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Relation and Function
CASE STUDY 1:
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}
1. 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 2. 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 3. Three friends F 1 , F2 and F 3 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
ONE – NATION
ONE – ELECTION
FESTIVAL OF
DEMOCRACY
GENERAL ELECTION –
4. 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 5. 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
ANSWERS
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
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}
CASE STUDY 3:
An organization conducted bike race under 2 different categories-boys and girls. Totally
there were 250 participants. Among all of them finally three from Category 1 and two from
Category 2 were selected for the final race. Ravi forms two sets B and G with these
participants for his college project.
Let B = {b 1 ,b 2 ,b 3 } G={g 1 ,g 2 } where B represents the set of boys selected and G the set
of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions
1. Ravi wishes to form all the relations possible from B to G. How many such relations are possible? a. 26 b. 25 c. 0 d. 23 2. Let R: B→B be defined by R = {(𝑥, 𝑦): 𝑥 and y are students of same sex}, Then this relation R is_______ a. Equivalence b. Reflexive only c. Reflexive and symmetric but not transitive d. Reflexive and transitive but not symmetric 3. Ravi wants to know among those relations, how many functions can be formed from B to G? a. 22 b. 212 c. 32 d. 23 4. Let 𝑅: 𝐵 → 𝐺 be defined by R = { (b 1 ,g 1 ), (b 2 ,g 2 ),(b 3 ,g 1 )}, then R is__________
a. Injective b. Surjective c. Neither Surjective nor Injective d. Surjective and Injective
5. Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible? a. 0 b. 2! c. 3! d. 0!
ANSWERS
1. (a) 2^6 2. (a) Equivalence 3. (d) 2^3 4. (b) Surjective 5. (a) 0
CASE STUDY 5:
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other
to one side of the playground ensuring that they had enough play area. Let us assume that
they planted one of the rows of the saplings along the line 𝑦 = 𝑥 − 4. Let L be the set of all
lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
1. Let relation R be defined by R = {(𝐿1, 𝐿2): 𝐿1 ║ 𝐿2 where L 1 ,L 2 € L} then R is______ relation a. Equivalence b. Only reflexive
CASE STUDY 5:
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji
found that the swing traced the path of a Parabola as given by 𝑦 = 𝑥^2.
Answer the following questions using the above information.
1. Let 𝑓: 𝑅 → 𝑅 be defined by 𝑓(𝑥) = 𝑥^2 is_________ a. Neither Surjective nor Injective b. Surjective c. Injective d. Bijective 2. Let 𝑓: 𝑁 → 𝑁 be defined by 𝑓(𝑥) = 𝑥^2 is ________ a. Surjective but not Injective b. Surjective c. Injective d. Bijective 3. Let f: {1,2,3,….}→{1,4,9,….} be defined by 𝑓(𝑥) = 𝑥^2 is _________ a. Bijective b. Surjective but not Injective c. Injective but Surjective d. Neither Surjective nor Injective 4. Let : 𝑁 → 𝑅 be defined by 𝑓(𝑥) = 𝑥^2. Range of the function among the following is _________ a. {1, 4, 9, 16,…} b. {1, 4, 8, 9, 10,…} c. {1, 4, 9, 15, 16,…} d. {1, 4, 8, 16,…} 5. The function f: Z→Z defined by 𝑓(𝑥) = 𝑥^2 is__________ a. Neither Injective nor Surjective
b. Injective c. Surjective d. Bijective
ANSWERS
1. (a) Neither Surjective nor Injective 2. (C) Injective 3. (a) Bijective 4. (a) {1, 4, 9, 16,…} 5. (a) Neither Injective nor Surjective
Inverse Trigonometric Function :
CASE STUDY1:
Two men on either side of a temple of 30 meters high observe its top at the angles of elevation
𝛼 and 𝛽 respectively. (as shown in the figure above). The distance between the two men is
40 √3 meters and the distance between the first person A and the temple is 30√3 meters.
Based on the above information answer the following:
a. sin−1^ (
2 √3)
b. sin−1^ ( 1 2 ) c. sin−1(2)
d. sin−1^ ( √ 2 )
2. ∠𝐶𝐴𝐵 = 𝛼 =
a. cos−1^ ( 1 5 )
b. cos−1^ ( 2 5 )
c. cos−1^ ( √ 2 )
ground level. For the viewer A, the angle of elevation of “D” is double the angle of elevation
of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer.
Look at the figure given and based on the above information answer the following:
1. Measure of ∠𝐶𝐴𝐵 = a. tan−1(2)
b. tan−1^ ( 1 2 ) c. tan−1( 1) d. tan−1( 3)
2. 𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓∠𝐷𝐴𝐵 =
a. tan−1^ ( 3 4 ) b. tan−1(3)
c. tan−1^ ( 4 3 ) d. tan−1(4)
3. 𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 ∠𝐸𝐴𝐵 = a. tan−1(11) b. tan−1^3
c. tan−1^ ( 2 11 )
d. tan−1^ ( 11 2 )
4. 𝐴|^ Is another viewer standing on the same line of observation across the road. If the width of the road is 5 meters, then the difference between ∠𝐶𝐴𝐵 and ∠𝐶𝐴′𝐵 Is a. tan-1(1/2)
b. tan-1^ (1/8)
c. tan−1^ ( 2 5 )
d. tan−1^ ( 11 21 )
5. Domain and Range of tan−1^ 𝑥 =
a. 𝑅+, (− 𝜋 2 ,^
𝜋 2 )
b. 𝑅−, (− 𝜋 2 ,^
𝜋 2 )
c. R , (− 𝜋 2 ,^
𝜋 2 )
d. R , (0 , 𝜋 2 ) ANSWERS
1. (b) tan−1^ ( 1 2 ) 2. (c) tan−1^ (
4 3 )
3. (d) tan−1^ ( 11 2 ) 4. (b) tan-1(1/8) 5. (c) R , (− 𝜋 2 ,^
𝜋 2 ) MATRICES
CASE STUDY1:
A manufacture produces three stationery products Pencil, Eraser and Sharpener which he
sells in two markets. Annual sales are indicated below
ANSWERS
1. Rs. 46, 2. Rs. 53, 3. RS.31, 4. (Rs.15, 000, Rs.17, 000) 5. Rs. 32,
CASE STUDY 2:
Amit, Biraj and Chirag were given the task of creating a square matrux of order 2.
