NCERT Solutions for Class 11 Maths Chapter 14 - Mathematical Reasoning

Exercise 14.1

1. Which of the following sentences are statements?

(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 numberis an even number.

(v) The sides of a quadrilateral have equal lengths.

(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.

Sol. (i) A month has 30 or 31 days. It is false to say that a month has 35 days. Hence, it is a
statement.

(ii) Mathematics may be difficult for one but may be easy for the other. Hence, it is not a statement.

(iii) It is true that sum of 5 and 7 is greater than 10. Hence, it is a statement.

(iv) The square of a number may be even or it may be odd. Square of an odd number is always odd and square of an even number is always even.

e.g., 4² = 16 (even) and 5² = 25 (odd). Hence, it is not a statement.

(v) A quadrilateral may have equal lengths as it may be a rhombus or a square or the quadrilateral may have unequal sides like parallelogram). Hence, it is not a statement.

(vi) It is an oder. Hence, it is not a statement.

(vii) It is false because product of (– 1) and 8 is not 8. It is – 8 hence, it is a statement.

(viii) It is true that sum of interior angles of a triangle is 180°. Hence, it is a statement.

(ix) It is a windy day. It is not clear that about which day it is said. Thus, it can’t be concluded whetherit is true or flase. Hence, it is not a statement.

(x) It is true that all real numbers are complex numbers. All real number can be expressed as a + ib. Hence it is a statement.

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

Sol. (i) Everyone in this room is bold. This is not a statement because from the context it is not clear which room is referred here and the term bold is not clearly defined.

(ii) She is an engineering student. This is also not a statement because who she is.

(iii) sin² θ is always greater than 1/2. This is not a statement because we can’t say whether the statement is true or not.

Exercise 14.2

1. Write the negations of the following statements:

(i) Chennai is the capital of Tamil Nadu.

$$\textbf{(ii)}\space\sqrt{\textbf{2}}\space\textbf{is not a complex number.}$$

(iii) All triangles are not equilateral triangle.

(iv) The number 2 is greater than 7.

(v) Every natural number is an integer.

Sol. Negations of the given statement are as follows:

(i) Chennai is not the capital of Tamil nadu.

$$\text{(ii)}\space\sqrt{2}\space\text{is a complex number.}$$

(iii) All triangles are equilateral triangle.

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

(v) Every natural number is not an integer.

2. 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.

Sol. (i) The negation of the statement. The number x is not a rational number i.e., The number x is a rational number. The second statement is the same as ‘x is not irrational’. Hence, both statements are negations of each other.

(ii) The negation of the statement. ‘The number x is not a rational number is ‘The number is x is not a rational number’ or we can say x is an irrational number. The second statement is same. Therefore, both are negations of each other.

3. 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.

Sol. (i) p : Number 3 is prime.

q : Number 3 is odd.

p, q are true.

(ii) p : All integers are positive.

q : All integers are negative.

Here both statements are false.

(iii) p : 100 is divisible by 3.

q : 100 is divisible by 11.

r : 100 is divisible by 5.

p is false, q is false and r is true.

∴ p, q both are false and r is true.

Exercise 14.3

1. 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 anintegerispositive 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.

Sol. (i) Connecting word is ‘and’.

p : All rational numbers are real.

q : All real numbers are not complex.

(ii) The connecting word is ‘or’.

p : The square of an integer is positive.

q : The square of an integer is negative.

(iii) Connecting word is ‘and’.

p : The sand heats up quickly in the sun.

q : The sand does not cool down fast at night.

(iv) Connecting word is ‘and’.

p : x = 2 is the root of equation 3x² – x – 10 = 0

q = : x = 3 is the root of equation 3x² – x – 10 = 0.

2. 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.

Sol. (i) Quantifier: The exists.

p : There exists a number which is equal to its square.

p(~ p): There does not exist a number which is equal to its square.

(ii) Quantifier: For every.

p : For every real number x, x is less than (x + 1).

~ p : There exists a real number x such that x is not less than (x + 1).

(iii) Quantifier: There exists.

p : There exists a capital for every state in India.

~ p : There does not exist a capital for every state in India.

3. Check whether the following pairs of statement 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.

Sol. Statement (i) and (ii) are not negation of each other

4. 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.

Sol. (i) When sun rises, the moon sets. one of the happenings will take place. Hence, ‘or’ is exclusive.

(ii) To apply for a driving licence either a rational card or a passport or both can be used.

∴ ‘or’ is inclusive.

(iii) All integers are positive or negative. An integers cannot be both positive or negative at a time.

∴ Here, ‘or’ is exclusive.

Exercise 14.4

1. 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.

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

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

(iii) If the square of a natural numberis not odd, then the natural number is also not odd.

(iv) For a natural number to be odd, its necessary that its square is odd.

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

2. Write the contrapositive and converse of the following statments:

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

(ii) If the two lines are parallel, then they do not interest 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) If x is an even number implies that x is divisible by 4.

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

Converse statement: If x is odd, then x is a prime number.

(ii) Contrapositive statement: If two straight lines interested ni a plane, then the lines are not parallel.

Converse statement: If two lines do not intersect in the same plane, then two lines are parallel.

(iii) Contrapositive statement: If the temperature of something is not low, then it is not cold.

Converse statement: If something has low temperature, then it is cold.

(iv) Contrapositive statement: If you know how to reason deductively, then you can comprehend geometry.

Converse statement: If you do not know how to reason deducting, then you can not comprehand geometry.

(v) Contrapositive statement: If x is not divisible by 4, then x is not an even number.

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

3. Write each of the following statements in the form ‘if-then’:

(i) You get a job implies that your credentials are good.

(ii) The bannana tree will bloom, if it stays warm for a month.

