#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 14: Mathematical Reasoning

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

#### Short Answer

Which of the following statements are compound statements

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

Which of the following statements are compound statements

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

Which of the following statements are compound statements

“Ram and Rahim are friends”

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

It is raining or the sun is shining.

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

2 is a positive number or a negative number.

Translate the following statements in symbolic form

2 and 3 are prime numbers

Translate the following statements in symbolic form

Tigers are found in Gir forest or Rajaji national park.

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.

Write negation of the statement

“Everyone who lives in India is an Indian”

Write the negation of the following statements:

p: All triangles are equilateral triangles.

Write the negation of the following statements:

q: 9 is a multiple of 4.

Write the negation of the following statements:

r: A triangle has four sides.

Write the negation of the following statements:

Suresh lives in Bhopal or he lives in Mumbai.

Write the negation of the following statements:

x + y = y + x and 29 is a prime number.

Rewrite the following statements in the form of conditional statements:

Mohan will be a good student if he studies hard.

Rewrite the following statements in the form of conditional statements:

Ramesh will get dessert only if he eats his dinner.

Rewrite the following statements in the form of conditional statements:

When you sing, my ears hurt.

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.

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.

Express in English, the statement p → q, where

p: It is raining today

q: 2 + 3 > 4

Translate the following statements in symbolic form:

If x = 7 and y = 4” then x + y = 11.

Form the biconditional of the following statements :

p: Today is 14^{th} of August

q: Tomorrow is Independence day

Translate the following biconditional into symbolic form:

“ABC is an equilateral triangle if and only if its each interior angle is 60°”

Identify the quantifiers and write the negation of the following statements:

There exists a number which is equal to its square.

Identify the quantifiers and write the negation of the following statements:

For all even integers x, x^{2} is also even.

Identify the quantifiers and write the negation of the following statements:

There exists a number which is a multiple of 6 and 9.

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.

Check the validity of the statements:

r: 100 is a multiple of 4 and 5.

Check the validity of the statements:

s: 60 is a multiple of 3 or 5.

#### Objective Type Questions from 16 to 18

Which of the following is a statement?

Roses are black

Mind your own business

Be punctual

Do not tell lies

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

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

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

#### Short Answer

Which of the following sentences are statements? Justify

A triangle has three sides.

True

False

Which of the following sentences are statements? Justify

0 is a complex number.

True

False

Which of the following sentences are statements? Justify

Sky is red.

True

False

Which of the following sentences are statements? Justify

Every set is an infinite set.

True

False

Which of the following sentences are statements? Justify

15 + 8 > 23

True

False

Which of the following sentences are statements? Justify

y + 9 = 7.

True

False

Which of the following sentences are statements? Justify

Where is your bag?

Which of the following sentences are statements? Justify

Every square is a rectangle.

True

False

Which of the following sentences are statements? Justify

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

True

False

Find the component statements of the following compound statements.

Number 7 is prime and odd.

Write the negation of the following simple statements

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

Find the component statements of the following compound statements.

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

Find the component statements of the following compound statements.

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

Find the component statements of the following compound statements.

Chandigarh is the capital of Haryana and U.P.

Find the component statements of the following compound statements.

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

Find the component statements of the following compound statements.

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

Find the component statements of the following compound statements.

Plants use sunlight, water and carbon dioxide for photosynthesis.

Write the negation of the following simple statements.

A leap year has 366 days.

Find the component statements of the following compound statements.

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

Find the component statements of the following compound statements.

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

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

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

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

2 is an even number and a prime number.

True

False

Write the negation of the following simple statements.

The number 17 is prime.

Write the negation of the following simple statements.

2 + 7 = 6.

Write the negation of the following simple statements.

Violets are blue.

Write the negation of the following simple statements.

`sqrt(5)` is a rational number.

Write the negation of the following simple statements.

2 is not a prime number.

Write the negation of the following simple statements.

Every real number is an irrational number.

Write the negation of the following simple statements.

Cow has four legs.

Write the negation of the following simple statements

A leap year has 366 days.

Write the negation of the following simple statements

All similar triangles are congruent.

Which of the following sentences are statements? Justify

sin^{2}x + cos^{2}x = 0

True

False

Translate the following statements into symbolic form

Rahul passed in Hindi and English.

Translate the following statements into symbolic form

x and y are even integers.

