NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning are part of NCERT Solutions for Class 11 Maths. Here we have given NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning.

Board	CBSE
Textbook	NCERT
Class	Class 11
Subject	Maths
Chapter	Chapter 14
Chapter Name	Mathematical Reasoning
Exercise	Ex 14.1, Ex 14.2, Ex 14.3, Ex 14.4, Ex 14.5
Number of Questions Solved	18
Category	NCERT Solutions

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

Chapter 14 Mathematical Reasoning Exercise 14.1

Question 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.
Solution:
(i) This sentence is false since the maximum number of days in a month can never exceed 31. Therefore, this sentence is a statement.
(ii) This sentence is subjective in the sense that for those who hate mathematics, it is difficult but for others it may not be. This means that this sentence is not always true. Hence it is not a statement.
(iii) This sentence is true as sum of 5 and 7 is greater than 10. Hence it is a statement.
(iv) This sentence is subjective in the sense that it depends on the number that is being squared. Hence it is not a statement.
(v) This sentence is sometimes true and sometimes false since sides in squares and rhombuses have equal length whereas rectangles and trapeziums have unequal length. Hence it is not a statement.
(vi) This sentence is an order and so, it is not a statement.
(vii) This sentence is false as product of (-1) and 8 is -8. So, it is a statement.
(viii) This sentence is true and therefore it is a statement.
(ix) It is not clear from the context which day is referred. Therefore, it is not a statement.
(x) All real numbers can be written in the form of complex numbers. So, this sentence is true and it is a statement.

Question 2.
Give three examples of sentences which are not statements. Give reasons for the answers.
Solution:
(i) Who are you?
This sentence is an interrogative sentence. Hence, it is not a statement.
(ii) May God bless you!
This sentence is an exclamatory sentence. Hence, it is not a statement.
(iii) How are you?
This sentence is an interrogative sentence. Hence, it is not a statement.

Chapter 14 Mathematical Reasoning Exercise 14.2

Question 1.
Write the negation of the following statements:
(i) Chennai is the capital of Tamil Nadu.
(ii) 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.
Solution:
(i) Negation of statement is : Chennai is not the capital of Tamil Nadu.
(ii) Negation of statement is: is a complex number.
(iii) Negation of statement is : All triangles are equilateral triangles.
(iv) Negation of statement is : The number 2 is not greater than 7.
(v) Negation of statement is : Every natural number is not an integer.

Question 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.
Solution:
(i) Let p: The number x is not a rational number.
q : The number x is not an irrational number.
Now, ~p : The number x is a rational number. ~q : The number x is an irrational number.
∴ ~p = q and ~q = p
Thus, p and q are negations of each other.

(ii) Let p: The number x is a rational number.
q : The number x is an irrational number.
Now, ~p: The number x is not a rational number.
~q: The number x is not an irrational number.
∴ ~p = q and ~q = p
Thus, p and q are negations of each other.

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

(ii) The component statements are:
p : All integers are positive.
q : All integers are negative.
Both the component 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.
The component statements p and q are false whereas r is true.

Chapter 14 Mathematical Reasoning Exercise 14.3

Question 1.
For each of thefollowing 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.
Solution:
(i) The compound statement has the connecting word ‘and’. Component statements are
p: All rational numbers are real.
q: All real numbers are not complex.

(ii) The compound statement has the connecting word ‘or’. Component statements are:
p: Square of an integer is positive.
q: Square of an integer is negative.

(iii) The compound 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 compound statement has the connecting word ‘and’. Component statements are:
p: x- 2 is a root of the equation 3x² – x – 10 = 0.
q: x = 3 is a root of the equation 3x² – x – 10 = 0.

Question 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.
Solution:
(i) Here the quantifier is ‘there exists’.
The negation of statement is : There does not exist a number which is equal to its square.
(ii) Here the quantifier is ‘for every’
The negation of statement is : For atleast one real number x, x is not less than x + 1.
(iii) Here the quantifier is ‘there exists’
The negation of statement is : There exists a state in India which does not have a capital.

