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

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, x² 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.

Which of the following is a statement?

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

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

Which of the following sentences are statements? Justify

A triangle has three sides.

Which of the following sentences are statements? Justify

0 is a complex number.

Which of the following sentences are statements? Justify

Sky is red.

Which of the following sentences are statements? Justify

Every set is an infinite set.

Which of the following sentences are statements? Justify

15 + 8 > 23

Which of the following sentences are statements? Justify

y + 9 = 7.

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.

Which of the following sentences are statements? Justify

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

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.

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.

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.

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.

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

sin²x + cos²x = 0

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 x² – 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 x² < 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 x² + 1 = 0.

Prove by direct method that for any integer ‘n’, n³ – 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 x² = y².

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

Which of the following is a statement?

Which of the following is not a statement>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Which of the following is the conditional p → q?

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

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

Which of the following statement is a conjunction?

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?