NCERT Class 11 Maths Chapter Mathematical Reasoning: Topics Covered, Questions, Preparation

Exercise 14.1

Q1. 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

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

A.1. (i) This sentence is false because there are less than 35 days in a month. Hence it is a statement.

(ii) This sentence differs to individual, as some may find it difficult and some may find it easy. This means that it is not always true. Hence it is not a statement.

(iii) This sentence is true as the sum of 5 and 7 is 12 which is greater than 10. Hence it is a statement.

(iv) This sentence can be correct and incorrect sometimes as the square of 2 is even but the square of 3 is an odd. Hence it is not a statement.

(v) This sentence is not always true. For instance, square and rhombus have sides of equal lengths but a rectangle has sides of unequal lengths. Hence it is not a statement.

(vi) This sentence is an order. So, it is not a statement.

(vii) This sentence is incorrect as the product of (–1) and 8 is –8. Hence, it is a statement.

(viii) The given sentence is true. Therefore, it is a statement.

(ix) This sentence has “today” in it which is a variable time. Hence it is not a statement.

(x) This sentence is always correct. So, it is a statement.

 

Q2. Give three examples of sentences which are not statements. Give reasons for the answers.

A.2. The three examples of sentences which are not statements are:

(1) He is an engineer.

Here, in this sentence, it is not evident as to whom ‘he’ is referred to. Hence, it is not a statement.

(2) Painting is easy for some people, painting can be easy and for some others, it can be difficult. Hence, it is not a statement.

(3) Where is my hat?

This sentence is a question which is not considered as a statement in mathematical language. Hence, it is not a statement.

 

Exercise 14.2

Q1. Write the negation of the following statements:

(i) Chennai is the capital of Tamil Nadu.

(iii) All triangles are not equilateral triangle.

(iv) The number 2 is greater than 7.

(v) Every natural number is an integer.

A.1. (i) Chennai is not the capital of Tamil Nadu.

(iii) All triangles are equilateral triangles.

(iv) The number 2 is not greater than 7.

(v) Every natural number is not an integer.

 

Q2. 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.

A.2. (i) The negation of the first statement is ‘the number x is a rational number’.

This is same as the second statement because if a number is not an irrational number then the number is a rational number.

Hence, the given statements are negations of each other.

(ii) The negation of the first statement is ‘the number x is not a rational number. This means that the number x is an irrational number which is same as the second statement.

Hence, the given statements are negations of each other.

 

Q3. 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.

A.3. (i) The component statements are

(a) Number 3 is prime

(b) Number 3 is odd

Here, both the statements are true.

(ii) The component statements are

(a) All integers are positive

(b) All integers are negative

Here, both the statements are false.

(iii) The component statements are

(a) 100 is divisible by 3

(b) 100 is divisible by 11

(c) 100 is divisible by 5

Here, the statements (a) and (b) are false and c) is true.

 

Exercise 14.3

Q1. 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.

A.1. (i) Here, ‘and’ is the connecting word

The component statements are–

(a) All rational numbers are real

(b) All real numbers are not complex

(ii) Here, ‘or’ is the connecting word

The component statements are–

(a) Square of an integer is positive

(b) Square of an integer is negative

(iii) Here, ‘and’ is the connecting word

The component statements are–

(a) The sand heats up quickly in the Sun

(b) The sand does not cool down fast at night

 

Q2. 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.

A.2. (i) Here, the quantifier is ‘there exists’.

The negation of this statement is

There does not exist a number which is equal to its square.

(ii) Here, the quantifier is ‘for every’.

The negation of the statement is

There exist a real number x, such that x is not less than x+1

(iii) Here, the quantifier is ‘there exists’.

The negation of this statement is

In India, there exists a state which does not have a capital.

 

Q3. Check whether the following pair of statements are negation of each other. Give reasons for your answer.

(i) x + y = y + x is true for every real numbers x and y.

(ii) There exists real numbers x and y for which x + y = y + x.