Question 3.
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.
Solution:
Let p: x + y = y + x is true for every real numbers x and y.
q: There exists real numbers x and y for which
x+y=y + x.
Now, ~p: There exists real numbers x and y for which x + y ≠ y + x.
Thus, ~p ≠ q.

Question 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.
Solution:
(i) This statement makes use of exclusive “or”. Since when sun rises, moon does not set during day-time.
(ii) This statement makes use of inclusive ‘or’. Since you can apply for a driving licence even if you have a ration card as well as a passport.
(iii) This statement makes use of exclusive ‘or’. Since a integer is either positive or negative, it cannot be both.

Chapter 14 Mathematical Reasoning Exercise 14.4

Question 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.
Solution:
(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.

Question 2.
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.
Solution:
(i) The contra positive of given statement is:
If a number 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) The contra positive of given statement is:
If two lines intersect in the same plane, then they are not parallel.
The converse of given statement is:
If two lines do not intersect in the same plane, then they are parallel.

(iii) The contra positive of given statement is:
If something is not at low temperature, then it is not cold.
The converse of given statement is:
If something is at low temperature, then it is cold.

(iv) The contra positive of given statement is:
If you know how 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 cannot comprehend geometry.

(v) The contra positive 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.

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

Question 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.
Solution:
(a)
(i) contrapositive
(ii) converse

(b)
(i) contrapositive
(ii) converse

Chapter 14 Mathematical Reasoning Exercise 14.5

Question 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.
Solution:
The given compound statement is of the form “if p then q”
p: x ϵ R such that x³ + 4x = 0
q: x = 0
(i) Direct method:
We assume that p is true, then
x ϵ R such that x³ + 4x = 0
⇒ x ϵ R such that x(x² + 4) = 0
⇒ x ϵ R such that x = 0 or x² + 4 = 0
⇒ x = 0 => 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 ϵ R such that x³ + 4x = 0
⇒ x ϵ R such that x(x² + 4) = 0
⇒ x ϵ R such that x = 0 or x² + 4 = 0
⇒ x = 0.
which is a contradiction. So, 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 ≠ 0
x ϵ R such that x³ + 4x = 0
⇒ x ϵ R such that x = 0 or x² + 4 = 0
∴ statement q is false, so x ≠ 0. So, we have,
x ϵ R such that x² = -2
Which is not true for any x ϵ R.
⇒ p is false
So, when q is false, p is false.
Thus, the given compound statement is true.

Question 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.
Solution:
The given compound statement is of the form “if p then q”
We assume that p is true, then a, b ⍷ R such that a² = b²
Let us take a = -3 and b = 3
Now, a² = b², but a ≠ b
So, when p is true, q is false.
Thus, the given compound statement is not true.

Question 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.
Solution:
The given compound statement is of the form “if p then q”
p: x ϵ Z and x² is even.
q: x is an even integer.
We assume that q is false, then x is not an even integer.
⇒ x is an odd integer.
⇒ x² is an odd integer.
⇒ p is false
So, when q is false, p is false.
Thus, the given compound statement is true.

Question 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.
Solution:
(i) Since the triangle is an obtuse angled triangle then 0 > 90°.
Let 0 = 100°
Also all the angles of the triangle is equal.
∴ Sum of all angles of the triangle is 300°, which is not possible.
Thus, the given compound statement is not true,

(ii) We see that x = 1 is a root of the equation x² – 1 = 0, which lies between 0 and 2. Thus, the given compound statement is not true.

Question 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 ofa 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. is a rational number.
Solution:
(i) A chord of a circle is a line whose two end points 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 if we put a = b, then we get an equation of circle.
Thus, the given statement is true.
(iv) It is given that x, y ϵ Z such that x > y. Multiplying both sides by negative sign, we have
x, y ϵ Z such that -x < -y.
Thus, the given statement is true.
(v) Since cannot be expressed in the form , where a and b are integers and b ≠ 0. Thus, the given statement is false.

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