## Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 14 Mathematical Reasoning

**Short Answer** **Type Questions **

**(Score 3)**

Question 1.

a. The truth values ofp,q and r for which (p ^{∧}q) v (~ r) has truth value F are respectively ………….

b. ~ [ (~ p) ^{∧} q ] is logically equivalent to

a. ~(pvq)

b. ~ [ p^{∧}(~ q) ]

c. p^{∧}(~q)

d. pv(~q)

c. Find the negation of (p v ~ q)^{∧}q

Answer:

a. F, F, T

b. p v (~ q)

c. (~ p ^{∧} q) v ~ q

Question 2.

Which one of the following is a statement?

a. close the door.

b. good evening sir.

c. Mumbai is the capital of India.

d. Bring the book.

b. Write the negation of the following statement and check whether the resulting statement is true: The sum of 3 and 4 is 9.

Answer:

a. Mumbai is the capital of India.

b. Given p: the sum of 3 and 4 is 9

~ p : the sum of 3 and 4 is not 9

We know that 3 + 4 = 7 ≠ 9.

So the above statement is true.

Question 3.

a. By giving an example, show that the following statement is false: “If n is an odd integer, then n is prime”,

b. Translate the following compound statement into symbolic form: “The sky is blue and grass is green”.

c. If p : 2 plus 3 is five and q: Delhi is the capital of India are two statements, then the statement “Delhi is the capital of India and it is not that 2 plus 3 is five” is

Answer:

a. We observe that 9 is an odd integer which is not prime. Hence the given statement is false.

b. p ^{∧}q; p: The sky is blue, q: Grass is green.‘and’ connective

c. ~ p ^{A} q

Question 4.

a. Show that the statement “for any real numbers a and b, a^{2}=b^{2} implies that a=b” is not true by giving a counter example.

b. Which of the following is not a logical statement?

[Delhi is the capital of Australia; There are 50 alphabets in English; Is it not interesting to watch the sunset? Mathematics has a lot of applications in Physics]

c. Write the negation of the statement : every living person is not 150 years old.

Answer:

a. We have (-2)^{2} = 2^{2} but -2 ≠ 2.

So the given statement is not true.

b. Is it not interesting to watch the sunset?

c. There exists a living person who is 150 years old.

Question 5.

a. Check the validity of the statement given below by contradiction method.“The sum of an irrational number and a rational number is irrational”

b. Identify the quantifier in the following statement: There exists a real number which is twice of itself

c. Write the converse of the statement: If two lines are parallel, then they do not intersect in the same plane.

Answer:

a. Let p be the given statement. If possible Let p be not true. Then p is false.

⇒ the sum of an irrational number and a rational number is not irrational.

⇒ there exists an irrational number x (say) and a rational number y (say) such that x + y is not irrational.

⇒ x + y is rational, say, z⇒ x + y = z ⇒ x = z – y ⇒ x is rational. But x is irrational. So we arrive at a contradiction. Thus our assumption is wrong. Hence p is true.

b. There exists.

c. If two lines do not intersect in the same plane, then they are parallel.

Question 6.

For the given statement identify the necessary and sufficient conditions: If a quadrilateral is equiangular, then it is a rectangle.

Answer:

’Let p and q denote the statements: p: A quadrilateral is equiangular q: A quadrilateral is a rectangle We know that the implications of “if p, then q” indicates that p is sufficient for q.

It also indicates that q is necessary for p. Sufficient condition is “a quadrilateral is equiangular” and the necessary condition is “a quadrilateral is a rectangle”.

Question 7.

Which of the following is not a statement?

a. It is not that the sky is blue.

b. Is the sky blue?

c. The sky is blue.

d. The sky is dark in the night.

b. Write the component statements of the following compound statement and check whether the compound statement is true or false: Chennai is the capital of Kerala or Tamil Nadu.

Answer:

a. Is the sky blue?

b. The component statements of the given statements are:

p : Chennai is the capital of Kerala

q : Chennai is the capital of Tamil Nadu.