A.3. Negation of statement (i) is:

There exists real numbers x and y for which x + y = y + x.

Now, this statement is not same as statement (ii).

Therefore, the given statements are not negation of each other.

 

Q4. 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.

A.4. (i) The ‘or’ in this statement is conclusive because it is not possible for the sun to rise and the moon to set together.

(ii) The ‘or’ in this statement is inclusive since a person can have both a ration card and a passport to apply for a driving license.

(iii) The ‘or’ in this statement is exclusive since all integers cannot be both positive and negative.

 

Exercise 14.4

Q1. Rewrite the following statement with “if-then” in five different ways conveying the same meaning.

If a natural number is odd, then its square is also odd.

A.1.Five different ways are–

(i) A natural number is odd implies that its square is odd.

(ii) A natural number is odd only if its square is odd.

(iii) For a natural number to be odd it is necessary that its square is odd.

(iv) For the square of a natural number to be odd it is sufficient that the number is odd.

(v) If the square of a natural number is not odd, then the natural number is not odd.

Q2. Write the contrapositive and converse of the following statements.

(i) If x is a prime number, then x is odd.

(ii) If 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.

A.2. (i) Contrapositive

If a number x is not odd, then x is not a prime number.

Converse

If a number x is odd, then it is a prime number.

(ii) Contrapositive

If two lines intersect in the same place, then the two lines are not parallel.

Converse

If two lines do not intersect in the same place, then they are parallel.

(iii) Contrapositive

If something does not have a low temperature, then it is not cold.

Converse

If something is at a low temperature, then it is cold.

(iv) Contrapositive

If you know how to reason deductively, then you can comprehend geometry.

Converse

If you do not know how to reason deductively, then you cannot comprehend geometry.

(v) Contrapositive

If x is not divisible by 4, then x is not an even number.

Converse

If x is divisible by 4, then x is an even number.

Q3. Write each of the following statements 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 an A+ in the class, it is necessary that you do all the exercises of the book.

A.3. (i) If you get a job, then your credentials are good.

(ii) If the Banana trees stay warm for a month, then the trees will bloom.

(iii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

(iv) If you want to score an A+ in the class, then you do all the exercises in the book.

Q.4.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.

A.4. (a) (i) Contra positive

(ii) Converse

(b) (i) Contra positive

(ii) Converse

 

Miscellaneous Exercise

Q1.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 exists a number x such that 0 < x < 1.

A.1 (i) The negation of statement p is as follows.

There exists a positive real number x, such that x – 1 is not positive.

(ii) The negation of statement q is

There exists a cat that does not scratch.

(iii) The negation of statement r is

There exist a real number x, such that neither x>1 nor x<1.

(iv) The negation of statement s is as follow.

There does not exist a number x, such that 0

 

Q2. 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.

A.2. (i) Statement p can be written as follows.

If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is given below

If a positive integer has no divisor other than/and itself, then it is prime.

The contrapositive of the statement is given below

If positive integer has divisors other than 1 and itself, then it is not prime.

(ii) The given statement can be written as follows.

If it is a sunny day, then I go to a beach.

The converse of the statement is given below.

If I go to a beach, then it is sunny day.

The contrapositive of statement is given below.

If I do not go to a beach, then it is not a sunny day.

(iii) The converse of statement is

If you fell thirsty, then it is not outside.

The contrapositive of statement ‘r’ is

If you do not feel thirsty, then it is not hot outside.

 

Q3. Write each of the statements in the form “if p, then q”

(i) p: It is necessary to have a password to log on to the server.

(ii) q: There is traffic jam whenever it rains.

(iii) r: You can access the website only if you pay a subscription fee.

A.3. (i) The statement p in the form ‘if then’ is as follows.

If you log on to the server, then you have password.

(ii) The statement q in the form ‘if then’ is as follows.

If it rains, then there is a traffic jam.

(iii) The statement r in the form ‘if them’ is as follows.

If you can access the website, then you pay a subscription fee.

