The quantifier is 'there exists' and the negation is There does not exist a number which is multiple of both 6 and 9

(i)Let us assume that p is not true.
\(\therefore \)Sum of an irrational and a rational number is not irrational.
\(\Rightarrow \)There exists an irrational number a and a rational number b such that a + b is not irrational.
\(\Rightarrow \)a + b = c (say) is a rational number.
\(\Rightarrow \)a=c-b
\(\Rightarrow \)a is rational.
But a is irrational, which is contradiction
So our supposition in wrong.
Thus p is true.
(ii) Let rand s be the statements given by
r: n is a real number with n > 3
s: n²> 9
If possible let q is not true then
\(\Rightarrow \)\(\sim \)q is true
\(\Rightarrow \)(r\(\Rightarrow \)s) is true
[\(\therefore \)q:rs]
\(\Rightarrow \)r and \(\sim \)s is true.
\(\Rightarrow \)n is a real number with n > 3 and n² < 9
which is contradiction So our supposition is wrong.
Thus q is true.

(i) The contrapositive of the given statement is: If a positive integer has divisors other than 1 and itself then it is not prime. The converse of the given statement is: If a positive integer has no divisors other than 1 and itself, then it is a prime
(ii) The contrapositive of the given statement is: If! do not go to a beach then it is not a sunny day. The converse of the given statement is: If I go to beach, then it is a sunny day.
(iii) The contrapositive of the given statement IS: If you do not feel thirsty, then it is not hot outside. The converse of the given statement is: If you feel thirsty, then it is hot outside.

The given compound statement is of the form "if p then q".
p: x \(\in \)R such that x³+ 7x = 0
q: x= 0
(i) Direct method:
We assume that p is true, then
x\(\in \)R such that.x³ + 7x = 0
\(\Rightarrow \)X\(\in \)R such that x (x²+ 7) = 0
\(\Rightarrow \)X\(\in \)R such that x = 0 or x²+ 7 = 0
\(\Rightarrow \)x=o
p is true, q is true.
So when p is true, q is true.
Thus the given compound statement is true.
(ii) Method of contradiction:
We assume that p is true and q is false, then
x\(\in \)R such that x³+ 7x = 0
\(\Rightarrow \)X\(\in \)R such that x (x² + 7) = o.
\(\Rightarrow \)X\(\in \)R such that x = 0 or x²+ 7 = 0
\(\Rightarrow \)x=o
which is a contradiction. Thus our assumption that X"#0 is false. Thus the given compound statement is true.
(iii) Method of contrapositive:
We assume that q is false then
x\(\neq \)0
\(\Rightarrow \)x\(\in \)R such that x³+ 7x\(\neq \)0
\(\Rightarrow \)p is false
So when q is false, p is false.
Thus the given compound statement is true.

(i) A chord of a circle is a line whose two endpoints lie on the circle and all the points on the line lie inside the circle. So the radius of a circle is not a chord of the circle. Thus the given statement is false.
(ii) The centre of a circle bisects chord of circle when the chord is diameter of circle. When the chord is other than diameter then centre of circle does not lie on the chord. Thus the given statement is false.
(iii) In the equation of an ellipse is we put a = b then we get an equation of circle. Thus the given statement is true.
(iv) It is given that x, y\(\in \) Z such that x > y. Multiplying both sides by negative sign, we have x,y\(\in \) Z such that - x < - y.
Thus the given statement is true.
(v) Since \(\sqrt { 11 } \) cannot be expressed in the form  \(\frac { a }{ b } \) where a and b are integers and b\(\neq \)0
Thus the given statement is false.

