Punjab State Board PSEB 11th Class Maths Book Solutions Chapter 14 Mathematical Reasoning Miscellaneous Exercise Questions and Answers.
PSEB Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Miscellaneous Exercise
Question 1.
Write the negation of the following statements :
(i) p : For every positive real number x, the number #-l is also positive.
(ii) q : All cats scratch.
(iii) r : For every real number x, either x > 1 or x < 1.
(iv) s : There exists a number# such that 0 < x < 1.
Answer.
(i) ~ p : There exists a positive real number x such that x -1 is not positive.
(ii) ~ q : There exist a cat which does not scratch.
(iii) ~ r : There exists a real number x such that neither x > 1 nor x < 1.
(iv) ~ s: There does not exist a number x such that 0 < x < 1.
Question 2.
State the converse and contrapositive of each of the following statements:
(i) p : A positive integer is prime only if it has no divisors other than 1 and itself.
(ii) q : I go to a beach whenever it is a sunny day.
(iii) r : If it is hot outside, then you feel thirsty.
Answer.
(i) Converse : If a positive integer has no divisor other than 1 and itself then it is a prime. Contrapositive : If a positive integer has no divisor other than 1 and itself then it is not prime.
(ii) Converse : If it is a sunny day, then I go to beach.
Contrapositive : If it is not sunny day, then I do not go to beach.
(iii) Converse : If you feel thirsty then it is hot outside.
Contrapositive : If you do not feel thirsty then it is not hot outside.
Question 3.
Write each of the statements in the form “if p, then q
(i) p : It is necessary to have a password to log on to the server,
(ii) q : There is traffic jam whenever it rains.
(iii) r : You can access the website only if you pay a subscription fee.
Answer.
(i) Statement p can be written as follows. If you log on to the server, then you have a password.
(ii) Statement q can be written as follows. If if rains, then there is a traffice jam.
(iii) Statement r can be written as follows. If you can access the website, then you pay a subscription fee.
Question 4.
Rewrite each of the following statements in the form “p if and only if q”.
(i) p : If you watch television, then your mind is free and if your mind is free, then you watch television.
(ii) q : For you to get an A grade, it is necessary and sufficent that you do all the homework regularly.
(iii) r : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.
Answer.
(i) You watch television if and only if your mind is free.
(ii) You get an A grade if and only if you do all the homework regularly.
(iii) A quadrilateral is equiangular if and only if it is a rectangle.
Question 5.
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.
Answer.
The compound statement with ‘And’ is “25 is a multiple of 5 and 8”. This is a false statement, since 25 is not a multiple of 8. The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”. This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.
Question 6.
Check the validity of the statements given below by the method given against it.
(i) p : The sum of an irrational number and a rational number is irrational (by contradiction method).
(ii) q : If n is a real number with n > 3, then n2 > 9 (by contradiction method).
Answer.
(i) Let √a be irrational number and b be a rational number.
Their sum = b + √a
Let it is not irrational. Therefore, it is a rational number. …(i)
b + √a = \(\frac{p}{q}\), where p, q are co-prime
√a = \(\frac{p}{q}\) – b ……………(i)
L.H.S. = √a = An irrational number
R.HS. = \(\frac{p}{q}\) – b = A rational number
It is a contradiction. Therefore, the sum of a rational and irrational number is irrational, which is a valid statement.
(ii) Let n > 3 and n2 < 9
Put n = 3 + a, we have
n2 = 9 + 6a + a2 = 9 + a (6 + a), which is a contradiction.
⇒ If n > 3, then n2 > 9, which is a valid statement.
Question 7.
Write the following statement in five different ways, conveying the same meaning.
p: If triangle is equiangular, then it is an obtuse angled triangle.
Answer.
(i) A triangle is equiangular implies that it is an obtuse angled triangle.
(ii) A triangle is equiangular only if it is an obtuse angled triangle.
(iii) For a triangle to be equiangular it is necessary that it is an obtuse angled triangle.
(iv) For a triangle to be obtuse angled triangle, it is sufficient that it is equiangular.
(v) If a triangle is not obtuse angled triangle, then it is not an equiangular triangle.