 

Q.4. Rewrite each of the following statements in the form “p if and only if q”

(i) p: If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

A.4. (i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

 

Q5.Given below are two statements

p : 25 is a multiple of 5.

q : 25 is a multiple of 8.

Write the compound statements connecting these two statements with “And” and

“Or”. In both cases check the validity of the compound statement.

A5. The compound statement with ‘And’ is as follows 25 is a multiple of 5 and 8.

This is false statement because 25 is not a multiple of 8.

The compound statement with ‘Or’ is as follows 25 is a multiple of 5 or 8.

This is true statement because 25 is not a multiple of 8 but it is a multiple of 5.

 

Q7. Write the following statement in five different ways, conveying the same meaning.

p: If a triangle is equiangular, then it is an obtuse angled triangle.

A.7. The given statement can be written in five different ways: –

(i) A triangle is equiangular implies that it is an obtuse angled triangle.

(ii) A triangle is equiangular only if the triangle is an obtuse angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse angled triangle.

(iv) For a triangle to be an obtuse angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is an obtuse angled triangle, then the triangle is not equiangular.

 

Exercise 14.1

Q1. 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°.

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

A.1. (i) This sentence is false because there are less than 35 days in a month. Hence it is a statement.

(ii) This sentence differs to individual, as some may find it difficult and some may find it easy. This means that it is not always true. Hence it is not a statement.

(iii) This sentence is true as the sum of 5 and 7 is 12 which is greater than 10. Hence it is a statement.

(iv) This sentence can be correct and incorrect sometimes as the square of 2 is even but the square of 3 is an odd. Hence it is not a statement.

(v) This sentence is not always true. For instance, square and rhombus have sides of equal lengths but a rectangle has sides of unequal lengths. Hence it is not a statement.

(vi) This sentence is an order. So, it is not a statement.

(vii) This sentence is incorrect as the product of (–1) and 8 is –8. Hence, it is a statement.

(viii) The given sentence is true. Therefore, it is a statement.

(ix) This sentence has “today” in it which is a variable time. Hence it is not a statement.

(x) This sentence is always correct. So, it is a statement.

 

Q2. Give three examples of sentences which are not statements. Give reasons for the answers.

A.2. The three examples of sentences which are not statements are:

(1) He is an engineer.

Here, in this sentence, it is not evident as to whom ‘he’ is referred to. Hence, it is not a statement.

(2) Painting is easy for some people, painting can be easy and for some others, it can be difficult. Hence, it is not a statement.

(3) Where is my hat?

This sentence is a question which is not considered as a statement in mathematical language. Hence, it is not a statement.

 

Exercise 14.2

Q1. Write the negation of the following statements:

(i) Chennai is the capital of Tamil Nadu.

(ii) √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.

A.1. (i) Chennai is not the capital of Tamil Nadu.

(ii) √2 $$ is a complex number.

(iii) All triangles are equilateral triangles.

(iv) The number 2 is not greater than 7.

(v) Every natural number is not an integer.

 

Q2. 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.

A.2.  (i) The negation of the first statement is ‘the number x is a rational number’.

This is same as the second statement because if a number is not an irrational number then the number is a rational number.

Hence, the given statements are negations of each other.

(ii) The negation of the first statement is ‘the number x is not a rational number. This means that the number x is an irrational number which is same as the second statement.

Hence, the given statements are negations of each other.

 

Q3. 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.

A.3.  (i) The component statements are

(a) Number 3 is prime

(b) Number 3 is odd

Here, both the statements are true.

(ii) The component statements are

(a) All integers are positive

(b) All integers are negative

Here, both the statements are false.

(iii) The component statements are

(a) 100 is divisible by 3

(b) 100 is divisible by 11

(c) 100 is divisible by 5

Here, the statements (a) and (b) are false and c) is true.

 

Exercise 14.3

Q1. 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 3x₂ – x – 10 = 0.

