NCERT Solutions for Class 11 Mathematics Chapter 14 

Q.1 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.

Ans

(i) This sentence is false because there is no month of 35 days. Hence, it is a statement.
(ii) Mathematics may be easy or difficult for some people. This sentence is not always true. So, it is not a statement.
(iii) It is true that the sum of 5 and 7 is greater than 10, so it is a statement.
(iv) It is false that square of a number is even number because square of even number is even and that of an odd number is odd. So, it is a statement.
(v) It is not true for all quadrilaterals. So, it is a statement.
(vi) It is an exclamation. So, it is not a statement.
(vii) It is false because product of (– 1) and 8 is – 8. So, it is a statement.
(viii) It is true because sum of interior angles of triangle is 180°. So, it is a statement.
(ix) This sentence is not a statement because this sentence is not correct for each day.
(x) It is true that all real numbers are complex numbers and it can be written in the form of a + i.0. So, it is a statement.

Q.2 Give three examples of sentences which are not statements. Give reasons for the answers.

Ans

The three examples of sentences which are not statements are given below:

1. Delhi is far from here.
Reason: The word ‘here’ does not convey a particular location.

2. There are few students in that school.
Reason: The word ‘that’ does not indicate a definite school.

3. It is raining today.
Reason: The word ‘today’ does not indicate a particular date.

4. Today is Saturday.

Q.3 Write the negation of the following statements:

(i) Chennai is the capital of Tamil Nadu.
(ii)

$\sqrt{2}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{complex}\mathrm{number}.$

(iii) All triangles are not equilateral triangle.
(iv) The number 2 is greater than 7.
(v) Every natural number is an integer.

Ans

(i) Negation: Chennai is not the capital of Tamil Nadu.
(ii) Negation:

$\sqrt{2}\mathrm{is}\mathrm{a}\mathrm{complex}\mathrm{number}.$

(iii) Negation: All triangles are equilateral triangles.
(iv) Negation: The number 2 is not greater than 7.
(v) Negation: Every natural number is not an integer.

Q.4 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.

Ans

(i) The negation of the given statement is “The number x is a rational number.”
And if a number is not a rational number it means that number is a rational number.
Thus, the given statements are negation of each other.

(ii) The negation of “The number x is a rational number” is “The number x is a irrational number”, which is same as second statement.
Thus, the given statements are negation of each other.

Q.5 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.

Ans

(i) The component statements are:
p: Number 3 is prime.
q: Number 3 is odd.
Both statements p and q are true.

(ii) The component statements are:
p: All integers are positive.
q: All integers are negative.
Both statements p and q are false.

(iii) The component statements are:
p: 100 is divisible by 3.
q: 100 is divisible by 11.
r: 100 is divisible by 5.
Statements p and q are false. Statement r is true.

Q.6 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.

Ans

(i) The connecting word of given components statements is “and”. The component statements are:
p: All rational numbers are real.
q: All real numbers are not complex.

(ii) The connecting word of given components statements is “or”. The component statements are:
p: Square of an integer is positive.
q: Square of an integer is negative.

(iii) The connecting word of given components statements is “and”. The component statements are:
p: The sand heats up quickly in the Sun.
q: The sand does not cool down fast at night.

(iv) The connecting word of given components statements is “and”. The component statements are:
p: x = 2 is the roots of the equation 3x² – x – 10= 0.
q: x = 3 is the roots of the equation 3x² – x – 10= 0.

Q.7 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.

Ans

(i) The quantifier is “There exists” in the given statement.
Negation: There does not exist a number which is equal to its square.

(ii) The quantifier is “For every” in the given statement.
Negation: For every real number x, x is not less than x + 1.

(iii) The quantifier is “There exists” in the given statement.
Negation: There does not exist a capital for every state in India.

Q.8 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.