We find that p is false and q is true.

∴ The compound statement is true.

Question 8.

a. Which of the following is not a statement?

a. The sky is dark in the night.

b. Is the sky blue?

c. The sky is blue.

d. It is not that the sky is blue.

b. Write negation of each of the following statements.

i. Ramesh is smart and healthy.

ii. Mridul is cruel or he is strict.

Answer:

a. Is the sky blue?

b.

i. Ramesh is neither smart nor healthy.

ii. Mridul is not cruel and he is not strict.

**Short Answer** **Type Questions **

**(Score 4)**

Question 1.

Rewrite the following statement with “if- then” in five different ways conveying the same meaning: If a natural number is odd, then its square is also odd.

Answer:

The component statement of the given statement are:

p: A natural number is odd

q: Square of a natural number is odd The given statement is: “If p, then q” It is same as each of the following statements.

- p⇒ q i.e, x is an odd natural number

⇒ x^{2}is an odd natural number - p is a sufficient condition for q

i..e, knowing that a natural number is odd is sufficient to conclude that its square is odd. - p only if q i.e, a natural number is odd only if its square is odd.
- q is necessary condition for p

i..e, when a natural number is odd, its square is necessarily odd - ~ q => ~ p i.e, If the square of a natural number is not odd, then the natural number is not odd.

Question 2.

For each of the following statements, determine whether an inclusive “OR” or exclusive “OR” is used. Give reasons for your answer.

i. All integers are positive or negative.

ii. The school is closed if it is a holiday or a Sunday.

Answer:

i. ‘OR’ is an exclusive OR, since all integers cannot be both positive as well as negative

ii. Here OR is inclusive since school is closed on holiday as well as on Sunday.

Question 3.

The truth value of the statement

“Ernakulam is a city in Kerala and 3 + 2 = 6” is

a. T

b. F

c. can’t say

d. T and F

b. Write the truth value of each of the following statements:

i. If 2 + 1 = 3, then is irrational number”.

ii. If 2 + 1 = 3, then is rational number”.

iii. If 3 is not an integer, then ^{ }rational number.

Answer:

a. F

b.

(i) True

(ii) False

(iii) True

Question 4.

a. The truth value of the negation of the statement “2 is a composite number” is

a. T

b. F

c. T and F

d. can’t say

b. Form the bi conditional statement of the following statements:

p: Δ ABC is equilateral,

q: Δ ABC is equiangular.

Answer:

a. T

b. The biconditional statement of p and q is given by ‘ Δ ABC is equilateral if and only if it is equiangular’.

Question 5.

Find out which of the following sentences are statements and which are not, justify your answer.

i. Listen to me, John!

ii. The moon revolves around the sun.

iii. Who are you?

Answer:

i. No, it is neither true nor false.

ii. Yes, it has truth value-true.

iii. No, it has no truth value.

Question 6.

i. “The number 6 has three prime factors”. Is it a statement? Justify your answer.

ii. Write the truth value of the statement: “Kerala is in India and 3 + 4 = 8”.

iii. Write the negation of the statement p: All squares are not rectangles.

Answer:

i. Yes, it is a statement. It has truth value- false

ii. False

iii. ~ p : All squares are rectangles.

Question 7.

i. Write the contrapositive of the following

statement: “If a triangle is equilateral, it is isosceles”.

ii. Check whether the following statement is true or false by contrapositive method: “If x and y are odd integers then xy is odd”.

Answer:

i. If a triangle is not isosceles, then it is not equilateral,

ii. Let p : xy is odd

q : both x and y are odd

~ q: It is false that both x and y are odd

⇒ x (or y) is even

Let x = 2n for some integer n.

Then xy = (2n) (y) = 2ny, n∈z ⇒ xy is even ⇒ ~ p is true

Then we have shown that ~ q⇒ ~ p.

Hence given statement is true.

Question 8.

