Week-1 Mathematics

 Which of the following are irrational numbers?

$(\sqrt{8}-\sqrt{2})(\sqrt{18}+\sqrt{2})$

${\displaystyle \frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}}$

${\displaystyle \frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}-\sqrt{2}}}$

${\displaystyle \frac{\sqrt{6}}{\sqrt{8}}}$

Yes, the answer is correct.
Score: 1

Accepted Answers:

${\displaystyle \frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}}$

${\displaystyle \frac{\sqrt{6}}{\sqrt{8}}}$

Suppose $f:D⟶\mathbb{R}$ is a function defined by $f(x)={\displaystyle \frac{\sqrt{x^2-36}}{x+6}}$, where $D\subset \mathbb{R}$. Let $A$ be the set of integers which are not in the domain of $f$, then find the cardinality of the set $A$.

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 12

1 point

Consider the set $S=\{a\mid a\in \mathbb{N},a\le 36\}$. Let $R_1\text{ and }{R}_2$ be relations from $S$ to $S$ defined as $R_1=\{(x,y)\mid x,y\in S,y=3x\}$ and $R_2=\{(x,y)\mid x,y\in S,y=x^2\}$. Find the cardinality of the set $R_1\setminus (R_1\cap R_2)$.

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 11.0

1 point

In a Zoo, there are 6 Bengal white tigers and $7$ Bengal royal tigers. Out of these tigers, 5 are males and 10 are either Bengal royal tigers or males. Find the number of female Bengal white tigers in the Zoo.

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 3

1 point

1 point

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers $(F)$, 245 ponds are polluted by pesticides $(P)$ and 257 ponds are polluted by pharmaceutical products $(Ph)$. 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.

                              
Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilisers and pharmaceutical products. Which of the following is/are true?

Relation is reflexive.

Relation is transitive.

Relation is symmetric.

This is an equivalence relation.

Yes, the answer is correct.
Score: 1

Accepted Answers:

Relation is transitive.

Relation is symmetric.

1 point

Which of the following relations is/are one-one function?

$R_1=\{(x,y)\mid x,y\in \mathbb{R},x+y>2\}$

$R_2=\{(x,y)\mid x,y\in \mathbb{R},x>y\}$

$R_3=\{(x,y)\mid x+y=12\}$

$R_4=\{(x,y)\mid x,y\in \mathbb{R},y=x^2\}$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$R_3=\{(x,y)\mid x+y=12\}$

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers $(F)$, 245 ponds are polluted by pesticides $(P)$ and 257 ponds are polluted by pharmaceutical products $(Ph)$. 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.

                              
Find the number of ponds polluted by all types of water contaminants.

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 52

1 point

1 point

Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh. This family tree is shown in the figure below. Let us define two relations, $R$ and $S$, on the set $M$, which is the collection of all family members, as follows,

$R:=\{(A,B)\mathrm{\mid }A\text{ and }B\text{ are cousins, i.e. their parents are siblings}\}$.

$S:=\{(A,B)\mathrm{\mid }A\text{ is son of }B\}$.




Let $f:=\{(A,B)\mathrm{\mid }A\text{ is son of }B\}\subset P\times Q$ , where $P$ and $Q$ are subsets of $M$.

Which of the following options are correct?

$f:P\to Q$ is a function, where $P=\{\text{Yashubh, Navrtna, Rathi, Rakesh, Mahesh}\}$ and $Q=\{\text{ Shubh, Rabi, Mahendra, Rajat}\}$.

If $f:P\to Q$ is a function, where $P=\{\text{ Yashubh, Navrtna, Rathi, Rakesh}\}$ and $Q=\{\text{Shubh, Rabi, Mahendra, Rajat}\}$, then $f$ is one-one.

If $f:P\to Q$ is a function, where $P=\{\text{Yashubh, Navrtna, Rathi, Rakesh}\}$ and $Q=\{\text{ Shubh, Rabi}\}$, then $f$ is onto.

If $f:P\to Q$ is a function, where $P=\{\text{Yashubh, Rathi}\}$ and $Q=\{\text{Shubh, Rabi}\}$, then $f$ is bijective.

Yes, the answer is correct.
Score: 1

Accepted Answers:

If $f:P\to Q$ is a function, where $P=\{\text{Yashubh, Navrtna, Rathi, Rakesh}\}$ and $Q=\{\text{ Shubh, Rabi}\}$, then $f$ is onto.

If $f:P\to Q$ is a function, where $P=\{\text{Yashubh, Rathi}\}$ and $Q=\{\text{Shubh, Rabi}\}$, then $f$ is bijective.

Consider the following sets.

• $A$= $\{x\in \mathbb{N}\mid x\text{ mod }2$=$0\text{ and }1\le x\le 10\}$
• $B$ = $\{x\in \mathbb{N}\mid x\text{ mod }5=0\text{ and }6\le x\le 25\}$
• $C=\{x\in \mathbb{N}\mid x\text{ mod }7=0\text{ and }7\le x\le 29\}$

What is the cardinality of $(A\setminus (B\cup C))\cup (B\setminus (C\cup A))\cup (C\setminus (B\cup A))$

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 11

1 point

A group of 180 people watched movies on Netflix, of which 95 watched Dabangg, 100 watched Avatar, and 100 watched RRR. 50 people watched Dabangg and Avatar, 40 watched Avatar and RRR, and 55 watched Dabangg and RRR. Find the number of people who watched only RRR and Avatar.

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 10