A.1. (i) Here, ‘and’ is the connecting word

The component statements are–

(a) All rational numbers are real

(b) All real numbers are not complex

(ii) Here, ‘or’ is the connecting word

The component statements are–

(a) Square of an integer is positive

(b) Square of an integer is negative

(iii) Here, ‘and’ is the connecting word

The component statements are–

(a) The sand heats up quickly in the Sun

(b) The sand does not cool down fast at night

(iv) Here, ‘and’ is the connecting word

The component statements are–

(a) n = 2 is the root of the equation 3x²–x– 10 = 0

(b) n = 3 is the root of the equation 3x²– x–10 = 0

 

Q2. 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.

A.2. (i) Here, the quantifier is ‘there exists’.

The negation of this statement is

There does not exist a number which is equal to its square.

(ii) Here, the quantifier is ‘for every’.

The negation of the statement is

There exist a real number x, such that x is not less than x+1

(iii) Here, the quantifier is ‘there exists’.

The negation of this statement is

In India, there exists a state which does not have a capital.

 

Q3. Check whether the following pair of statements are negation of each other. Give reasons for your answer.

(i) x + y = y + x is true for every real numbers x and y.

(ii) There exists real numbers x and y for which x + y = y + x.

A.3. Negation of statement (i) is:

There exists real numbers x and y for which x + y = y + x.

Now, this statement is not same as statement (ii).

Therefore, the given statements are not negation of each other.

 

Q4. 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.

A.4. (i) The ‘or’ in this statement is conclusive because it is not possible for the sun to rise and the moon to set together.

(ii) The ‘or’ in this statement is inclusive since a person can have both a ration card and a passport to apply for a driving license.

(iii) The ‘or’ in this statement is exclusive since all integers cannot be both positive and negative.

 

Exercise 14.4

Q1. Rewrite the following statement with “if-then” in five different ways conveying the same meaning.

If a natural number is odd, then its square is also odd.

A.1. Five different ways are–

(i) A natural number is odd implies that its square is odd.

(ii) A natural number is odd only if its square is odd.

(iii) For a natural number to be odd it is necessary that its square is odd.

(iv) For the square of a natural number to be odd it is sufficient that the number is odd.

(v) If the square of a natural number is not odd, then the natural number is not odd.

 

Q2. Write the contrapositive and converse of the following statements.

(i) If x is a prime number, then x is odd.

(ii) If 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.

A.2. (i) Contrapositive

If a number x is not odd, then x is not a prime number.

Converse

If a number x is odd, then it is a prime number.

(ii) Contrapositive

If two lines intersect in the same place, then the two lines are not parallel.

Converse

If two lines do not intersect in the same place, then they are parallel.

(iii) Contrapositive

If something does not have a low temperature, then it is not cold.

Converse

If something is at a low temperature, then it is cold.

(iv) Contrapositive

If you know how to reason deductively, then you can comprehend geometry.

Converse

If you do not know how to reason deductively, then you cannot comprehend geometry.

(v) Contrapositive

If x is not divisible by 4, then x is not an even number.

Converse

If x is divisible by 4, then x is an even number.

 

Q3. Write each of the following statements 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 an A+ in the class, it is necessary that you do all the exercises of the book.

A.3. (i) If you get a job, then your credentials are good.

(ii) If the Banana trees stay warm for a month, then the trees will bloom.

(iii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

(iv) If you want to score an A+ in the class, then you do all the exercises in the book.

 

Q.4.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.

A.4. (a) (i) Contra positive

(ii) Converse

(b) (i) Contra positive

(ii) Converse

 

Exercise 14.5

Q1. Show that the statement p: “If x is a real number such that x³ + 4x = 0, then x is 0” is true by

(i) direct method, (ii) method of contradiction, (iii) method of contrapositive

A.1. (i) Direct method:

Let  $x^3+4x=0,x\in R$

$\Rightarrow x\left(x^2+4\right)=0$

$\Rightarrow$ Either x=0 or x²+4=0

But x² + 4 > 4 because $x\in R$ and hence  $\ne 0.$

Therefore x=0

$\therefore x^3+4x=0,x\in R\Rightarrow x\ne 0$

This, p is a true statement.