The given compound statement is of the form "if p then q".
p: x \(\in \) R such that.x³ + 4x = 0
q: x= 0
(i) Direct method:
We assume that p is true then
x \(\in \) R such that x³ + 4x = 0
\(\Rightarrow \) x \(\in \) R such x (x²+ 4) = 0
\(\Rightarrow \) x \(\in \) R such that x = 0 or x²+ 4 = 0
\(\Rightarrow \)x = 0
\(\Rightarrow \)q is true
So when p is true, q is true.
Thus the given compound statement is true.
(ii) Method of contradiction:
We assume that p is true and q is false. then
x \(\in \) R such that x³+4x=0
\(\Rightarrow \) x \(\in \) R such that x(x² + 4) = 0
\(\Rightarrow \) x \(\in \) R such that x=0 or x² +4=0
\(\Rightarrow \) x = 0
which is a contradiction. So our assumption that x \(\neq \) 0 is false. Thus the given compound statement is true.
(iii) Method of contrapositive:
We assume that q is false, then
X \(\neq \) 0
\(\Rightarrow \) x \(\in \) R such that x³+ 4 x \(\neq \) 0
\(\Rightarrow \) p is false
So when q is false, p is false.
Thus the given compound statement is true.

(i) Here p: x is a prime number.
q: x is odd.
Now ~ p: x is not a prime number.
\(\sim \)q: x is not odd.
The contrapositive of given statement is:
If x is not odd then x is not a prime number.
The converse of given statement is:
If x is an odd number then x is a prime
number.
(ii)Here p: Two lines are parallel.
q: They do not intersect in the same plane.
Now \(\sim \): Two line are not parallel.
\(\sim \)q: They intersect in the same plane.
The contrapositive of given statement is:
if two lines intersect in the same plane then they are not parallel.
The converse of given statement is:
if the two lines do not intersect in the same plane then they are parallel.
(iii) Here p: Something is cold.
q: It has low temperature.
Now\(\sim \)p: Something is not cold.
\(\sim \)q: It has not low temperature.
The contrapositive of given statement is:
If something does not have low temperature then it is not cold.
The converse of given statement is:
If something has low temperature then it is cold.
(iv) Here p: You can not comprehend geometry:
q: You do not know how to reason deductively.
Now\(\sim \)p: You can comprehend geometry.
\(\sim \)q: You know how to reason deductively.
The contrapositive of given statement is:
If you know to reason deductively then you can comprehend geometry.
The converse of given statement is:
If you do not know how to reason deductively then you can not comprehend geometry.
(v) Here p: x is an even number.
q: x is divisible by 4.
Now\(\sim \)p: x is not an even number.
\(\sim \)q: x is not divisible by 4.
The contrapositive of given statement is:
If x is not divisible by 4 then x is not an even number.
The converse of given statement is:
if x is divisible by 4 then x is an even number.

(l) The component statement has the connecting word 'and' component statements are
p: All rational numbers are real.
q: All real numbers are not complex.
(ii) The component statement has the connecting words 'or' component statements are
p: Square of an integer is positive.
q: Square of an integer is negative.
(iii) The component statement has the connecting word 'and' component statements are
p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.
(iv) The component statement has the connecting word 'and' component statements are
p: x = 2 is a root of the equation 3\({ x }^{ 2 }\)- x - 10 = 0.
q: x = 3 is a root of the equation 3\({ x }^{ 2 }\)- x - 10 = 0.

The three examples of sentences, which are not statements, are as follows.
(i) He is a doctor.
It is not evident from the sentence as to whom ‘he’ is referred to. Therefore, it is not a statement.
(ii) Geometry is difficult.
This is not a statement because for some people, geometry can be easy and for some others, it can be difficult.
(iii) Where is she going?
This is a question, which also contains ‘she’, and it is not evident as to who ‘she’ is. Hence, it is not a statement.

(i) No month has 35 days. Thus the sentence is false declarative sentence. Hence it is a statement. (ii)Here the correctness of the sentence depends upon the observer. It may be easy for someone and may be difficult for other one. Hence it is not a statement.
(iii)The sentence is true. Hence it is a statement.
(iv) Here the correctness of the sentence depends upon the number that is squared. Hence it is not a statement.
(v) This sentence is sometimes true and sometimes false. For example sides in squares and rhombuses have equal length whereas, in a rectangle and trapezium, they have unequal length. Hence it is not a statement.
(vi) This sentence is an order. Hence it is not a statement.
(vii)The sentence is true. Hence it is a statement.
(viii) The sentence is false. Hence it is a statement.
(ix) It is not clear from the context which day is referred. Hence it is not a statement.
(x) The sentence is true because all real numbers can be written in the form a + i x O.Hence it is a statement.