Write the negation of the following statement:“Both the diagonals of a rectangle have the same length”, h. Verily by the method of contradiction that 7i is irrational.

Answer:

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

b. 275 is a perfect square,

Let is 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 9.

Consider the following statements:

p : it is raining

q: it is cloudy

i. Express the following sentences in symbolic form using p and q.

a. A necessary condition for raining is that it is cloudy.

b. A sufficient condition for raining is that it is cloudy.

c. A necessary and sufficient condition for raining is that it is cloudy.

ii. If p is true and q is true, find the truth values of the above statements.

Answer:

i.

(a) p → q

(b) q → p

(c) p ↔ q

ii. Since p and q are true,

p → q and q → p are true Since p and q have the same truth value, p ↔ is true

**Long Answer** **Type Questions **

**(Score 6)**

Question 1.

a. Check whether the following statement is true or not: “If x and y are odd integers, then xy is an odd integer”.

b. Show that the following statement is true by the method of contrapositive: “If x^{2} is even, x∈Z, then x is also even”.

Answer:

a. We may check the validity of the given statement by direct method as follows: Let p: x and y are odd integers, q: xy is odd.

Assume p is true. Then x = 2m + 1, y = 2n+ 1,m,n∈z xy = (2m+ 1)(2n+ 1)

= 2 (2mn + m + n) + 1 ⇒ xy is odd ⇒ q is true

Hence “If p, then q” is a true statement.

b. Let p : x^{2} is even, x∈Z

q : x is even

If possible, let q be false. Then x is not an even integer, i.e, x is an odd integer.

⇒ x = (2k+ 1),k ∈ z

⇒ x^{2}=(2k+ 1)^{2}=4k^{2}+4k+1=2 (2k^{2}+2k)+1

⇒ x^{2} is odd

⇒ x^{2} is not even

⇒ p is false.

Thus q is false ⇒ p is false

i..e, “If~q then~p

∴ “If p, then q”is a true statement.

Question 2.

Check the validity of the statement given below by contradiction method.“The sum of an irrational number and a rational number is irrational”.

Answer:

Let p be the given statement. If possible Let p be not true. Then p is false.

⇒ the sum of an irrational number and a rational number is not irrational.

⇒ there exists an irrational number x (say) and a rational number y (say) such that x + y is not irrational.

⇒ x + y is rational, say, z ⇒ x + y = z

⇒ x = z-y⇒ xis rational. But x is irrational. So we arrive at a contradiction. Thus our assumption is wrong. Hence p is true.

Question 3.

a. Check the validity of the bi conditional statement: “a glass of water is half empty if and only if it is half full”.

b. Verify by method of contradiction is irrational.

Answer:

a.

p: a glass of water is half empty

q: a glass of water is half full

p ⇒ q : If a glass of water is half empty, then it is half full (True)

q ⇒ p: If a glass of water is half full, then it is half empty (True)

Hence the given statement is true.

b.

Let the given statement be false.

i..e, is rational.

It means , where p and q are prime.

On squaring both sides, we get

It means 11 divides p.

Thus, there exists an integer r such that

From Eqs. (i) and (ii), we get 11q^{2}=121r^{2} ⇒q^{2}=11r^{2 }It means 11 divides q.

From Eqs. (ii) and (iv), we get 11 divides p and q.

It means 11 is a common factor of p and q which contradicts our assumption that p and q have no common factor.Hence, is rational is false.

Question 4.

a. Check the validity of the statement given below by contradiction method. ‘p: The sum of an irrational number and a rational number is irrational”

b. Determine the truth value of each of the following statements.

i. Kolkata is in India and 2+2=4.

ii. Nepal is in Asia or 5 + 5 = 11.

Answer:

a. If possible, let p be not true. Then, p is false.

The sum of an irrational number and a rational number is not irrational.

There exists an irrational number x ( say) and a rational number y(say) such that x + y is not irrational.

⇒ x + y is rational, say z.

⇒ x + y=z ⇒ x = z-y ⇒ x is rational.

