In mathematics, we come across many statements of the form “if p then q”, “p only if q”, and “if and only if”. Such statements are called implications. In this section, we shall discuss such statements. The solutions are prepared and reviewed by the subject matter experts, and they are revised according to the latest syllabus. For students to excel in the board examinations, these solutions will increase their confidence level, as the concepts are clearly explained and well-structured. RD Sharma Class 11 Maths Solutions are widely used by students who wish to excel in board exams as it provides a vast number of questions to solve. Students can refer to and download the PDF of RD Sharma Solutions, which is available in the links given below.
RD Sharma Solutions for Class 11 Maths Exercise 31.5 Chapter 31 – Mathematical Reasoning
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EXERCISE 31.5 PAGE NO: 31.21
1. Write each of the following statements in the form “if p, then q.”
(i) You can access the website only if you pay a subscription fee.
(ii) There is traffic jam whenever it rains.
(iii) It is necessary to have a passport to log on to the server.
(iv) It is necessary to be rich in order to be happy.
(v) The game is cancelled only if it is raining.
(vi) It rains only if it is cold.
(vii) Whenever it rains, it is cold.
(viii) It never rains when it is cold.
Solution:
(i) If you access the website, then you pay a subscription fee.
(ii) If it rains, then there is a traffic jam.
(iii) If you log on to the server, then you must have a passport.
(iv) If he is happy, then he is rich.
(v) If it is raining, then the game is cancelled.
(vi) If it rains, then it is cold.
(vii) If it rains, then it is cold.
(viii) If it is cold, then it never rains.
2. State the converse and contrapositive of each of the following statements:
(i) If it is hot outside, then you feel thirsty.
(ii) I go to a beach whenever it is a sunny day.
(iii) A positive integer is prime only if it has no divisions other than 1 and itself.
(iv) If you live in Delhi, then you have winter clothes.
(v) If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Solution:
(i) If it is hot outside, then you feel thirsty.
Converse: If you feel thirsty, then it is hot outside.
Contrapositive: If you do not feel thirsty, then it is not hot outside.
(ii) I go to a beach whenever it is a sunny day.
Converse: If I go to a beach, then it is a sunny day.
Contrapositive: If I do not go to a beach, then it is not a sunny day.
(iii) A positive integer is prime only if it has no divisions other than 1 and itself.
Converse: If an integer has no divisor other than 1 and itself, then it is prime.
Contrapositive: If an integer has some divisor other than 1 and itself, then it is prime.
(iv) If you live in Delhi, then you have winter clothes.
Converse: If you have winter clothes, then you live in Delhi.
Contrapositive: If you do not have winter clothes, then you do not live in Delhi.
(v) If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Contrapositive: If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.
3. Rewrite each of the following statements in the form “p if and only is q.”
(i) p: If you watch television, then your mind is free, and if your mind is free, then you watch television.
(ii) q: If a quadrilateral is equiangular, then it is a rectangle, and if a quadrilateral is a rectangle, then it is equiangular.
(iii) r: For you to get an A grade, it is necessary and sufficient that you do all the homework you regularly.
(iv) s: If a tumbler is half empty, then it is half full, and if a tumbler is half full, then it is half empty.
Solution:
(i) You watch television if and only if your mind is free.
(ii) A quadrilateral is a rectangle if and only if it is equiangular.
(iii) You get an A grade if and only if you do all the homework regularly.
(iv) A tumbler is half empty if and only if it is half full.
4. Determine the Contrapositive of each of the following statements:
(i) If Mohan is a poet, then he is poor.
(ii) Only if Max studies will he pass the test.
(iii) If she works, she will earn money.
(iv) If it snows, then they do not drive the car.
(v) It never rains when it is cold.
(vi) If Ravish skis, then it snowed.
(vii) If x is less than zero, then x is not positive.
(viii) If he has courage he will win.
(ix) It is necessary to be strong in order to be a sailor.
(x) Only if he does not tire will he win.
(xi) If x is an integer and x2 is odd, then x is odd.
Solution:
(i) If Mohan is a poet, then he is poor.
Contrapositive: If Mohan is not poor, then he is not a poet.
(ii) Only if Max studies will he pass the test.
Contrapositive: If Max does not study, then he will not pass the test.
(iii) If she works, she will earn money.
Contrapositive: If she does not earn money, then she does not work.
(iv) If it snows, then they do not drive the car.
Contrapositive: If then they do not drive the car, then there is no snow.
(v) It never rains when it is cold.
Contrapositive: If it rains, then it is not cold.
(vi) If Ravish skis, then it snowed.
Contrapositive: If it did not snow, then Ravish will not ski.
(vii) If x is less than zero, then x is not positive.
Contrapositive: If x is positive, then x is not less than zero.
(viii) If he has courage, he will win.
Contrapositive: If he does not win, then he does not have courage.
(ix) It is necessary to be strong in order to be a sailor.
Contrapositive: If he is not strong, then he is not a sailor
(x) Only if he does not tire will he win.
Contrapositive: If he tries, then he will not win.
(xi) If x is an integer and x2 is odd, then x is odd.
Contrapositive: If x is even, then x2 is even.
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