# Class 11 RD Sharma Solutions – Chapter 31 Derivatives – Exercise 31.5

**Question 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 canceled 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 pay a subscription fee, then you can access the website .

(ii)If it rains, then there is a traffic jam.

(iii)If you log on 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 canceled.

(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.

**Question 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.The converse and contrapositive are as follows :

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.The converse and contrapositive are as follows :

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.The converse and contrapositive are as follows :

Converse:If an integer has no divisor other that 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.The converse and contrapositive are as follows :

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.The converse and contrapositive are as follows :

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.

**Question 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.

**Question 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.The contrapositive is as follows :

Contrapositive: If Mohan is not poor, then he is not a poet.

(ii)Only if Max studies will he pass the test.The contrapositive is as follows :

Contrapositive: If Max does not study, then he will not pass the test.

(iii)If she works, she will earn money.The contrapositive is as follows :

Contrapositive: If she does not earn money, then she does not work.

(iv)If it snows, then they do not drive the car.The contrapositive is as follows :

Contrapositive: If then they do not drive the car, then there is no snow.

(v)It never rains when it is cold.The contrapositive is as follows:

Contrapositive: If it rains, then it is not cold.

(vi)If Ravish skis, then it snowed.The contrapositive is as follows :

Contrapositive: If it did not snow, then Ravish will not ski.

(vii)If x is less than zero, then x is not positive.The contrapositive is as follows:

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

(viii)If he has courage he will win.The contrapositive is as follows :

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.The contrapositive is as follows :

Contrapositive: If he is not strong, then he is not a sailor

(x)Only if he does not tire will he win.The contrapositive is as follows :

Contrapositive: If he tries, then he will not win.

(xi)If x is an integer and x2 is odd, then x is odd.The contrapositive is as follows :

Contrapositive: If x is even, then x2 is even.

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