Important Questions: Mathematical Reasoning | Mathematics for Grade 11 PDF Download

Q1: Write a component statement for the following compound statements:
50 is a multiple of both 2 and 5.
Ans: Given compound statement: 50 is a multiple of both 2 and 5.
p: 50 is multiple of 2
q: 50 is multiple of 5.

Q2: Write the contrapositive of the given if-then statements:
(i) If a triangle is equilateral, then it is isosceles
Ans: Contrapositive statement: If a triangle is not isosceles, then it is not equilateral.

(ii) If a number is divisible by 9, then it is divisible by 3.
Ans: Contrapositive statement: If a number is not divisible by 3, then it is not divisible by 9.

Q3: Find the component statements for the following given statements and check whether it is true or false:
(i) A square is a quadrilateral and its four sides are equal
Ans: The component statements are:
P: A square is a quadrilateral
Q: A square has all its sides equal.
In this statement, the connecting word is “and”
We know that a square is a quadrilateral
So, the statement P is true.
Also, it is known that all the four sides of a square are equal.
Hence, the statement “Q” is also true.
Therefore, both the component statements are true.

(ii) All prime numbers are either even or odd
Ans: The component statements are:
P: All the prime numbers are odd numbers
Q: All the prime numbers are even numbers
In this statement, the connecting word is “or”
We know that all the prime numbers are not odd numbers
So, the statement P is false.
Also, it is known that all the prime numbers are not even numbers.
Hence, the statement “Q” is also false.
Therefore, both the component statements are not true.

Q4: Write the negation of the following statements
(i) The number 3 is less than 1.
Ans: The number 3 is not less than 1 (or) The number 3 is more than 1.

(ii) Every whole number is less than 0.
Ans: Every whole number is not less than 0 (or) Every whole number is more than 0.

(iii) The sun is cold
Ans: The sun is not cold (or)The sun is hot.

Q5: Identify the quantifier in the following statement.
There exists a real number which is twice itself.
Ans: Given statement:
There exists a real number which is twice itself.
For the given statement, the quantifier is “There exists”.

Q6: Show that the statement, p: if a is a real number such that a3 + 4a =0, then a is 0″, is true by direct method?
Ans: Let q and r are the statements given by q: a is a real number such that a3 + 4a = 0
r: a is 0.
let q be true then
a is a real number such that a3 + 4a = 0
a is a real number such that a(a2 + 4) = 0
a = 0
r is true
So, q is true and r is true, so p is true.

Q7: Which of the following sentences are statements? Justify your answer.
(i) Answer this question
Ans: Since it is an order, the given sentence is not a statement.

(ii) All the real numbers are complex numbers
Ans: We know that all the real numbers can be written in the form: a+i0
Where a is a real number. Hence, it always true, the given sentence is a statement

(iii) Mathematics is difficult
Ans: Mathematics is a subject that can be easy for some people and difficult for some people.
So, the given sentence can be both true or false.
Hence, it is not a statement.