Ans

No, the following pair of statements are not negation of each other. The negation of statement (i) is:

$\mathrm{x}+\mathrm{y}\ne \mathrm{y}+\mathrm{x}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{every}\mathrm{real}\mathrm{numbers}\mathrm{x}\mathrm{and}\mathrm{y}.$

While the sense of given statement (ii) is different.

Q.9 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.

Ans

(i) Here “Or” is exclusive because both events do not happen simultaneously.
(ii) Here “Or” is inclusive because for applying a driving licence, a person can have both a ration card and a passport.
(iii) Here “Or” is exclusive because all integers are neither positive nor negative.

Q.10 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.

Ans

The given statement can be written in 5 different ways as follows:

(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.

Q.11 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.

Ans

(i) Contrapositive:
If x is not odd, then x is not a prime number.
Converse: If the number x is odd, then x is a prime number.

(ii) Contrapositive:
If the two lines intersect in the same plane, then they are not parallel.
Converse: If two lines do not intersect in the same plane, then they are parallel.

(iii) Contrapositive:
If something has not low temperature, then it is not cold.
Converse: If something is at low temperature, then it is cold.

(iv) Contrapositive:
If you do 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:
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.

Q.12 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.

Ans

(i) If your credentials are good, then you get a job.
(ii) If it stays warm for a month, then the banana trees will bloom.
(iii) If diagonals of a quadrilateral bisect each other, then it is a parallelogram.
(iv) If you do all the exercises of the book, the you will get A⁺ in the class.

Q.13 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.

Ans

(a) (i) Given statement is Contrapositive of statement (a).
(ii) Given statement is Converse of statement (a).

(b) (i) Given statement is Contrapositive of statement (b).
(ii) Given statement is Converse of statement (b).

Q.14 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.

Ans

p: If x is an integer and x² is even, then x is also even.
Let q: x is an integer and x² is even.
r: x is even.
Let x is not even.
x is not even then x² is also not even.
Therefore, statement q is false.
Thus, the given statement p is true.

Q.15 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.

Ans

(i) The given statement is of the form “If p then r”.
q: All the angles of a triangle are equal.
r: The triangle is an obtuse-angled triangle.
The statement p has to be proved false.
It is known that the sum of all angles of a triangle is 180°.
Therefore, if all the three angles are equal then each angle is 60°, which is not an obtuse angle.
In an equilateral triangle, the measurement of all angles is equal. There is no obtuse angle.
So, this triangle is not an obtuse angled-triangle.
Thus, it is concluded that the given statement p is false.

(ii) The given statement is as:
q: The equation x² –1 = 0 does not have a root lying between 0 and 2.
Let x² – 1 = 0
x² = 1
x = ± 1
One root of the equation x² – 1= 0 is 1, which lies between 0 and 2.
Thus, the given statement q is false.

Q.16 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 : √1 is a rational number.

Ans

(i) False, because chord is a line segment which joins the two points on the circumference,
while radius is a line segment, which joins centre of the circle to any point on the circumference.

(ii) False, centre of circle bisects diameter only. It does not bisect each chord because other chords do not pass through the centre.

(iii) True, If the length of major axis and minor axis is same, then ellipse is called circle.

$\begin{array}{l}\mathrm{For}\text{example:}\\ \mathrm{Equation}\text{of ellipse is:}\\ \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1\\ \mathrm{If}\text{a=b, then}\\ \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{a}}^2}=1\\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2\\ \mathrm{Which}\text{is the equation of circle.}\end{array}$

(iv) True,
For example: Let two numbers are 4 and 5 as 5 > 4, then – 4 > – 5.

(v) False, 11 is a prime number so its factors are 1 and 11. So, square root of prime number is irrational number.
Therefore,

$\sqrt{11}$

is an irrational number.
Thus, the given statement t is false.

Q.17 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.

Ans

(i) Negation (p): There exists a real positive number x, such that (x – 1) is not a positive.

(ii) Negation (q): All cats do not stratch.

(iii) Negation (r): For every real number x, neither x > 1 nor x < 1.