But x is irrational. So, we arrive at a contradiction. Thus, our supposition is wrong.

Hence, p is true.

b.

i. True, because both statements are true.

ii. True, because first sub-statement “Nepal is in Asia” is true and 5 + 5 = 11 is false.

**NCERT Questions and Answers**

Question 1.

Check whether the following sentences are statements. Give reasons for your answer.

i. 8 is less than 6.

ii. Every set is a finite set.

iii. The sun is a star.

iv. Mathematics is fun.

v. There is no rain without clouds.

vi How far is Chennai from here?

Answer:

i. This sentence is false because 8 is greater than,6. Hence it is a statement.

ii. This sentence is also false since there are sets which are not finite. Hence it is a statement.

iii. It is a scientifically established fact that sun is a star and, therefore, this sentence is always true. Hence it is a statement.

iv. This sentence is subjective in the sense that for those who like mathematics, it may be fun but for others it may not be. This means that this sentence is not always true. Hence it is not a statement.

v. It is a scientifically established natural phenomenon that cloud is formed it rains. Therefore, this sentence is always true. Hence it is a statement,

vi. This is a question which also contains the word “Here”. Hence it is not a statement.

Question 2.

Which of the following sentences are statements? Give reasons for your answer.

i. There are 35 days in a month.

ii. Mathematics is difficult.

iii. The sum of 5 and 7 is greater than 10.

iv. The square of a number is an even number.

v. The sides of a quadrilateral have equal length.

vi Answer this question.

vii. The product of (-1) and 8 is 8.

viii. The sum of all interior angles of a triangle is 180°.

ix. Today is a windy day.

x. All real numbers are complex numbers.

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

Answer:

a.

i. This sentence is always false because the maximum number of days in a month is 31. Therefore, it is a statement.

ii. This is not a statement because for some people mathematics can be easy and for some others it can be difficult.

iii. This sentence is always true because the sum is 12 and it is greater than 10. Therefore, it is a statement.

iv. This sentence is sometimes true and sometimes not true. For example the square of 2 is even number and the square of 3 is an odd number. Therefore, it is not a statement.

v. This sentence is sometimes true and sometimes false. For example, squares and rhombus have equal length whereas rectangles and trapezium have unequal length. Therefore, it is not a statement.

vi. It is an order and therefore, is not a statement.

vii. This sentence is false as the product is (-8). Therefore, it is a statement.

viii. This sentence is always true and therefore, it is a statement.

ix. It is not clear from the context which day is referred and therefore, it is not a statement.

x. This is a true statement because all real numbers can be written in the form a + ixO.

b.

i. Everyone in this room is bold. This is not a statement because from the context it is not clear which room is referred here and the term bold is not precisely defined.

ii. She is an engineering student. This is also not a statement because who ‘she’ is.

iii. “cos^{2} θ is always greater than 1/2”. Unless, we know what Q is, we cannot say whether the sentence is true or not.

Question 3.

Write the negation of the following statements and check whether the resulting statements are true,

i. Australia is a continent.

ii. There does not exist a quadrilateral which has all its sides equal.

iii. Every natural number is greater than 0.

iv. The sum of 3 and 4 is 9.

Answer:

i. The negation of the statement is

It is false that Australia is a continent.

This can also be rewritten as Australia is not a continent.

We know that this statement is false.

ii. The negation of the statement is

It is not the case that there does not exist a quadrilateral which has all its sides equal. This also means the following:

There exists a quadrilateral which has all its sides equal.

This statement is true because we know that square is a quadrilateral such that its four sides are equal.

iii. The negation of the statement is

It is false that every natural number is greater than 0.

This can be rewritten as ‘There exists a natural number which is not greater than 0.

This is a false statement.

iv. The negation is

It is false that the sum of 3 and 4 is 9.

This can be written as

The sum of 3 and 4 is not equal to 9.

This statement is true

#### Plus One Maths Chapter Wise Questions and Answers