# NCERT solutions for Mathematics Exemplar Class 11 chapter 14 - Mathematical Reasoning [Latest edition]

## Chapter 14: Mathematical Reasoning

Solved ExamplesExercise
Solved Examples [Pages 254 - 261]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 14 Mathematical Reasoning Solved Examples [Pages 254 - 261]

Solved Examples | Q 1.(i) | Page 254

Which of the following statements are compound statements

“2 is both an even number and a prime number”

Solved Examples | Q 1.(ii) | Page 254

Which of the following statements are compound statements

“9 is neither an even number nor a prime number”

Solved Examples | Q 1.(iii) | Page 254

Which of the following statements are compound statements

“Ram and Rahim are friends”

Solved Examples | Q 2.(a) | Page 255

Identify the component statements and the connective in the following compound statements.

It is raining or the sun is shining.

Solved Examples | Q 2.(b) | Page 255

Identify the component statements and the connective in the following compound statements.

2 is a positive number or a negative number.

Solved Examples | Q 3.(i) | Page 255

Translate the following statements in symbolic form

2 and 3 are prime numbers

Solved Examples | Q 3.(ii) | Page 255

Translate the following statements in symbolic form

Tigers are found in Gir forest or Rajaji national park.

Solved Examples | Q 4 | Page 256

Write the truth value of the following statements.
(i) 9 is an even integer or 9 + 1 is even.
(ii) 2 + 4 = 6 or 2 + 4 = 7
(iii) Delhi is the capital of India and Islamabad is the capital of Pakistan.
(iv) Every rectangle is a square and every square is a rectangle
(v) The sun is a star or sun is a planet.

Solved Examples | Q 5 | Page 256

Write negation of the statement
“Everyone who lives in India is an Indian”

Solved Examples | Q 6.(a) | Page 256

Write the negation of the following statements:
p: All triangles are equilateral triangles.

Solved Examples | Q 6.(b) | Page 256

Write the negation of the following statements:
q: 9 is a multiple of 4.

Solved Examples | Q 6.(c) | Page 256

Write the negation of the following statements:
r: A triangle has four sides.

Solved Examples | Q 7.(i) | Page 257

Write the negation of the following statements:
Suresh lives in Bhopal or he lives in Mumbai.

Solved Examples | Q 7.(ii) | Page 257

Write the negation of the following statements:
x + y = y + x and 29 is a prime number.

Solved Examples | Q 8.(i) | Page 257

Rewrite the following statements in the form of conditional statements:

Mohan will be a good student if he studies hard.

Solved Examples | Q 8.(ii) | Page 257

Rewrite the following statements in the form of conditional statements:

Ramesh will get dessert only if he eats his dinner.

Solved Examples | Q 8.(iii) | Page 257

Rewrite the following statements in the form of conditional statements:

When you sing, my ears hurt.

Solved Examples | Q 8.(iv) | Page 257

Rewrite the following statements in the form of conditional statements:

A necessary condition for Indian team to win a cricket match is that the selection committee selects an all-rounder.

Solved Examples | Q 8.(v) | Page 257

Rewrite the following statements in the form of conditional statements:

A sufficient condition for Tara to visit New Delhi is that she goes to the Rashtrapati Bhawan.

Solved Examples | Q 9 | Page 259

Express in English, the statement p → q, where
p: It is raining today
q: 2 + 3 > 4

Solved Examples | Q 10 | Page 259

Translate the following statements in symbolic form:
If x = 7 and y = 4” then x + y = 11.

Solved Examples | Q 11 | Page 259

Form the biconditional of the following statements :
p: Today is 14th of August
q: Tomorrow is Independence day

Solved Examples | Q 12 | Page 259

Translate the following biconditional into symbolic form:
“ABC is an equilateral triangle if and only if its each interior angle is 60°”

Solved Examples | Q 13.(i) | Page 259

Identify the quantifiers and write the negation of the following statements:
There exists a number which is equal to its square.

