Q. Give three examples of sentences which are not statements Give reasons for the answers

Q. Which of the following sentences are statements? Give reasons for your answer
(i) There are $35$ days in a month
(ii) Mathematics is difficult
(iii) The sum of $5$ and $7$ is greater than $10$
(iv) The square of a number is an even number
(v) The sides of a quadrilateral have equal length
(vi) Answer this question
(vii) The product of $(-1)$ and $8$ is $8$
(viii) The sum of all interior angles of a triangle is ${180}^{\circ }$
(ix) Today is a windy day
(x) All real numbers are complex numbers

Q. Are the following pairs of statements negations of each other?
(i) The number $x$ is not a rational number
The number $x$ is not an irrational number
(ii) The number $x$ is a rational number
The number $x$ is an irrational number

Q. Find the component statements of the following compound statements and check whether they are true or false
(i) Number $3$ is prime or it is odd
(ii) All integers are positive or negative
(iii)$100$ is divisible by $3,11$ and $5$

Q. Write the negation of the following statements:
(i) Chennai is the capital of Tamil Nadu
(ii) $\sqrt{2}$ is not a complex number
(iii) All triangles are not equilateral triangle
(iv) The number $2$ is greater than $7$
(v) Every natural number is an integer

Q. Identify the quantifier in the following statements and write the negation of the statements
(i) There exists a number which is equal to its square
(ii) For every real number $x,x$ is less than $x+1$
(iii) There exists a capital for every state in India

Q. Check whether the following pair of statements is negation of each other. Give reasons for the answer
(i) $x+y=y+x$ is true for every real numbers $x$ and $y$
(ii) There exists real number $x$ and $y$ for which $x+y=y+x$

Q. For each of the following compound statements first identify the connecting words and then break it into component statements
(i) All rational numbers are real and all real numbers are not complex
(ii) Square of an integer is positive or negative
(iii) The sand heats up quickly in the Sun and does not cool down fast at night
(iv) $x=2$ and $x=3$ are the roots of the equation $3x^2-x-10=0$

Q. State whether the " Or" used in the following statements is "exclusive "or" inclusive Give reasons for your answer
(i) Sun rises or Moon sets
(ii) To apply for a driving licence you should have a ration card or a passport
(iii) All integers are positive or negative

Q. Rewrite the following statement in five different ways conveying the same meaning.

Q. Given statements in (a) and (b) Identify the statements given below as contrapositive or converse of each other
(a) If you live in Delhi then you have winter clothes
(i) If you do not have winter clothes then you do not live in Delhi
(ii) If you have winter clothes then you live in Delhi
(b) If a quadrilateral is a parallelogram then its diagonals bisect each other
(i) If the diagonals of a quadrilateral do not bisect each other then the quadrilateral is not a parallelogram
(ii) If the diagonals of a quadrilateral bisect each other then it is a parallelogram

Q. Write each of the following statement in the form "if-then"
(i) You get a job implies that your credentials are good
(ii) The Banana trees will bloom if it stays warm for a month
(iii) A quadrilateral is a parallelogram if its diagonals bisect each other
(iv) To get $A^{+}$ in the class it is necessary that you do the exercises of the book

Q. Write the contrapositive and converse of the following statements
(i) If $x$ is a prime number then $x$ is odd
(ii) It the two lines are parallel then they do not intersect in the same plane
(iii) Something is cold implies that it has low temperature
(iv) You cannot comprehend geometry if you do not know how to reason deductively
(v) $x$ is an even number implies that $x$ is divisible by $4$

Q. Show that the following statement is true by the method of contrapositive
p : If $x$ is an integer and $x^2$ is even then $x$ is also even

Q. Which of the following statements are true and which are false? In each case give a valid reason for saying so
(i) p : Each radius of a circle is a chord of the circle
(ii) q : The centre of a circle bisects each chord of the circle
(iii) r : Circle is a particular case of an ellipse
(iv) s : If x and y are integers such that $x>y$ then $-x<-y$
(v) t : $\sqrt{11}$ is a rational number

Q. Show that the statement
$p$ : "If $x$ is a real number such that $x^3+4x=0$ then $x$ is $0$" is true by
(i) direct method
(ii) method of contradiction
(iii) method of contrapositive

Q. By giving a counter example show that the following statements are not true
(i) p : If all the angles of a triangle are equal then the triangle is an obtuse angled triangle.
(ii) q : The equation $x^2-1$ $=0$ does not have a root lying between $0$ and $2$.

Q. Show that the statement "For any real numbers a and b, $a^2=b^2$ implies that $a=b$" is not true by giving a counter-example

Q. Write the negation of the following statements:
(i) p : For every positive real number $x$ the number $x-1$ is also positive
(ii) q : All cats scratch
(iii) r : For every real number $x$, either $x>1$ or $x<1$
(iv) s : There exist a number $x$ such that $0<x<1$

Q. State the converse and contrapositive of each of the following statements:
(i) p : A positive integer is prime only if it has no divisors other than $1$ and itself
(ii) q : I go to a beach whenever it is a sunny day
(iii) r : If it is hot outside then you feel thirsty