## MP Board Class 11th Maths Important Questions Chapter 14 Mathematical Reasoning

### Mathematical Reasoning Important Questions

**Mathematical Reasoning Objective Type Questions**

(A) Choose the correct option :

Question 1.

If p and H are statements then determine of equivalance statement is :

(a) p ⇔ q

(b) p ∨ q

(c) p ∧ q

(d) None of these.

Answer:

(a) p ⇔ q

Question 2.

Examine of the following sentence is not statement:

(a) The sum of interior angles of a quadrilateral is 360°

(b) There are only three sides in a triangle

(c) You have long live

(d) The sum of three and three is 6.

Answer:

(c) You have long live

Question 3.

Examine that which of the following is correct part of a negative statement:

(a) 3 + 5 > 9

(b) Each square is a rectangle

(c) Equilateral triangle is an isosceles triangle

(d) None of these.

Answer:

(a) 3 + 5 > 9

Question 4.

Which of the following is a statement:

(a) n is a real numbers

(b) Let us go

(c) Switch of the fan

(d) 3 is a natural number.

Answer:

(d) 3 is a natural number.

Question 5.

The connective in the statement “3 + 5 > 9 or 3 + 5 < 9” is :

(a) >

(b) <

(c) or

(d) and.

Answer:

(c) or

Question 6.

Examine which of the following sentence statement:

(a) Each square is a rectangle

(b) Door closed

(c) God bless you

(d) Oh ! I passed.

Answer:

(a) Each square is a rectangle

Question 7.

The opposite statement of the statement p => q is :

(a) ~ q ⇔ p

(b) q ⇒ P

(c) ~ p ⇒ q

(d) None of these.

Answer:

(b) q ⇒ P

Question 8.

The negative of the statement “51 is not a multiple of 3” is :

(a) 51 is an odd number

(b) 51 is not an odd number

(c) 51 is a multiple of 2

(d) 51 is a multiple of 3.

Answer:

(d) 51 is a multiple of 3.

Question 9.

The contrapositive of the statement “ If p, then q ” is :

(a) If q, then ~ P

(b) If ~ p, then ~ q

(c) If p, then ~ q

(d) If ~ q, then ~ p

Answer:

(d) If ~ q, then ~ p

(B) Write answer in one word / sentence :

1. Write the component statements of the compound statement:

“13 is an odd number and a prime number.”

Answer:

p : Number 13 is prime; q: Number 13 is odd.

2. Write the negation of the statement:

“Everyone who lives in India is an Indian.”

Answer:

Every one who live in India is an Indian

3. Identify the connective in the following compound statement:

“It is raining or the Sun is shining.”

Answer:

or.

4. From the biconditional statement p ⇔ q, where :

p ≡ A triangle is an equilateral.

q ≡ All three sides of a triangle are equal.

Answer:

A triangle is an equilateral triangle if and only if all three sides of the triangle are equal.

5. If p ≡ Mathematics is hard, q ≡ 4 is even number, then write the formula p ∨ q in logical sentences.

Answer:

Mathematics is hard or 4 is even number.

6. If p ≡ the question paper is hard, q ≡ I will fail in the examination, then write the statement in symbolically, “The question paper is hard if and only if I will fail in the examination.”

Answer:

p ⇔ q.

7. State whether the statement is exclusive or inclusive :

“All integers are positive or negative.”

Answer:

Exclusive.

**Mathematical Reasoning Very Short Answer Type Questions**

Question 1.

Which of the following sentences are statements? Give reasons for your answer: (NCERT)

- There are 35 days in a month.
- Mathematics is difficult.
- The sum of 5 and 7 is greater than 10.
- The square of a number is an even number.
- The sides of a quadrilateral have equal lengths.
- Answer this question.
- The product of (- 1) and 8 is 8.
- The sum of all interior angles of a triangle is 180°.
- Today is a windy day.
- All real numbers are complex numbers.

Answer:

- A month has 30 or 31 days. It is false to say that a month has 35 days, hence it is a statement.
- Mathematics may be difficult for one but may be easy for the others. Hence it is not a statement.
- It is true that sum of 5 and 7 is greater than 10. Hence it is a statement.
- The square of a number may be even or it may be odd. Squaring of an odd number is always odd and square of even number is always even. Hence it is not a statement.
- A quadrilateral may have equal lengths as it may be a rhombus or a square or the quadrilateral may have unequal sides
- like parallelogram, hence it is not a statement.
- It is an order, hence it is not a statement.
- It is false because the product of (- 1) and 8 is – 8, hence it is a statement. It is true because sum of three angles of a triangles is 180°. Hence it is a statement.
- It is a windy day. It is not clear that about which day it is said. Thus it can’t be concluded whether it is true or false. Hence it is not a statement.
- It is true that all real numbers are complex numbers. All real numbers can be expressed as a + ib, hence it is a statement.

Question 2.

State whether the ‘or’ used in the following statements is exclusive or inclusive. Give reasons for your answer: (NCERT)

- Sun rises or Moon sets.
- To apply for driving licence, you should have a ration card or a passport.
- All integers are positive or negative.

Answer:

- When Sun rises, the Moon sets. One of happening will take place, hence ‘or’ is exclusive.
- To apply for a driving licence either a ration card or a passport or both can be used, hence ‘or’ is inclusive.
- All integers are positive or negative. An integers cannot be both positive or negative, hence ‘or’ is exclusive.

Question 3.