Solved Examples | Q 13.(ii) | Page 259

Identify the quantifiers and write the negation of the following statements:
For all even integers x, x2 is also even.

Solved Examples | Q 13.(iii) | Page 259

Identify the quantifiers and write the negation of the following statements:
There exists a number which is a multiple of 6 and 9.

Solved Examples | Q 14 | Page 260

Show that the following statement is true.
p: For any real numbers x, y if x = y, then 2x + a = 2y + a when a ∈ Z.

Solved Examples | Q 15.(i) | Page 260

Check the validity of the statements:
r: 100 is a multiple of 4 and 5.

Solved Examples | Q 15.(ii) | Page 260

Check the validity of the statements:
s: 60 is a multiple of 3 or 5.

#### Objective Type Questions from 16 to 18

Solved Examples | Q 16 | Page 260

Which of the following is a statement?

• Roses are black

• Be punctual

• Do not tell lies

Solved Examples | Q 17 | Page 261

The negation of the statement “It is raining and weather is cold.” is ______.

• It is not raining and weather is cold

• It is raining or weather is not cold

• It is not raining or weather is not cold

• It is not raining and weather is not cold

Solved Examples | Q 18 | Page 261

Which of the following is the converse of the statement?
“If Billu secure good marks, then he will get a bicycle.”

• If Billu will not get bicycle, then he will not secure good marks

• If Billu will get a bicycle, then he will secure good marks

• If Billu will get a bicycle, then he will not secure good marks

• If Billu will not get a bicycle, then he will secure good marks

Exercise [Pages 261 - 269]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 14 Mathematical Reasoning Exercise [Pages 261 - 269]

Exercise | Q 1.(i) | Page 261

Which of the following sentences are statements? Justify

A triangle has three sides.

• True

• False

Exercise | Q 1.(ii) | Page 261

Which of the following sentences are statements? Justify

0 is a complex number.

• True

• False

Exercise | Q 1.(iii) | Page 261

Which of the following sentences are statements? Justify

Sky is red.

• True

• False

Exercise | Q 1.(iv) | Page 261

Which of the following sentences are statements? Justify

Every set is an infinite set.

• True

• False

Exercise | Q 1.(v) | Page 261

Which of the following sentences are statements? Justify

15 + 8 > 23

• True

• False

Exercise | Q 1.(vi) | Page 261

Which of the following sentences are statements? Justify

y + 9 = 7.

• True

• False

Exercise | Q 1.(vii) | Page 261

Which of the following sentences are statements? Justify

Exercise | Q 1.(viii) | Page 261

Which of the following sentences are statements? Justify

Every square is a rectangle.

• True

• False

Exercise | Q 1.(ix) | Page 261

Which of the following sentences are statements? Justify

Sum of opposite angles of a cyclic quadrilateral is 180°.

• True

• False

Exercise | Q 2.(i) | Page 262

Find the component statements of the following compound statements.

Number 7 is prime and odd.

Exercise | Q 2.(x) | Page 262

Write the negation of the following simple statements

Area of a circle is same as the perimeter of the circle.

Exercise | Q 2.(ii) | Page 262

Find the component statements of the following compound statements.

Chennai is in India and is the capital of Tamil Nadu.

Exercise | Q 2.(iii) | Page 262

Find the component statements of the following compound statements.

The number 100 is divisible by 3, 11 and 5.

Exercise | Q 2.(iv) | Page 262

Find the component statements of the following compound statements.

Chandigarh is the capital of Haryana and U.P.

Exercise | Q 2.(v) | Page 262

Find the component statements of the following compound statements.

sqrt(7) is a rational number or an irrational number.

Exercise | Q 2.(vi) | Page 262

Find the component statements of the following compound statements.

0 is less than every positive integer and every negative integer.

Exercise | Q 2.(vii) | Page 262

Find the component statements of the following compound statements.

Plants use sunlight, water and carbon dioxide for photosynthesis.