(iii) A quadrilateral is a parallelogram, if its diagonal bisect each other.

(iv) To get an A+ in the class, it is necessary that you do all the exercises of the book.

Sol. (i) If you get a job, then your credientials are good.

(ii) If the bannana tree stays warm form a month. Then the tree will bloom.

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

(iv) If you get on A⁺ in the class then, you do all the exercises of the book.

4. Give 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 quadraterial 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 paralleogram.

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

Sol. (a) (i) Contrapositive statement

(ii) Converse statement

(b) (i) Contrapositive statement

(ii) Converse statement

Exercise 14.5

1. 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.

Sol. (i) Direct method

x³ + 4x = 0 or x(x² + 4) = 0

⇒ x = 0, x² + 4 ≠ 0, x ∈ R

(ii) Method of contradiction

Let x ≠ 0 and let it be x = p, p ∈ R and p is a
root of x³ + 4x = 0.

∴ p³ + 4p = 0

⇒ p(p² + 4) = 0

p ≠ 0

Also, p² + 4 ≠ 0

⇒ p = 0

(iii) Method of contrapositive

Let x = 0 is not true and x = p ≠ 0.

∴ p³ + 4p = 0,

p being the root of x² + 4 < 0

or p(p² + 4) = 0

Now, p = 0, also p² + 4 = 0

⇒ p(p + 4) ≠ 0, if p is not true.

∴ x = 0 is the root of x³ + 4x = 0.

2. 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.

Sol. Let a = 1, b = – 1, a² = b² = 1 but a ≠ b

Here, the given statement is not true.

3. 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.

Sol. Let x is not even i.e.,

x = 2n + 1

∴ x² = (2n + 1)² = 4n² + 4n + 1

= 4(n² + n) + 1

∴ 4(x² + x) + 1 is odd.

i.e., ‘If q is not true, then p is not true’ is proved.

Hence, the given statement is true.

4. 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.

Sol. (i) Let an angle of triangle is 90° + α.

∴ Sum of the angles = 3(90° + α) = 270° + 3α which is greater than 180°.

∴ A triangle having equal angles cannot be obtuse angled triangle.

(ii) The equation x² – 1 = 0 has the root x = 1, which lies between 0 and 2.

∴ The given statement is not true.

5. 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 chord of the circle.

(ii) q : The centre of a circle bsiects 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.

$$\textbf{(v)}\space\textbf{t :}\sqrt{\textbf{11}}\space\textbf{is a rational number.}$$

Sol. (i) False: The end points of radius do not lie on the circle, therefore it is not a chord.

(ii) False: Only diameters are bisected at the centre. Other chords do not pass through the centre. Therefore, centre cannot bisect them.

(iii) True: Equation of ellipse is

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$

When b = a, the equation becomes = 1

⇒ x² + y² = a²

which is the equation of circle.

(iv) True: By the rule of inequality, if x and y are integers and x > y, then – x < – y, e.g., 3 > 2
⇒ – 3 < – 2.

(v) False: Since, 11 is prime number therefore,

$$\sqrt{\textbf{11}}\space\text{is rational}.$$

(·.· Prime numbers are not perfect squares)

Miscellaneous Exercise

1. 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.

Sol. (i) ~ p : There exists atleast one positive real number x, such that (x – 1) is not positive.

(ii) ~ q : All cats do not scratch.

(iii) ~ r : There exist atleast one number x such that neither x > 1 nor x < 1.

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

2. 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 beach whenever it is a sunny day.

(iii) r : If it is hot outside, then you feel thirsty.

Sol. (i) Contrapositive: If a positive integer has divisor other than 1 and itself, then it is not prime.

Converse: If a positive intger has no divisors other than 1, then it is prime.

(ii) Contrapositive: If it is not a sunny day, then I do not go to beach.

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

(iii) Contrapositive: If you do not feel thirsty, then it is not hot outside.

Converse: If you feel thristy, then it is not outside.

3. Write each of the statement in the form ‘if p then q’.

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

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

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

Sol. (i) If you log on the server, then you have a password.

(ii) If it rains, then there is a traffic jam.

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

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, if it is necessary and sufficient that you do all the home work regularly.

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

Sol. (i) You watch a television if and only if your mind is free.

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

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

5. 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 the cases, check the validity of the compound statement.

Sol. (i) Compound statement with ‘and’ 25 is a multiple of 5 and 8.

This is a false statement, since p and q both are not true.

[∵ 25 is divisible by 5 but not divisible by 8]

(ii) Compound statement with ‘or’ 25 is a multiple of 5 or it not is a multiple of 8.

This is a true statement.

6. 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).

$$\textbf{Sol.}\space\text{(i) Let}\space\sqrt{p}\space\text{be irrational number and}\\\text{ q be rational number.}\\\text{ Their sum =}q + \sqrt{p}$$

Let it is not irrational. Then, it is a rational number.

$$\text{i.e\space}\space q+\sqrt{p}\space \text{= where a and b are co-prime.}$$

$$\Rarr\space\sqrt{p}=\frac{a}{b}-q\\\therefore\space\text{L.H.S}=\sqrt{p}=\text{An irrational number}\\\text{R.H.S}=\frac{a}{b}-q\space\text{= A rational number}$$

It is a contradiction.

Hence, the sum of rational and irrational number is irrational.

(ii) Let n > 3 and n² ≤ 9

Putting n = 3 + p, p ∈ R⁺

⇒ n² = (3 + p)²

⇒ n² = 9 + 6p + p² = 9 + p(6 + p)

⇒ n² > 9

which is a contradiction.

Therefore, the given statement is true.

∴ If n > 3, then n² > 9

7. 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.