## Kerala Plus One Maths Chapter Wise Previous Questions Chapter 14 Mathematical Reasoning

Question 1.

a. Which one of the following sentences is a STATEMENT?

i. 275 is a perfect square.

ii. Mathematics is a difficult subject.

iii. Answer this question.

iv. Today is a rainy day.

b. Verify by method of contradiction:

[March – 2018]

Answer:

a. 275 is a pefect square

b. Let is a rational

;p and q have no common factor

p^{2} = 2q^{2},

i.e., 2 divides p^{2} is 2 divides p

p = 2k, p^{2} = 4k^{2}

2q^{2} = 4k^{2}, q^{2} = 2k^{2}

2 divides q^{2} is 2 divides q.

p and q have a common factor 2.

Question 2.

a. Write the contra positive of the statement “If a number is divisible by 9, then it is divisible by 3”.

b. Prove by the method of contradiction, “P : is irrational”. [March-2017]

Answer:

a. Contra positive statement:

If a number is not divisible by 3, it is not divisible by 9.

b. Let us assume that is a rational number.

∴ where a and b are co-prime.

i.e., a and b have no common factors other than 1.

5b^{2}= a^{2} ⇒ 5 divides a.

there exists an integer ‘k’ such that a = 5k

a^{2}= 25k^{2} ⇒ 5b^{2} = 25k2 ⇒ 5k^{2} = b^{2}

⇒ 5 divides b.

i.e., 5 divides both a and b, which is contradiction to our assumption that a and b are prime.

our assumption is wrong.

is an irrational number.

Question 3.

a Write the negation of the statement:

“ Every natural number is greater than zero”

b. Verify by the method of contradiction: “ P: is irrational ” [March-2016]

Answer:

a. Let p : “Every natural number is greater than zero”.

~p: “ Every natural number is not greater than zero”.

b. Let us assume that is a rational number , where a and b are co-prime, i.e., a and b have no common factors, which implies that 13b^{2}= a^{2} => 13 divides

a. There exists an integer ‘k’ such that a= 13k

a^{2}= 169 k^{2} ⇒13b^{2}= 169 k^{2} ⇒ 13k^{2} ⇒ 13 divides b.

i.e., 13 divides both a and b which is contradiction to our assumption that a and b have no common factor.

∴ Our supposition is wrong.

∴ is an irrational number.

Question 4.

a. Write the negation of the statement “ is rational”.

b. Prove that “ is irrational” by the method of contradiction. [March-2015]

Answer:

a. is rational

b. In this method, we assume that the given statement is false, ie, is rational.

∴ ; where a and b have no common factor by squaring on both sides.

a^{2} = 7b^{2} ⇒ 7

divides a then consider an integer c, such that, i.e., a = 7c, a^{2} =49c^{2 }and we know that

7b^{2} = a^{2} .’. 7b^{2} = 49c^{2}

b^{2} = 7c^{2} ⇒7 divides b

.’. a and b has no common factor 7, so, our assumption, ‘a and b has no common factor’, is wrong. so we can say the assumption is rational also wrong i…..e, is irrational.

Question 5.

a.Write the negation of the following statement:

” is not a complex number.”

b. Verify by the method of contradiction:” P^{:} is irrational number.”[March-2014]

Answer:

a. Negotiation of statement can be written as “ is a complex number”.

b. Assume that is a rational number

It can be written as where a and b are two comprime numbers.

a= b

a^{2} = 2b^{2}………….. (1)

Let a = 2c

Therefore (1) becomes,

(2c)^{2} = 2b^{2}

4c^{2} = 2b^{2 }2c^{2}=b^{2}…….. (2)

From 1 and 2 it is clear that ‘a’ and ‘b’ has a common factor 2. This is a contradiction.

∴ Our assumption that rational is wrong. is irrational.

Question 6.

a. Write the negation of the following statement: “All triangle are not equilateral triangles”.

b. Verify by the method of contradiction p: is irrational [March-2013]

Answer:

a. It is false. All triangles are equilateral triangles.

b. In this method, we assume that the given statement is false, ie, is rational.

∴ ; where a and b have no common factor by squaring on both sides.

a^{2} = 7b^{2} ⇒ 7

divides a then consider an integer c, such that, i.e., a = 7c, a^{2} =49c^{2 }and we know that

7b^{2} = a^{2} .’. 7b^{2} = 49c^{2}

b^{2} = 7c^{2} ⇒7 divides b

.’. a and b has no common factor 7, so, our assumption, ‘a and b has no common factor’, is wrong. so we can say the assumption is rational also wrong i…..e, is irrational.

Question 7.

Verify by the method of contradiction p: is irrational. =[February-2013]

Answer:

Assume that is a rational number

It can be written as where a and b are two comprime numbers.

a= b

a^{2} = 2b^{2}………….. (1)

Let a = 2c

Therefore (1) becomes,

(2c)^{2} = 2b^{2}

4c^{2} = 2b^{2 }2c^{2}=b^{2}…….. (2)

From 1 and 2 it is clear that ‘a’ and ‘b’ has a common factor 2. This is a contradiction.

∴ Our assumption that rational is wrong. is irrational.

Question 8.

Consider the statement, “ If x is an integer and x^{2} is even, then x is also even”.

a. Write the converse of this statement.

b. Prove the statement by the contrapositive method. [March-2012]

Answer:

a. If x is even then x^{2} is even.

b. If x is odd,x = 2k+1

x^{2}= 4k^{2} + 4k + 1 is not even

i.e., x^{2} is odd. then if r is false ⇒ q is false

p : if is q is true then r is true

Question 9.

i. Write the converse of the statement: “If a number n is even, then n^{2} is even”.

ii. Verify the method of contradiction: “ is irrational”. [March-2011]

Answer:

i. If n^{2} is even then n is even.

ii. Assume that is a rational number

It can be written as where a and b are two comprime numbers.

a= b

a^{2} = 2b^{2}………….. (1)

Let a = 2c

Therefore (1) becomes,

(2c)^{2} = 2b^{2}

4c^{2} = 2b^{2 }2c^{2}=b^{2}…….. (2)

From 1 and 2 it is clear that ‘a’ and ‘b’ has a common factor 2. This is a contradiction.

∴ Our assumption that rational is wrong. is irrational.

Question 10.

i. Write the negation of the statement. “Both the diagonals of a rectangle have the same length”.

ii. Prove the statement, “Product of two odd integers is odd,” by proving its contra positive. [March-2010]

Answer:

i. It is false that both the diagonals of a rectangle have the same length.

ii. p: x and y are odd integers q: xy is odd

Suppose ~ q: xy is even this is possible only either x or y is even.

i.e., p is not true

i.e.,~ q => ~p

Question 11.

a. Write the converse of the statement.

P: If a divides b, then b is a multiple of a.

b Consider the compound statement.

p: 2 +2 is equal to 9 4 or 6.

1. Write the component statement.

2. Is the compound statement true? why? [September-2010]

Answer:

a. Convers of P is if b is a multiple of a, then a divides b.

b. 1. p : 2 + 2 is equal to 4 or 6. component statement are

q : 2 + 2 = 4

r:2 + 2 = 6

2. True