Exercise | Q 2.(viii) | Page 262

Write the negation of the following simple statements.

A leap year has 366 days.

Exercise | Q 2.(viii) | Page 262

Find the component statements of the following compound statements.

Two lines in a plane either intersect at one point or they are parallel.

Exercise | Q 2.(ix) | Page 262

Find the component statements of the following compound statements.

A rectangle is a quadrilateral or a 5-sided polygon.

Exercise | Q 3.(i) | Page 262

Write the component statements of the following compound statements and check whether the compound statement is true or false.

57 is divisible by 2 or 3.

• True

• False

Exercise | Q 3.(ii) | Page 262

Write the component statements of the following compound statements and check whether the compound statement is true or false.

24 is a multiple of 4 and 6.

• True

• False

Exercise | Q 3.(iii) | Page 262

Write the component statements of the following compound statements and check whether the compound statement is true or false.

All living things have two eyes and two legs.

• True

• False

Exercise | Q 3.(iv) | Page 262

Write the component statements of the following compound statements and check whether the compound statement is true or false.

2 is an even number and a prime number.

• True

• False

Exercise | Q 4.(i) | Page 262

Write the negation of the following simple statements.

The number 17 is prime.

Exercise | Q 4.(ii) | Page 262

Write the negation of the following simple statements.

2 + 7 = 6.

Exercise | Q 4.(iii) | Page 262

Write the negation of the following simple statements.

Violets are blue.

Exercise | Q 4.(iv) | Page 262

Write the negation of the following simple statements.

sqrt(5) is a rational number.

Exercise | Q 4.(v) | Page 262

Write the negation of the following simple statements.

2 is not a prime number.

Exercise | Q 4.(vi) | Page 262

Write the negation of the following simple statements.

Every real number is an irrational number.

Exercise | Q 4.(vii) | Page 262

Write the negation of the following simple statements.

Cow has four legs.

Exercise | Q 4.(viii) | Page 262

Write the negation of the following simple statements

A leap year has 366 days.

Exercise | Q 4.(ix) | Page 262

Write the negation of the following simple statements

All similar triangles are congruent.

Exercise | Q 4.(x) | Page 261

Which of the following sentences are statements? Justify

sin2x + cos2x = 0

• True

• False

Exercise | Q 5.(i) | Page 262

Translate the following statements into symbolic form

Rahul passed in Hindi and English.

Exercise | Q 5.(ii) | Page 262

Translate the following statements into symbolic form

x and y are even integers.

Exercise | Q 5.(iii) | Page 262

Translate the following statements into symbolic form

2, 3 and 6 are factors of 12

Exercise | Q 5.(iv) | Page 263

Translate the following statements into symbolic form

Either x or x + 1 is an odd integer.

Exercise | Q 5.(v) | Page 263

Translate the following statements into symbolic form

A number is either divisible by 2 or 3.

Exercise | Q 5.(vi) | Page 263

Translate the following statements into symbolic form

Either x = 2 or x = 3 is a root of 3x 2 – x – 10 = 0

Exercise | Q 5.(vii) | Page 263

Translate the following statements into symbolic form

Students can take Hindi or English as an optional paper.

Exercise | Q 6.(i) | Page 263

Write down the negation of following compound statements

All rational numbers are real and complex.

Exercise | Q 6.(ii) | Page 263

Write down the negation of following compound statements

All real numbers are rationals or irrationals.

Exercise | Q 6.(iii) | Page 263

Write down the negation of following compound statements

x = 2 and x = 3 are roots of the Quadratic equation x2 – 5x + 6 = 0.

Exercise | Q 6.(iv) | Page 263

Write down the negation of following compound statements

A triangle has either 3-sides or 4-sides.

Exercise | Q 6.(v) | Page 263

Write down the negation of following compound statements

35 is a prime number or a composite number

Exercise | Q 6.(vi) | Page 263

Write down the negation of following compound statements

All prime integers are either even or odd.