(ii) Method of contradiction

Let  $x^3+4x=0,x\in R$

Suppose  $x\ne 0$

$\Rightarrow x^2>0$

$\Rightarrow x^2+4>0$

$\Rightarrow x^2+4\ne 0$

Now  $x\ne 0$ and  $x^2+4\ne 0$

$\Rightarrow x\left(x^2+4\right)\ne 0$

$\Rightarrow x^3+4x\ne 0$ which is a contradiction to given.

$\therefore$ Our supposition is wrong and hence x = 0

This, p is a true statement.

(iii) Method of contrapositive: The components of the give if… then statement p are:

Let  $q:x\in R$ andx³ + 4x = 0 and r:x = 0

$\therefore$ The given statement p is q  $\Rightarrow$ r

Its contra positive is  $\sim r\Rightarrow \sim q.$

Let  $\sim r$ be true i.e., x is a non-zero real number

Now  $x\ne 0,x\in R$

$\Rightarrow x^2>0$

$\Rightarrow x^2+4>4$

$\Rightarrow x^2+4\ne 0$

$\therefore x\left(x^2+4\right)\ne 0\Rightarrow x^3+4x\ne 0$

$\therefore \sim q$ is true

i.e.,  $\sim r\Rightarrow \sim q$

$\therefore q\Rightarrow r$ is true

This, p is a true statement.

 

Q2. Show that the statement “For any real numbers a and b, a² = b² implies that a = b” is not true by giving a counter-example.

A.2. Let a = 2, b = –2 be two real numbers.

Clearly, a² = b² (=4) but $a\ne b.$

Thus, the given statement is not true.

 

Q3. Show that the following statement is true by the method of contrapositive. p: If x is an integer and x2 is even, then x is also even.

A.3. Let p: If x is an integer and x² is even, then x is also even.

Let q: x be an integer and x² be even

Let r: x is even

We have to prove, using the method of contra positive whether  $q\Rightarrow r$ is true.

[i.e., its contra positive  $\sim r\Rightarrow \sim q$ is true]

Let  $\sim r$ be true i.e., r be false

Let us assume that x is not even (integer) i.e., x is odd.

$\therefore x=2m+1$ where m is an integer

$\therefore x^2{\left(2m+1\right)}^2=4m+1+4m$

$=4m^2+4m+1$

$=2\left(2m^2+2m+1\right)$  $=2t+1t=2m^2+2m$ is an integer.

$\Rightarrow$ x² is also odd $\Rightarrow$ x is not even

$\Rightarrow q\text{\hspace{0.17em}isfalse}\left[\text{\hspace{0.17em}Bydefof}q\right]\Rightarrow \sim q\text{\hspace{0.17em}istrue}$

$\therefore \sim r\Rightarrow \sim q$ is true

Thus, by the method of contrapositive  $q\Rightarrow r$ is true

 

Q4. 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 x² – 1 = 0 does not have a root lying between 0 and 2.

A.4. (i) Let us consider an equilateral triangle ABC where  $\angle A=\angle B=\angle C=60°,$ then triangle ABC is not obtuse angled since all its angles are equal

$\therefore$ The statement p is not true.

(ii) 1 lies between 0 and 2. Also, 1 satisfies the equation x² – 1 = 0

$\left[\because 1^2-1=1-1=0\right]$

$\therefore$ The statement q is not true.

 

Q5.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: √11 $$ is a rational number.

A.5. (i) False: As per the definition of a chord, it should intersect the circle at two distinct points.

(ii) False: The centre of a circle only bisects the diameter, which is the chord of the circle.

(iii) True: The equation of an ellipse is  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If we put a = b = 1, then

x² + y² = 1, which is an equation of a circle

(iv) True: Given x > y. Multiplying by –1, –x < – y.

(v) False: Since | | is a prime number, therefore √11 $$ is irrational.

