RD Sharma solutions for Mathematics [English] Class 11 chapter 31 - Mathematical reasoning [Latest edition]

Find out the following  sentence is a statement and is not. Justify your answer.

Listen to me, Ravi !

Find out the sentence are statement and are not. Justify your answer.

 Every set is a finite set.

Find out the sentence are statement and are not. Justify your answer.

Two non-empty sets have always a non-empty intersection.

Find out the sentence are statement and are not. Justify your answer.

The cat pussy is black.

Find out the sentence are statement and are not. Justify your answer.

 Are all circles round?

Find out the sentence are statement and are not. Justify your answer.

 All triangles have three sides.

Find out the sentence are statement and are not. Justify your answer.

Every rhombus is a square.

Find out the sentence are statement and are not. Justify your answer.

 x² + 5 | x | + 6 = 0 has no real roots.

Find out the sentence are statement and are not. Justify your answer.

 This sentence is a statement.

Find out the sentence are statement and are not. Justify your answer.

Is the earth round?

Find out the sentence are statement and are not. Justify your answer.

 Go !

Find out the sentence are statement and are not. Justify your answer.

 The real number x is less than 2.

Find out the sentence are statement and are not. Justify your answer.

There are 35 days in a month.

Find out the sentence are statement and are not. Justify your answer.

 Mathematics is difficult.

Find out the sentence are statement and are not. Justify your answer.

All real numbers are complex numbers.

Find out the sentence are statement and are not. Justify your answer.

Give three examples of sentences which are not statements. Give reasons for the answers.

Write the negation of the statement:
Banglore is the capital of Karnataka.

Write the negation of the statement:

 It rained on July 4, 2005.

Write the negation of the statement:

 Ravish is honest.

Write the negation of the statement:

 The earth is round.

Write the negation of the statement:

All birds sing.

 Some even integers are prime.

There is a complex number which is not a real number.

 I will not go to school.

 Both the diagonals of a rectangle have the same length.

Are the pair of statement are negation of each other:
The number x is not a rational number.
The number x is not an irrational number.

Are the pair of statement are negation of each other:

 The number x is not a rational number.
The number x is an irrational number.

Write the negation of the statement:

 p : For every positive real number x, the number (x − 1) is also positive.

Write the negation of the statement:

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

Write the negation of the statement:

Check whether the following pair of statements are negation of each other. Give reasons for your answer.
a + b = b + a is true for every real number a and b.
 There exist real numbers a and b for which a + b = b + a.

Find the component statement of the compound statement:
 The sky is blue and the grass is green.

Find the component statement of the compound statement:

The earth is round or the sun is cold.

Find the component statement of the compound statement:

All rational numbers are real and all real numbers are complex.

Find the component statement of the compound statement:

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

 Students can take Hindi or Sanskrit as their third language.

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

 To entry a country, you need a passport or a voter registration card.

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

A lady gives birth to a baby boy or a baby girl.

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

Write the component statement of the compound statement and check whether the compound statement is true or false:

 To enter into a public library children need an identity card from the school or a letter from the school authorities.

Write the component statement of the compound statement and check whether the compound statement is true or false:

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

Write the component statement of the compound statement and check whether the compound statement is true or false:

 Square of an integer is positive or negative.i

Write the component statement of the compound statement and check whether the compound statement is true or false:

 x = 2 and x = 3 are the roots or the equation 3x² − x − 10 = 0.

Write the component statement of the compound statement and check whether the compound statement is true or false:

Determine whether the compound statement are true or false:

 Delhi is in India and 2 + 2 = 4.
 

Determine whether the compound statement are true or false:

 Delhi is in England and 2 + 2 = 4.
 

Determine whether the compound statement are true or false: 

 Delhi is in India and 2 + 2 = 5.

Determine whether the compound statement are true or false: 

Write the negation of  statement:

For every x ϵ N, x + 3 < 10

Write the negation of  statement:

Negate  statement:
 All the students completed their homework.

Negate of the  statement :

Write of the statement in the form "if p, then q". 

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

Write of the statement in the form "if p, then q". 

There is traffic jam whenever it rains.

Write of the statement in the form "if p, then q". 

 It is necessary to have a passport to log on to the server.

Write of the statement in the form "if p, then q". 

 It is necessary to be rich in order to be happy.

Write of the statement in the form "if p, then q". 

 The game is cancelled only if it is raining.

Write of the statement in the form "if p, then q". 

 It rains only if it is cold.

Write of the statement in the form "if p, then q". 

 Whenever it rains it is cold.

Write of the statement in the form "if p, then q". 

State the converse and contrapositive of  statement:

If it is hot outside, then you feel thirsty.

State the converse and contrapositive of  statement:

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

State the converse and contrapositive of  statement:

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

State the converse and contrapositive of  statement:

If you live in Delhi, then you have winter clothes.

State the converse and contrapositive of  statement:

Rewrite of the  statement 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.

Rewrite of the  statement in the form "p if and only if q".

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

Rewrite of the  statement in the form "p if and only if q".

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

Rewrite of the  statement in the form "p if and only if q".

 s : If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.

Determine the contrapositive of the statement:

 If Mohan is a poet, then he is poor.

Determine the contrapositive of the statement:

 Only if Max studies will he pass the test.

Determine the contrapositive of the statement:

If she works, she will earn money.

Determine the contrapositive of the statement:

If it snows, then they do not drive the car.

Determine the contrapositive of the statement:

 It never rains when it is cold.

Determine the contrapositive of the statement:

 If Ravish skis, then it snowed.

Determine the contrapositive of the statement:

If x is less than zero, then x is not positive.

Determine the contrapositive of the statement:

 If he has courage he will win.

Determine the contrapositive of the statement:

 It is necessary to be strong in order to be a sailor.

Determine the contrapositive of the statement:

Only if he does not tire will he win.

Determine the contrapositive of the statement:

Check the validity of the statement:

 p : 100 is a multiple of 4 and 5.

Check the validity of the statement:

q : 125 is a multiple of 5 and 7.

Check the validity of the statement:

Check whether the statement are true or not: 

p : If x and y are odd integers, then x + y is an even integer.

Check whether the statement are true or not: 

 q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.

Show that the statement
p : "If x is a real number such that x³ + x = 0, then x is 0"
is true by
(i) direct method
(ii) method of contrapositive
(iii) method of contradition.

Show that the following statement is true by the method of contrapositive
p : "If x is an integer and x² is odd, then x is also odd" 

Show that the following statement is true
"The integer n is even if an only if n² is even"

By giving a counter example, show that the following statement is not true.
p : "If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle".

 statement are true and false? In each case give a valid reason for saying so

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

 statement are true and false? In each case give a valid reason for saying so

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

 statement are true and false? In each case give a valid reason for saying so

r : Circle is a particular case of an ellipse.

 statement are true and 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.

 statement are true and false? In each case give a valid reason for saying so

 t :  \[\sqrt{11}\]  is a rational number. 

Determine whether the argument used to check the validity of the following statement is correct:
p : "If x² is irrational, then x is rational"
The statement is true because the number x² = π² is irrational, therefore x = π is irrational.