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

Which of the following sentences are statements? Give reasons for your answer.

All real numbers are complex numbers.

Which of the following sentences are statements? Give reasons for your answer.

There are 35 days in a month.

Which of the following sentences are statements? Give reasons for your answer.

Mathematics is difficult.

Which of the following sentences are statements? Give reasons for your answer.

The sum of 5 and 7 is greater than 10.

Which of the following sentences are statements? Give reasons for your answer.

The square of a number is an even number.

Which of the following sentences are statements? Give reasons for your answer.

The sides of a quadrilateral have equal length.

Which of the following sentences are statements? Give reasons for your answer.

Answer this question.

Which of the following sentences are statements? Give reasons for your answer.

The product of (–1) and 8 is 8.

Which of the following sentences are statements? Give reasons for your answer.

The sum of all interior angles of a triangle is 180°.

Which of the following sentences are statements? Give reasons for your answer.

Today is a windy day.

Write the negation of the following statements:

Chennai is the capital of Tamil Nadu.

Write the negation of the following statements:

`sqrt2` is not a complex number.

Write the negation of the following statements:

All triangles are not equilateral triangle.

Write the negation of the following statements:

The number 2 is greater than 7.

Write the negation of the following statements:

Every natural number is an integer.

Are the following pairs of statements negations of each other?

The number x is not a rational number.

The number x is not an irrational number.

Are the following pairs of statements negations of each other?

The number x is a rational number.

Find the component statements of the following compound statements and check whether they are true or false.

Number 3 is prime or it is odd.

Find the component statements of the following compound statements and check whether they are true or false.

All integers are positive or negative.

Find the component statements of the following compound statements and check whether they are true or false.

 100 is divisible by 3, 11 and 5.

For each of the following compound statements first identify the connecting words and then break it into component statements.

All rational numbers are real and all real numbers are not complex.

For each of the following compound statements first identify the connecting words and then break it into component statements.

Square of an integer is positive or negative.

For each of the following compound statements first identify the connecting words and then break it into component statements.

The sand heats up quickly in the Sun and does not cool down fast at night.

For each of the following compound statements first identify the connecting words and then break it into component statements.

x = 2 and x = 3 are the roots of the equation 3x² – x – 10 = 0.

Identify the quantifier in the following statements and write the negation of the statements.

There exists a number which is equal to its square.

Identify the quantifier in the following statements and write the negation of the statements.

For every real number x, x is less than x + 1.

Identify the quantifier in the following statements and write the negation of the statements.

There exists a capital for every state in India.

Check whether the following pair of statements is negation of each other. Give reasons for the answer.

(i) x + y = y + x is true for every real numbers x and y.

(ii) There exists real number x and y for which x + y = y + x

State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.

Sun rises or Moon sets.

State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.

To apply for a driving licence, you should have a ration card or a passport.

State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.

All integers are positive or negative.

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.

Write the contrapositive and converse of the following statements.

If x is a prime number, then x is odd

Write the contrapositive and converse of the following statements.

It the two lines are parallel, then they do not intersect in the same plane.

Write the contrapositive and converse of the following statements.

Something is cold implies that it has low temperature.

Write the contrapositive and converse of the following statements.

You cannot comprehend geometry if you do not know how to reason deductively.

Write the contrapositive and converse of the following statements.

x is an even number implies that x is divisible by 4

Write each of the following statement in the form “if-then”.

You get a job implies that your credentials are good.

Write each of the following statement in the form “if-then”.

The Banana trees will bloom if it stays warm for a month.

Write each of the following statement in the form “if-then”.

A quadrilateral is a parallelogram if its diagonals bisect each other.

Write each of the following statement in the form “if-then”.

To get A⁺ in the class, it is necessary that you do the exercises of the book.

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.

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

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.

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.

By giving a counter example, show that the following statements are not true.

p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.

By giving a counter example, show that the following statements are not true.

q: The equation x² – 1 = 0 does not have a root lying between 0 and 2.

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

p: Each radius of a circle is a chord of the circle.

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

q: The centre of a circle bisects each chord of the circle.

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

r: Circle is a particular case of an ellipse.

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

s: If x and y are integers such that x > y, then –x < –y.

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

t`sqrt11` is a rational number.

Write the negation of the following statements:

p: For every positive real number x, the number x – 1 is also positive.

Write the negation of the following statements:

q: All cats scratch.

Write the negation of the following statements:

r: For every real number x, either x > 1 or x < 1.

Write the negation of the following statements:

s: There exists a number x such that 0 < x < 1.

State the converse and contrapositive of each of the following statements:

p: A positive integer is prime only if it has no divisors other than 1 and itself.

State the converse and contrapositive of each of the following statements:

q: I go to a beach whenever it is a sunny day.

State the converse and contrapositive of each of the following statements:

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

Write the statement in the form “if p, then q”.

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

Write the statement in the form “if p, then q”.

q: There is a traffic jam whenever it rains.

Write the statement in the form “if p, then q”.

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

Re write each of the following statements in the form “p if and only if q”.

p: If you watch television, then your mind is free and if your mind is free, then you watch television.

Re write each of the following statements in the form “p if and only if q”.

q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

Re write each of the following statements in the form “p if and only if q”.

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

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 cases check the validity of the compound statement.

Check the validity of the statements given below by the method given against it.

p: The sum of an irrational number and a rational number is irrational (by contradiction method).

Check the validity of the statements given below by the method given against it.

q: If n is a real number with n > 3, then n² > 9 (by contradiction method).

Write the following statement in five different ways, conveying the same meaning.

p: If triangle is equiangular, then it is an obtuse angled triangle.