Rewrite the following statement ‘if then’ in five different ways conveying the same meaning:

If a natural number is odd than its square is odd. (NCERT)

Answer:

- A natural number is odd implies that its square is odd.
- A natural number is odd only if its square is odd.
- If the square of a natural number is not odd, then the natural number is also not odd.
- For a natural number to be odd, its necessary that its square is odd.
- For a square of a natural number to be odd, if it is sufficient that the number is odd.

Question 4.

Give statement in (a) and (b) identify the statements given below as contra – positive or converse of each other: (NCERT)

(a) If you live in Delhi, then you have winter clothes.

(i) If you do not have winter clothes, then you do not live in Delhi.

(ii) If you have winter clothes, then you live in Delhi.

(b) If a quadrilateral is parallelogram, then its diagonal bisect each other.

(i) If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.

(ii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram

Answer:

(a) (i) Contrapositive statement, (ii) Converse statement.

(b) (i) Contrapositive statement, (ii) Converse statement.

Question 5.

Which of the following sentence is a statement:

- New Delhi is a capital of India.
- How are you?

Answer:

- Its a statement.
- Its not a statement.

Question 6.

Find the component statement of the following statements :

1. The sky is blue and grass is green.

2. All the rational numbers are real number and all the real numbers are complex number.

Solution:

1. The components are:

p : The sky is blue.

q : The grass is green.

2. The component statement are as follows:

p : All the rational numbers are real number.

q : All the real numbers are complex number.

Question 7.

If p = It is 7 o’clock

q = The train is late,

then write the following symbols in the form of statement:

- q ∨ ~ p
- p ∧ q
- ~ (p ∧ q)
- p ∧ ~ q

Answer:

- Either the train is late or it is not 7 o’clock.
- It is 7 o’clock, but train is late.
- It is not true that it is 7 o’clock and the train is late.
- It is 7 o’clock, but train is not late.

Question 8.

If p ≡ He is intelligent and q ≡ He is strong, then write the following statement symbolically with the help of logical connectives:

- He is intelligent and strong.
- He is intelligent but not strong.
- He is neither intelligent nor strong.

Answer:

- p ∧ q
- p ∧ ~ q
- ~ q ∧ ~ q

**Mathematical Reasoning Short Answer Type Questions**

Question 1.

Show that the statement p : If x is a real number such that x3 + 4x = 0, then x is ‘0’ is true by :

- Direct method
- Method of contradiction
- Method of contrapositive

Solution:

1. Direct method :

Given :

x^{3} + 4x = 0 ⇒ x(x^{2} + 4) = 0

x = 0 or x^{3} + 4 = 0

x is real number.

∴ x^{2} + 4 ≠ 0

∴ x = 0.

2. Method of contradiction :

Let x ≠ 0

Let x = p, (where p is a real number)

p is one the root of equation x^{3} + 4x = 0.

∴ p^{3} + 4p = 0

⇒ P(P^{2} + 4) = 0

⇒ P = o

and p^{2} + 4 ≠ 0, (It is a real number)

∴ p = 0.

3. Method of contrapositive :

Let x = 0 is not true and x = p ≠ 0

∴p being the root of x^{3} + 4x = 0

∴ p^{3} + 4 p = 0

⇒ P(p^{2} + 4) = 0

p = 0 and p^{2} + 4 = 0

⇒ P(p^{2} + 4) ≠ 0, if p is not true

∴ x = 0 is one root of equation x^{3} + 4x = 0.

Question 2.

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

Solution:

Let a = 3 and b = -3 then a, b are real numbers.

Here a^{2} = b^{2} but a b

Hence a, b ∈ R and a^{2} = b^{2}

⇒ a = b, statement is not true.

Question 3.

Find the components of the following compound statement and check whether they are true or false.

(i) The number 3 is prime or odd.

(ii) All the integers are positive or negative.

(iii) Number 100 is divisible by the numbers 3, 11 and 5.

Answer:

(i) p : Number 3 is prime.

q : Number 3 is odd.

Here p and q both are true.

(ii) p : All the integers are positive. q : All the integers are negative.

Clearly p and q both are false.

(iii) p : The number 100 is divisible by 3.

q : The number 100 is divisible by 11.

r : The number 100 is divisible by 5.

p is false, q is false and r is true. p, q and r are false statement.

Question 4.

Show that the following statement is true by the method of contrapositive :

p : If x is an integer and x^{2} is even, then x is also even.

Solution:

P – If x is an integer and x^{2} is even, then x is also even.

Let q : x is an integer and x2 is even.

r : x is even.

To prove that P is true by contrapositive method, we assume that r is false and prove that q is also false.

Let x is not even.

To prove that q is false, it has to be proved that

x is not an integer or x^{2} is not even.

x is not even implies that x^{2} is also not even.

Therefore, statement q is false.

Thus, the given statement p is true.

Question 5.

For each of the following compound statements first identify the connecting words and then break it into compound statements: (NCERT)

(i) All rational numbers are real and all real numbers are not complex.

(ii) x = 2 and x = 3 are the roots of the equation 3x^{2} – x – 10 = 0.

Solution:

(i) The connecting word in the compound statement is ‘and’.

p : All the rational numbers are real number.

q: AH the real numbers are not complex number.

(ii) The connecting word in the compound statement is ‘and’.

p : x = 2 is the root of equation 3x^{2} – x – 10 = 0.

q : x = 3 is the root of equation 3x^{2} – x – 10 = 0.