Exercise | Q 6.(vii) | Page 263

Write down the negation of following compound statements

|x| is equal to either x or – x.

Exercise | Q 6.(viii) | Page 263

Write down the negation of following compound statements

6 is divisible by 2 and 3.

Exercise | Q 7.(i) | Page 263

Rewrite the following statements in the form of conditional statements

The square of an odd number is odd.

Exercise | Q 7.(ii) | Page 263

Rewrite the following statements in the form of conditional statements

You will get a sweet dish after the dinner.

Exercise | Q 7.(iii) | Page 263

Rewrite the following statements in the form of conditional statements

You will fail, if you will not study.

Exercise | Q 7.(iv) | Page 263

Rewrite the following statements in the form of conditional statements

The unit digit of an integer is 0 or 5 if it is divisible by 5.

Exercise | Q 7.(v) | Page 263

Rewrite the following statements in the form of conditional statements

The square of a prime number is not prime.

Exercise | Q 7.(vi) | Page 263

Rewrite the following statements in the form of conditional statements

2b = a + c, if a, b and c are in A.P.

Exercise | Q 8.(i) | Page 263

Form the biconditional statement p ↔ q, where
p: The unit digit of an integer is zero.
q: It is divisible by 5.

Exercise | Q 8.(ii) | Page 263

Form the biconditional statement p ↔ q, where
p: A natural number n is odd.
q: Natural number n is not divisible by 2.

Exercise | Q 8.(iii) | Page 263

Form the biconditional statement p ↔ q, where
p: A triangle is an equilateral triangle.
q: All three sides of a triangle are equal.

Exercise | Q 9.(i) | Page 263

Write down the contrapositive of the following statements:

If x = y and y = 3, then x = 3.

Exercise | Q 9.(ii) | Page 264

Write down the contrapositive of the following statements:

If n is a natural number, then n is an integer.

Exercise | Q 9.(iii) | Page 264

Write down the contrapositive of the following statements:

If all three sides of a triangle are equal, then the triangle is equilateral.

Exercise | Q 9.(iv) | Page 264

Write down the contrapositive of the following statements:

If x and y are negative integers, then xy is positive.

Exercise | Q 9.(v) | Page 264

Write down the contrapositive of the following statements:

If natural number n is divisible by 6, then n is divisible by 2 and 3.

Exercise | Q 9.(vi) | Page 264

Write down the contrapositive of the following statements:

If it snows, then the weather will be cold.

Exercise | Q 9.(vii) | Page 264

Write down the contrapositive of the following statements:

If x is a real number such that 0 < x < 1, then x2 < 1.

Exercise | Q 10.(i) | Page 264

Write down the converse of following statements:

If a rectangle ‘R’ is a square, then R is a rhombus.

Exercise | Q 10.(ii) | Page 264

Write down the converse of following statements:

If today is Monday, then tomorrow is Tuesday.

Exercise | Q 10.(iii) | Page 264

Write down the converse of following statements:

If you go to Agra, then you must visit Taj Mahal.

Exercise | Q 10.(iv) | Page 264

Write down the converse of following statements:

If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right-angled.

Exercise | Q 10.(v) | Page 264

Write down the converse of following statements:

If all three angles of a triangle are equal, then the triangle is equilateral.

Exercise | Q 10.(vi) | Page 264

Write down the converse of following statements:

If x : y = 3 : 2, then 2x = 3y.

Exercise | Q 10.(vii) | Page 264

Write down the converse of following statements:

If S is a cyclic quadrilateral, then the opposite angles of S are supplementary

Exercise | Q 10.(viii) | Page 264

Write down the converse of following statements:

If x is zero, then x is neither positive nor negative.

Exercise | Q 10.(ix) | Page 264

Write down the converse of following statements:

If two triangles are similar, then the ratio of their corresponding sides are equal.

Exercise | Q 11.(i) | Page 264

Identify the Quantifiers in the following statements.

There exists a triangle which is not equilateral.