Below are the matrices created by them. A, B , C are the matrices created by Amit, Biraj
and Chirag respectively.
A = [^1
] B= [^4
] C= [^2
]
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. [
]
b. [
]
c. [^7 1 6
]
d. [^2 7 6
]
2. (𝐴𝑇)𝑇^ is equal to
a. [
]
b. [
]
c. [
]
d. [^2 −1 1
]
3. (𝑏𝐴)𝑇^ is equal to
a. [−2^ − 2 −
]
b. [−2^2 −4 −
]
c. [−2^2 −6 −
]
d. [−6^ − 2 4
]
4. AC−𝐵𝐶 is equal to
a. [−4^ − −4 4
]
b. [−4^ − 4 −
]
c. [−4^ − −6 4
]
d. [−6^4 −4 −
]
5. (𝑎 + 𝑏)𝐵 is equal to
a. [^0 10 2
]
b. [^2 8 0
]
c. [^8 2 10
]
d. [^2 8 10
]
Answers
1. (c) [^7 1 6
]
2. (a) [^1 −1 3
]
3. (b) [−2^2 −4 −
]
4. (c) [−4^ − −6 4
]
5. (c ) [^8 2 10
]
CASE STUDY 2:
Three schools DPS, CVC and KVS decided to organize a fair for collecting money for
helping the flood victims. They sold handmade fans, mats and plates from recycled material
at a cost of Rs. 25, Rs.100 and Rs. 50 each respectively. The numbers of articles sold are
given as
4. If the number of handmade fans and plates are interchanged for all the schools, then what is the total money collected by all schools? a. Rs. 18, b. Rs. 6, c. Rs. 5, d. Rs. 21, 5. How many articles (in total) are sold by three schools? a. 230 b. 130 c. 430 d. 330
ANSWERS
1. (b) 7000 2. (a) 14000 3. (c) Rs. 4. (d) 21250 5. (d) 330
CASE STUDY 3:
On her birth day, Seema decided to donate some money to children of an orphanage home.
If there were 8 children less, everyone would have got Rs.10 more. However, if there were
16 children more, everyone would have got Rs. 10 less. Let the number of children be x
and the amount distributed by Seema for one child be y (in Rs.).
Based on the information given above, answer the following questions:
1. The equations in terms x and y are a. 5x-4y = 40
5x-8y = -
b. 5x-4y = 40
5x-8y = 80
c. 5x-4y = 40
5x+8 y= -
d. 5x+4y = 40
5x-8y = -
2. Which of the following matrix equations represent the information given above? 1. [^5 5 8
] [
𝑦] = [
]
2. [^5 −
] [
𝑦] = [
]
3. [
] [
𝑦] = [
]
4. [^5
] [
𝑦] = [
]
3. The number of children who were given some money by Seema, is
a. 30
b. 40
c. 23
d. 32
4. How much amount is given to each child by Seema?
a. Rs. 32
b. Rs. 30
c. Rs. 62
d. Rs. 26
5. How much amount Seema spends in distributing the money to all the students of
the Orphanage?
a. Rs. 609
b. Rs. 960
c. Rs. 906
d. Rs. 690
ANSWERS
1. (a) 5x-4y = 40
5x-8y = -
c. A> 𝐵
d. A< 𝐵
2. What is the value of 𝐴 23?
a. 10000
b. 20000
c. 30000
d. 40000
3. The decrease in sales from September to October is given by _______.
a. A+B
b. A-B
c. A> 𝐵
d. A< 𝐵
4. If Ramkishan receives 2% profit on gross sales, compute his profit for each
variety sold in October.
a. Rs. 100, Rs. 200 and Rs. 120
b. Rs. 100, Rs. 200 and Rs. 130
c. Rs. 100, Rs. 220 and Rs. 120
d. Rs. 110, Rs. 200 and Rs. 120
5. If Gurucharan receives 2% profit on gross sales, compute his profit for each variety
sold in September.
a. Rs. 100, Rs. 200, Rs. 120
b. Rs. 1000 , Rs. 600, Rs. 200
c. Rs. 400, Rs. 200, Rs. 120
d. Rs. 1200, Rs. 200, Rs. 120
ANSWERS
1. (a) A+B 2. (a) 10000 3. (b) A-B 4. (a) Rs. 100, Rs. 200 and Rs. 120 5. (b) Rs. 1000, Rs. 600, Rs. 200
Determinants
CASE STUDY 1:
Manjit wants to donate a rectangular plot of land for a school in his village. When he was
asked to give dimensions of the plot, he told that if its length is decreased by 50 m and
breadth is increased by 50m, then its area will remain same, but if length is decreased by
10m and breadth is decreased by 20m, then its area will decrease by 5300 m^2
Based on the information given above, answer the following questions:
1. The equations in terms of X and Y are a. x-y=50, 2x-y= b. x-y=50, 2x+y= c. x + y = 50, 2x + y= d. x +y = 50, 2x + y= 2. Which of the following matrix equation is represented by the given information
a.
y
x
b.
y
x
c.
y
x
d.
y
x
3. The value of x (length of rectangular field) is a. 150m b. 400m c. 200m
y
x