Translate the following statements into symbolic form

2, 3 and 6 are factors of 12

Translate the following statements into symbolic form

Either x or x + 1 is an odd integer.

Translate the following statements into symbolic form

A number is either divisible by 2 or 3.

Translate the following statements into symbolic form

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

Translate the following statements into symbolic form

Students can take Hindi or English as an optional paper.

Write down the negation of following compound statements

All rational numbers are real and complex.

Write down the negation of following compound statements

All real numbers are rationals or irrationals.

Write down the negation of following compound statements

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

Write down the negation of following compound statements

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

Write down the negation of following compound statements

35 is a prime number or a composite number

Write down the negation of following compound statements

All prime integers are either even or odd.

Write down the negation of following compound statements

|x| is equal to either x or – x.

Write down the negation of following compound statements

6 is divisible by 2 and 3.

Rewrite the following statements in the form of conditional statements

The square of an odd number is odd.

Rewrite the following statements in the form of conditional statements

You will get a sweet dish after the dinner.

Rewrite the following statements in the form of conditional statements

You will fail, if you will not study.

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.

Rewrite the following statements in the form of conditional statements

The square of a prime number is not prime.

Rewrite the following statements in the form of conditional statements

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

Form the biconditional statement p ↔ q, where

p: The unit digit of an integer is zero.

q: It is divisible by 5.

Form the biconditional statement p ↔ q, where

p: A natural number n is odd.

q: Natural number n is not divisible by 2.

Form the biconditional statement p ↔ q, where

p: A triangle is an equilateral triangle.

q: All three sides of a triangle are equal.

Write down the contrapositive of the following statements:

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

Write down the contrapositive of the following statements:

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

Write down the contrapositive of the following statements:

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

Write down the contrapositive of the following statements:

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

Write down the contrapositive of the following statements:

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

Write down the contrapositive of the following statements:

If it snows, then the weather will be cold.

Write down the contrapositive of the following statements:

If x is a real number such that 0 < x < 1, then x^{2} < 1.

Write down the converse of following statements:

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

Write down the converse of following statements:

If today is Monday, then tomorrow is Tuesday.

Write down the converse of following statements:

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

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.

Write down the converse of following statements:

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

Write down the converse of following statements:

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

Write down the converse of following statements:

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

Write down the converse of following statements:

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

Write down the converse of following statements:

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

Identify the Quantifiers in the following statements.

There exists a triangle which is not equilateral.

Identify the Quantifiers in the following statements.

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

Identify the Quantifiers in the following statements.

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

Identify the Quantifiers in the following statements.

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

Identify the Quantifiers in the following statements.

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

Identify the Quantifiers in the following statements.

There exists a triangle which is not an isosceles triangle.

Identify the Quantifiers in the following statements.

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

Identify the Quantifiers in the following statements.

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

Identify the Quantifiers in the following statements.

There exists a even prime number other than 2.

Identify the Quantifiers in the following statements.

There exists a real number x such that x^{2} + 1 = 0.

Prove by direct method that for any integer ‘n’, n^{3} – n is always even.

Check the validity of the following statement.

p: 125 is divisible by 5 and 7.

Check the validity of the following statement.

q: 131 is a multiple of 3 or 11

Prove the following statement by contradication method.

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

Prove by direct method that for any real numbers x, y if x = y, then x^{2} = y^{2}.

Using contrapositive method prove that if n^{2} is an even integer, then n is also an even integers.

#### Objective Type Questions from 17 to 36

Which of the following is a statement?

x is a real number

Switch off the fan

6 is a natural number

Let me go

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

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

And

Or

<

>

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

Or

Earth

Sun

And

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

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

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

The negation of the statement “Plants take in CO_{2} and give out O_{2} ” is ______.

Plants do not take in CO

_{2}and do not give out O_{2}Plants do not take in CO

_{2}or do not give out O_{2 }Plants take in CO

_{2}and do not give out O_{2}Plants take in CO

_{2}or do not give out O_{2}

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

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

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

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

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

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

The statement “If x^{2} is not even, then x is not even” is converse of the statement ______.

If x

^{2}is odd, then x is evenIf x is not even, then x

^{2}is not evenIf x is even, then x

^{2}is evenIf x is odd, then x

^{2}is even

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

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

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

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

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

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

## 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.

Using NCERT Class 11 solutions Mathematical Reasoning exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

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