Exercise | Q 11.(ii) | Page 264

Identify the Quantifiers in the following statements.

For all real numbers x and y, xy = yx.

Exercise | Q 11.(iii) | Page 264

Identify the Quantifiers in the following statements.

There exists a real number which is not a rational number.

Exercise | Q 11.(iv) | Page 264

Identify the Quantifiers in the following statements.

For every natural number x, x + 1 is also a natural number.

Exercise | Q 11.(v) | Page 264

Identify the Quantifiers in the following statements.

For all real numbers x with x > 3, x 2 is greater than 9.

Exercise | Q 11.(vi) | Page 264

Identify the Quantifiers in the following statements.

There exists a triangle which is not an isosceles triangle.

Exercise | Q 11.(vii) | Page 264

Identify the Quantifiers in the following statements.

For all negative integers x, x 3 is also a negative integers.

Exercise | Q 11.(viii) | Page 264

Identify the Quantifiers in the following statements.

There exists a statement in above statements which is not true.

Exercise | Q 11.(ix) | Page 264

Identify the Quantifiers in the following statements.

There exists a even prime number other than 2.

Exercise | Q 11.(x) | Page 264

Identify the Quantifiers in the following statements.

There exists a real number x such that x2 + 1 = 0.

Exercise | Q 12 | Page 265

Prove by direct method that for any integer ‘n’, n3 – n is always even.

Exercise | Q 13.(i) | Page 265

Check the validity of the following statement.
p: 125 is divisible by 5 and 7.

Exercise | Q 13.(ii) | Page 265

Check the validity of the following statement.
q: 131 is a multiple of 3 or 11

Exercise | Q 14 | Page 265

Prove the following statement by contradication method.
p: The sum of an irrational number and a rational number is irrational.

Exercise | Q 15 | Page 265

Prove by direct method that for any real numbers x, y if x = y, then x2 = y2.

Exercise | Q 16 | Page 265

Using contrapositive method prove that if n2 is an even integer, then n is also an even integers.

#### Objective Type Questions from 17 to 36

Exercise | Q 17 | Page 265

Which of the following is a statement?

• x is a real number

• Switch off the fan

• 6 is a natural number

• Let me go

Exercise | Q 18 | Page 265

Which of the following is not a statement>

• Smoking is injurious to health

• 2 + 2 = 4

• 2 is the only even prime number

• Come here

Exercise | Q 19 | Page 265

The connective in the statement “2 + 7 > 9 or 2 + 7 < 9” is ______.

• And

• Or

• <

• >

Exercise | Q 20 | Page 266

The connective in the statement “Earth revolves round the Sun and Moon is a satellite of earth” is ______.

• Or

• Earth

• Sun

• And

Exercise | Q 21 | Page 266

The negation of the statement “A circle is an ellipse” is ______.

• An ellipse is a circle

• An ellipse is not a circle

• A circle is not an ellipse

• A circle is an ellipse

Exercise | Q 22 | Page 266

The negation of the statement “7 is greater than 8” is ______.

• 7 is equal to 8

• 7 is not greater than 8

• 8 is less than 7

• None of these

Exercise | Q 23 | Page 266

The negation of the statement “72 is divisible by 2 and 3” is ______.

• 72 is not divisible by 2 or 72 is not divisible by 3

• 72 is not divisible by 2 and 72 is not divisible by 3

• 72 is divisible by 2 and 72 is not divisible by 3

• 72 is not divisible by 2 and 72 is divisible by 3

Exercise | Q 24 | Page 266

The negation of the statement “Plants take in CO2 and give out O2 ” is ______.

• Plants do not take in CO2 and do not give out O2

• Plants do not take in CO2 or do not give out O

• Plants take in CO2 and do not give out O2

• Plants take in CO2 or do not give out O2

Exercise | Q 25 | Page 267

The negation of the statement “Rajesh or Rajni lived in Bangalore” is ______.