 

Miscellaneous Exercise

Q1. 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 exists a number x such that 0 < x < 1.

A.1 (i) The negation of statement p is as follows.

There exists a positive real number x, such that x – 1 is not positive.

(ii) The negation of statement q is

There exists a cat that does not scratch.

(iii) The negation of statement r is

There exist a real number x, such that neither x>1 nor x<1.

(iv) The negation of statement s is as follow.

There does not exist a number x, such that 0

 

Q2. 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.

A.2. (i) Statement p can be written as follows.

If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is given below

If a positive integer has no divisor other than/and itself, then it is prime.

The contrapositive of the statement is given below

If positive integer has divisors other than 1 and itself, then it is not prime.

(ii) The given statement can be written as follows.

If it is a sunny day, then I go to a beach.

The converse of the statement is given below.

If I go to a beach, then it is sunny day.

The contrapositive of statement is given below.

If I do not go to a beach, then it is not a sunny day.

(iii) The converse of statement is

If you fell thirsty, then it is not outside.

The contrapositive of statement ‘r’ is

If you do not feel thirsty, then it is not hot outside.

 

Q3. Write each of the statements in the form “if p, then q”

(i) p: It is necessary to have a password to log on to the server.

(ii) q: There is traffic jam whenever it rains.

(iii) r: You can access the website only if you pay a subscription fee.

A.3. (i) The statement p in the form ‘if then’ is as follows.

If you log on to the server, then you have password.

(ii) The statement q in the form ‘if then’ is as follows.

If it rains, then there is a traffic jam.

(iii) The statement r in the form ‘if them’ is as follows.

If you can access the website, then you pay a subscription fee.

 

Q.4. Rewrite each of the following statements in the form “p if and only if q”

(i) p: If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

A.4. (i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

 

Q5. Given below are two statements

p : 25 is a multiple of 5.

q : 25 is a multiple of 8.

Write the compound statements connecting these two statements with “And” and

“Or”. In both cases check the validity of the compound statement.

A5. The compound statement with ‘And’ is as follows 25 is a multiple of 5 and 8.

This is false statement because 25 is not a multiple of 8.

The compound statement with ‘Or’ is as follows 25 is a multiple of 5 or 8.

This is true statement because 25 is not a multiple of 8 but it is a multiple of 5.

 

Q6. Check the validity of the statements given below by the method given against it.

(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).

(ii) q: If n is a real number with n > 3, then n² > 9 (by contradiction method).

A.6. (i) The given statement is as follows.

p : the sum of an irrational number and a rational number is irrational.

Let us assume that, p is false.

i.e. The sum of an irrational number and rational number is rational.

Hence, √a $+\frac{b}{c}=\frac{d}{e},$ where √a $$ is irrational and b, c, d, e are integers.

$\frac{d}{e}-\frac{b}{c}$ is a rational number and √a $$ is an irrational number.

This is a contradiction. Therefore, our assumption is wrong.

Therefore, the sum of an irrational number and a rational number is rational.

Thus, the given statement is true.

(ii) The given statement, q is as follows.

If n is a real number with n>3, then n²>9.

Let us assume than n is a real number with n>3, but n²>9 is not true.

i.e. n² < 9

Then n > 9 and n is real number.

Squaring on both sides, we get

n²>(3)²

$\Rightarrow n^2>9,$ which is contradiction.

Since we have assumed that n²<9.

Thus, the given statement is true. That is, if n is a real number with n>3, then n²>9.

 

Q7. Write the following statement in five different ways, conveying the same meaning.

p: If a triangle is equiangular, then it is an obtuse angled triangle.

A.7. The given statement can be written in five different ways: –

(i) A triangle is equiangular implies that it is an obtuse angled triangle.

(ii) A triangle is equiangular only if the triangle is an obtuse angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse angled triangle.

(iv) For a triangle to be an obtuse angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is an obtuse angled triangle, then the triangle is not equiangular.

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