(iv) Negation (s): There does not exist a number x such that 0 < x < 1.

Q.18 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

Ans

(i) Converse: If a positive integer has no divisors other than 1 and itself, then it is prime number only.
Contrapositive: If a positive number has divisors other than 1 and itself, then it is not a prime number.

(ii) Converse: If it is a sunny day, then I go to a beach.
Contrapositive: If it is not a sunny day, then I don’t go to a beach.

(iii) Converse: You feel thirsty if it is hot outside.
Contrapositive: You don’t feel thirsty if it is not hot outside.

Q.19 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 subsciption fee.

Ans

(i) If you log on to the server, then you have a password.
(ii) If it rains, then there is a traffic jam.
(iii) If you can access the website, then you pay a subscription fee.

Q.20 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.

Ans

(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.

Q.21 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.

Ans

Compound statement by using ‘And’:
25 is a multiple of 5 and 8.
It is false because 25 is not a multiple of 8.
Compound statement by using ‘Or’:
25 is a multiple of 5 or 8.
It is true because 25 is multiple of 5.

Q.22 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).

Ans

$\begin{array}{l}\left(\mathrm{i}\right)\text{p}:\text{The sum of an irrational number and a}\mathrm{}\text{\hspace{0.17em}rational number}\\ \text{s irrational.}\\ \mathrm{Let}\text{p is false, then the sum of an irrational number and a}\\ \text{rational numbver is rational.}\\ \text{Then,}\\ \frac{\sqrt{3}}{2}+\frac{1}{2}=\frac{\mathrm{a}}{\mathrm{b}}\left(\mathrm{Let}\right)\end{array}$ $\begin{array}{l}\Rightarrow \frac{\sqrt{3}}{2}=\frac{\mathrm{a}}{\mathrm{b}}-\frac{1}{2}\\ \mathrm{Since},\sqrt{3}\text{is irrational number.So,}\\ \frac{\sqrt{3}}{2}\ne \frac{\mathrm{a}}{\mathrm{b}}-\frac{1}{2}\\ \mathrm{Therefore},\text{our assumption is wrong.}\\ \text{Thus, p is true i.e., the sum of an irrational number and a}\mathrm{}\\ \text{\hspace{0.17em}rational number is irrational.}\end{array}$

(ii) q: If n is a real number with n > 3, then n² > 9
Let q be false statement and it is as given below:
If n is a real number with n>3 then n² > 9 is not true.
i.e., n² < 9
Since, n > 3
Squaring both sides, we get
n² > 3²
i.e., n² > 9
This is a contradiction. So, our assumption as
n² < 9 is wrong.
Thus, the given statement q is true i.e., If n is a real number with n > 3, then n² > 9.

Q.23 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.

Ans

The given statement p can be written in 5 different ways as follows:

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.
(ii) A triangle is equiangular if it is an obtuse-angled triangle.
(iii) A triangle is not equiangular if it is not an obtuse-angled triangle.
(iv) A triangle is an obtuse-angled triangle if and only if it is an equilateral triangle.
(v) A triangle is not an obtuse-angled triangle if it is not an equilateral triangle.

Q.24 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.

Ans

The given statement p can be written in 5 different ways as follows:

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.
(ii) A triangle is equiangular if it is an obtuse-angled triangle.
(iii) A triangle is not equiangular if it is not an obtuse – angled triangle.
(iv) A triangle is an obtuse-angled triangle if and only if it is an equilateral triangle.
(v) A triangle is not an obtuse-angled triangle if it is not an equilateral triangle.

Q.25 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.

Ans

The given statement p can be written in 5 different ways as follows:

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.
(ii) A triangle is equiangular if it is an obtuse-angled triangle.
(iii) A triangle is not equiangular if it is not an obtuse – angled triangle.
(iv) A triangle is an obtuse-angled triangle if and only if it is an equilateral triangle.
(v) A triangle is not an obtuse-angled triangle if it is not an equilateral triangle.