• Rajesh did not live in Bangalore or Rajni lives in Bangalore

• Rajesh lives in Bangalore and Rajni did not live in Bangalore

• Rajesh did not live in Bangalore and Rajni did not live in Bangalore

• Rajesh did not live in Bangalore or Rajni did not live in Bangalore

Exercise | Q 26 | Page 267

The negation of the statement “101 is not a multiple of 3” is ______.

• 101 is a multiple of 3

• 101 is a multiple of 2

• 101 is an odd number

• 101 is an even number

Exercise | Q 27 | Page 267

The contrapositive of the statement “If 7 is greater than 5, then 8 is greater than 6” is ______.

• If 8 is greater than 6, then 7 is greater than 5

• If 8 is not greater than 6, then 7 is greater than 5

• If 8 is not greater than 6, then 7 is not greater than 5

• If 8 is greater than 6, then 7 is not greater than 5

Exercise | Q 28 | Page 267

The converse of the statement “If x > y, then x + a > y + a” is ______.

• If x < y, then x + a < y + a

• If x + a > y + a, then x > y

• If x < y, then x + a > y + a

• If x < y, then x + a > y + a

Exercise | Q 29 | Page 267

The converse of the statement “If sun is not shining, then sky is filled with clouds” is ______.

• If sky is filled with clouds, then the sun is not shining

• If sun is shining, then sky is filled with clouds

• If sky is clear, then sun is shining

• If sun is not shining, then sky is not filled with clouds

Exercise | Q 30 | Page 268

The contrapositive of the statement “If p, then q”, is ______.

• If q, then p

• If p, then ~ q

• If ~ q, then ~ p

• If ~ p, then ~ q

Exercise | Q 31 | Page 268

The statement “If x2 is not even, then x is not even” is converse of the statement ______.

• If x2 is odd, then x is even

• If x is not even, then x2 is not even

• If x is even, then x2 is even

• If x is odd, then x2 is even

Exercise | Q 32 | Page 268

The contrapositive of statement ‘If Chandigarh is capital of Punjab, then Chandigarh is in India’ is ______.

• If Chandigarh is not in India, then Chandigarh is not the capital of Punjab

• If Chandigarh is in India, then Chandigarh is Capital of Punjab

• If Chandigarh is not capital of Punjab, then Chandigarh is not capital of India

• If Chandigarh is capital of Punjab, then Chandigarh is not in India

Exercise | Q 33 | Page 268

Which of the following is the conditional p → q?

• q is sufficient for p

• p is necessary for q

• p only if q

• if q, then p

Exercise | Q 34 | Page 268

The negation of the statement “The product of 3 and 4 is 9” is ______.

• It is false that the product of 3 and 4 is 9

• The product of 3 and 4 is 12

• The product of 3 and 4 is not 12

• It is false that the product of 3 and 4 is not 9

Exercise | Q 35 | Page 269

Which of the following is not a negation of “A natural number is greater than zero”?

• A natural number is not greater than zero

• It is false that a natural number is greater than zero

• It is false that a natural number is not greater than zero

• None of the above

Exercise | Q 36 | Page 269

Which of the following statement is a conjunction?

• Ram and Shyam are friends.

• Both Ram and Shyam are tall

• Both Ram and Shyam are enemies

• None of the above

Exercise | Q 37 | Page 269

State whether the following sentences are statements are not.
(i) The angles opposite to equal sides of a triangle are equal.
(ii) The moon is a satellite of earth.
(iii) May God bless you!
(iv) Asia is a continent.
(v) How are you?

## Chapter 14: Mathematical Reasoning

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 11 chapter 14 - Mathematical Reasoning

NCERT solutions for Mathematics Exemplar Class 11 chapter 14 (Mathematical Reasoning) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 14 Mathematical Reasoning are Mathematically Acceptable Statements, New Statements from Old, Special Words Or Phrases, Contrapositive and Converse, Introduction of Validating Statements, Validation by Contradiction, Difference Between Contradiction, Converse and Contrapositive, Consolidating the Understanding.

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