?

a. 30 km/hr

b. 20 km/hr

c. 25 km/hr

d. 35 km/hr

iii. What is the Displacement from Village to Zoo?

a. 20 km

b. 10 km

c. 15 km

d. 25 km

iv. What is the angle in degree in Final displacement?

a. tan^-1(3/4)

b. tan^-1(4/3)

c. 45^o

d. 60^o

v. What is the displacement from temple to Mall?

a. 40 km

b. 30 km

c. 1 km

d. 20 km

CBSE Class 12 Mathematics Case Study 15

15. A plane started from airport O with a velocity of 120 m/s towards east. Air is blowing at a velocity of 50 m/s towards the north As shown in the figure. The plane travelled 1 hr in OA direction with the resultant velocity. From A and B travelled 1 hr with keeping velocity of 120 m/s and finally landed at B.

i. What is the resultant velocity from O to A?

a. 100 m/s

b. 130 m/s

c. 120 m/s

d. 170 m/s

ii. What is the direction of travel of plane O to A with east?

a. tan^-1(5/12)

b. tan^-1(12/13)

c. 40^o

d. 30^o

iii. What is the total displacement from O to A?

a. 500 km

b. 468 km

c. 432 km

d. 400 km

iv. What is the resultant velocity from A to B?

a. 120 m/s

b. 70 m/s

c. 170 m/s

d. 200 m/s

v. What is the displacement from A to B?

a. 550 km

b. 432 km

c. 600 km

d. 612 km

CBSE Class 12 Mathematics Case Study 16

16. Coloured balls are distributed in three bags as shown in the following table:

Find the following:

i. A bag is selected at random and then two balls are randomly drawn from the selected They happen to be black and red. What is the probability that they come from bag I?

a. 213/551

b. 231/551

c. 110/551

d. 551/110

ii. What is the probability that red and black balls are drawn from bag III?

a. 1/3

b. 11/2

c. 3/11

d. 2/11

iii. A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that they come from the bag II?

a. 110/551

b. 231/551

c. 188/551

d. 551/110

iv. If the events A and B are independent, then P(AB) is equal to

a. P(A) + P(B)

b. P(A).P(B)

c. P(A) - P(B)

d. P(A)/P(B)

v. If A and B are two events and A ≠ ϕ, B ≠ ϕ, then

a. P(A/B) = P (A) ⋅ P (B)

b. P(A/B) = P(A∩B) / P (B)

c. P (A/B) ⋅ P (B/A) = 1

d. P(A/B) = P(A)/P(B)

CBSE Class 12 Mathematics Case Study 17

17. In an office, three employees Vinay, Sonia and Iqbal process incoming copies of a certain 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. 0210

b. 0.04

c. 47

d. 0.06

ii. The probability that Sonia processed the form and committed an error is:

a. 0.005

b. 006

c. 0.008

d. 0.68

iii. The total probability of committing an error in processing the form is

a. 0

b. 047

c. 234

d. 1

iv. The manager of the company wants to do a quality During the 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

v. Let A be the event of committing an error in processing the form and let E₁, E₂ and E₃ be the events that Vinay, Sonia and Iqbal processed the form. The value of ³∑_i=1 P (E_i=1 ∣ A) is

a. 0

b. 03

c. 0.06

d. 1

CBSE Class 12 Mathematics Case Study 18

18. Three bags contain a number of red and white balls as follow:

Bag 1: 3 red balls,

Bag 2: 2 red balls and 1 white ball

Bag 3: 3 white balls.

The probability that bag i will be chosen and a ball is selected from it is: where i = 1,2,3.

i. What is the probability that a red ball will be selected?

a. 5/9

b. 7/18

c. 5/18

d. 11/18

ii. What is the probability that a white ball is selected?

a. 18/11

b. 11/18

c. 13/18

d. 7/18

iii. If a white ball is selected, what is the probability that it came from Bag 2?

a. 2/11

b. 11/2

c. 12/11

d.  7/11

iv. If a white ball is selected, what is the probability that it came from Bag 3

a. 5/11

b. 11/9

c. 7/11

d.9/11

v. If a white ball is selected, what is the probability that it came from Bag 1

a. 1

b. 11/18

c. 2/11

d. 0

CBSE Class 12 Mathematics Case Study 19

19. An item is manufactured by three machines A, B and Out of the total number of items manufactured during a specified period, 50% are manufactured on A, 30% on B and 20% on C.
2% of the items produced on A and 2% of items produced on B are defective, and 3% of these produced on C are defective. All the items are stored at one godown.

i. One item is drawn at random and is found to be What is the probability that it was manufactured on machine A?

a. 3/11

b. 2/11

c. 5/11

d. 11/3

ii. One item is drawn at random and is found to be What is the probability that it was manufactured on machine B?

a. 3/11

b. 2/11

c. 5/11

d. 4/11

iii. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine C?

a. 11/3

b. 3/11

c. 3/5

d. 5/11

iv. If two events are independent then

a. they must be mutually exclusive

b. the sum of their probabilities must be equal to 1

c. both (a) and (b) is correct

d. none of the above is correct

v. If A and B are such events that P(A) > 0 and P(B) ≠ 1 then P(A'/B') equals to

a. 1 − P (A/B)

b. 1 − P(A'/B)

c. 1−P(A∪B) /P( B′ )
′
d. P( A')/P( B′ )

CBSE Class 12 Mathematics Case Study 20

20. A shopkeeper sells three types of flower seeds A₁, A₂, and A₃. They are sold as a mixture where the proportions are 4:4:2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35%.

Based on the above information answer the following questions:

i. The probability of a randomly chosen seed to germinate:

a. 0.69

b. 39

c. 0.49

d. 0.59

ii. The probability that the seed will not germinate given that the seed is of type A₃:

a. 15/100

b. 65/100

c. 75/100

d. 55/100

iii. The probability that the seed is of the type A2 given that a randomly chosen seed does not germinate.

a. 22/51

b. 55/51

c. 51/16

d. 16/51

iv. Calculate the probability that it is of the type A1 given that a randomly chosen seed does not

a. 51/22

b. 22/51

c. 16/51

d. 7/51

v. The probability that it will not germinate given that the seed is of type A₁:

a. 55/100

b. 65/100

c. 35/100

d. 45/100