(i) $p$ : Number 7 is prime.

$q$ : Number 7 is odd.

(ii) P : Chennai is in India.

$q$ : Chennai is capital of Tamil Nadu.

(iii) $p: 100$ is divisible by 3 .

$q$ : 100 is divisible by 11 .

$r: 100$ is divisible by 5 .

(iv) $p$ : Chandigarh is capital of Haryana.

$q$ : Chandigarh is capital of UP.

(v) $p: \sqrt{7}$ is a rational number.

$q: \sqrt{7}$ is an irrational number.

(vi) $p: 0$ is less than every positive integer.

$q: 0$ is less than every negative integer.

(vii) $p$ : Plants use sunlight for photosysthesis.

$q$ : Plants use water for photosynthesis.

$r$ : Plants use carbon dioxide for photosysthesis.

(viii) $p$ : Two lines in a plane intersect at one point.

$q$ : Two lines in a plane are parallel.

(ix) $p$ : A rectangli, is a quadrilateral.

$q$ : A rectangle is a 5 -sided polygon.

(i) Given compound statement is of the form ' $p v q$ '. Since, the statement ' $p v q$ ' has the truth value $T$ whenever either $p$ or $q$ or both have the truth value $T$.

So, it is true statement.

Its component statements are

$p: 57$ is divisible by 2 . [false]

$q: 57$ is divisible by 3 . [true]

(ii) Given compound statement is of the form ' $p \wedge q$ '. Since, the statement ' $p \wedge q$ ' have the truth value $T$ whenever both $p$ and $q$ have the truth value $T$. So, it is a true statement.

Its component statements are

$p: 24$ is multiple of 4 [true]

$q: 24$ is multiple of 6 . [true]

(iii) It is a false statement. Since ' $p \wedge q$ ' has truth value $F$ whenever either $p$ or $q$ or both have the truth value $F$.

Its component statements are

$p$ : All living things have two eyes. [false]

$q$ : All living things have two legs. [false]

(iv) It is a true statement.

Its component statements are

$p: 2$ is an even number. [true]

$q: 2$ is a prime number. [true]

(i) The number 17 is not prime.

(ii) $2+7 \neq 6$.

(iii) Violets are not blue.

(iv) $\sqrt{5}$ is not a rational number.

(v) 2 is a prime number.

(vi) Every real number is not an irrational number.

(vii) Cow has not four legs.

(viii) A leap year has not 366 days.

(ix) There exist similar triangles which are not congruent.

(x) Area of a circle is not same as the perimeter of the circle.

(i) $p$ : Rahul passed in Hindi.

$q$ : Rahul passed in English.

$p \wedge q$ : Rahul passed in Hindi and English.

(ii) $p: x$ is even integers.

$q: y$ is even integers.

$p \cap q: x$ and $y$ are even integers.

(iii) $p: 2$ is factor of 12 .

$q: 3$ is factor of 12 .

$r: 6$ is factor of 12 .

$p \wedge q \wedge r: 2,3$ and 6 are factor of 12 .

(iv) $p: x$ is an odd integer.

$q:(x+1)$ is an odd integer.

$p \vee q$ : Either $x$ or $(x+1)$ is an odd integer.

(v) $p$ : A number is divisible by 2 .

$q$ : A number is divisible by 3.

$p \vee q$ : A number is either divisible by 2 or 3 .

(vi) $p: x=2$ is a root of $3 x^2-x-10=0$.

$q: x=3$ is a root of $3 x^2-x-10=0$.

$p \vee q$ : Either $x=2$ or $x=3$ is a root of $3 x^2-x-10=0$.

(vii) $p$ : Students can take Hindi as an optional paper.

$q$ : Students can take English as an optional subject.

$p \vee q$ : Students can take Hindi or English as an optional paper.

(i) Let $p$ : All rational numbers are real.

$q$ : All rational numbers are complex.

$\sim p$ : All rational number are not real.

$\sim q$ : All rational numbers are not complex.

$\sim(p \wedge q):$ All rational numbers are not real or not complex. $\quad[\because \sim(p \wedge q)=\sim p \vee \sim q]$

(ii) Let $p$ : All real numbers are rationals.

$q$ : All real numbers are irrational.

Then, the negation of the above statement is given by

$\sim(p \vee q)$ : All real numbers are not rational and all real numbers are not irrational.

$$[\because \sim(p \vee q)=\sim p \wedge \sim q]$$

(iii) Let $p$ : $x=2$ is root of quadratic equation $x^2-5 x+6=0$.

$q: x=3$ is root of quadratic equation $x^2-5 x+6=0$.

Then, the negation of conjunction of above statement is given by

$\sim(p \wedge q): x=2$ is not a root of quadratic equation $x^2-5 x+6=0$ or $x=3$ is not a root of the quadratic equation $x^2-5 x+6=0$.

(iv) Let $p$ : A triangle has 3 -sides.

$q$ : A triangle has 4 -sides.

Then, negation of disjunction of the above statement is given by $\sim(p \vee q)$ : A triangle has neither 3-sides nor 4-sides.

(v) Let $p: 35$ is a prime number.

$q: 35$ is a composite number.

Then, negation of disjunction of the above statement is given by $\sim(p \vee q): 35$ is not a prime number and it is not a composite number.

(vi) Let $p$ : All prime integers are even.

$q$ : All prime integers are odd.

Then negation of disjunction of the above statement is given by

$\sim(p \vee q)$ : All prime integers are not even and all prime integers are not odd.

(vii) Let $p:|x|$ is equal to $x$.

$q:|x|$ is equal to $-x$.

Then negation of disjunction of the above statement is given by

$\sim(p \vee q):|x|$ is not equal to $x$ and it is not equal to $-x$.

(viii) Let $p: 6$ is divisible by 2 .

$q: 6$ is divisible by 3 .

Then, negation of conjunction of above statement is given by

$\sim(p \wedge q): 6$ is not divisible by 2 or it is not divisible by 3