RBSE Class 11 Maths Important Questions Chapter 14 Mathematical Reasoning

Rajasthan Board RBSE Class 11 Maths Important Questions Chapter 14 Mathematical Reasoning Questions and Answers.

RBSE Class 11 Maths Chapter 14 Important Questions Mathematical Reasoning

Question 1.
Check whether ‘or’ used in the following compound statement Is exclusive or inclusive? Write the component statement of the compound statement and use them to check whether the compound statement is true or not. Justify your answer.
t: “You are wet when it rains or you are in a river”.
Answer:
‘Or’ used in the given statement is inclusive Because it is possible that it rains and you are in the river.
The component statement of the given statement are
p : you are wet when it rains. .
q : you are wet when you are in a river.
Here, both the component statements are true.
Therefore, the compound statement is true.

Question 2.
Write the negation of the following statements:
(i) p: Lucknow is the capital of Uttar Pradesh.
(ii) q : All birds have wings.
(iii) r : There exists a rational number x, such that x² = 3
Answer:
(i) p: Lucknow is the capital of Uttar pradesh.
Negation ~ p: Lucknow is not the capital of Uttar Pradesh. or p: It is not true that Lucknow is the capital of Uttar Pradesh.
or ~ p: There exist no Lucknow city which is capital of Uttar Pradesh.

(ii) q : All birds have wings.
The negation of the statement is:
~ q : There exists a bird which have no wings.

(iii) r: There exists a rational number x such that x² = 3 the negation of the statement is:
~ r: There exists no rational number x such that x² = 3
or ~ r: For all rational number x, x² ≠ 3.

Question 3.
Write the negation of the following sentences:
(i) p: For each real number x, x² > x.
(ii) q : There exist a rational number x such that x² = 3
(iii) r : Each animal have four legs.
Answer:
(i) Negation of p ~ p.
~ p : There exist a real number for which x² < x.
[∵ x² >x ⇒ x ≠ 0, x ≠ 1]

(ii) Negation of q ~ q ,
q : For each rational number x² ≠ 3

(iii) ≠ r: There exist an animal which does not have four legs.

Question 4.
Using the words ‘Neesary and Sufficient’ rewrite the statement “The integer n is even if and only if n² is even”. Also check whether the statements is true.
Answer:
The necessary and sufficient condition that the integer n be even is n² must be even.
p and q denote the statements.
p: the integer n is even.
q : n² in even.
To check the validity of ”p if and only if q”,
we have to check whether “if p then q” and “if q then p” is true.

Case I. If p, then q
If p, then q is the statement.
If the integer n is even, then n² is even.
We have to check whether this statement is true.
Let us assume that n is even.
Then, n = 2k, where k is an integer.
Thus, n² = (2k)² = 4k² = 2(2k²)
Thus, n² is even, since coefficient of 2k² is 2.

Case II. If q, then p
If q, then p is the statement:
If n is an integer and n² is even,
then n is odd.
We have to check whether this statement is true. We check this by contrapositive method.
The contrapositive of the given statement is:
‘If n is an odd integer, then n² is also an odd integer’
n is odd implies that n = 2k + 1, where k is an integer
∴ n² = (2k + 1)² = 4k² + 4k + 1
= 4k (k +1) + 1
which is odd.

Question 5.
For the given statements identify the necessary and sufficient conditions:
t: if you did not wear halmet while driving the bike you will get a fine.
Answer:
Let p and q denote the statements:
p : If you did not wear halmet while driving the bike
q: You will get a fine.
The implication if p then q indicates that p is sufficient for q.
i.e. To get fine, driving the bike without helmet is sufficient.
Here, ‘not wearing heLment’ is sufficient condition.
Thus, if p, then q also indicates that q is necessary for p.
i.e. when you did not wear helmet while driving the bike then you will necessarily get a fine.
Here, the necessary condition is ‘getting a fine’.

Multiple Choice Questions

Question 1.
The contrapositive of the statement “If you are born in India, then you are a citizen of India”, is:
(a) If you are not a citizen of India, then you are not born in India.
(b) If you are a citizen of India, then you are born in India.
(c) If you are born in India, then you are not a citizen of India.
(d) If you are not born in India, then you are not a citizen of India.
Answer:
(a) If you are not a citizen of India, then you are not born in India.

Question 2.
Contrapositive of the statement “If two numbers are not equal, then their squares are not equal” is:
(a) If the squares of two numbers are not equal, then the numbers are not equal.
(b) If the squares of two numbers are equal, then the numbers are equal.
(c) If the squares of two numbers are not equal, then the numbers are equal.
(d) If the squares of two numbers are equal, then the numbers are not equal.
Answer:
(b) If the squares of two numbers are equal, then the numbers are equal.

Question 3.
The negation of the statement:
“If I become a teacher, then I will open a school”, is :
(a) Twill become a teacher and I will not open a school.
(b) Either I will not become a teacher or I will not open a school.
(c) Neither I will become a teacher nor I will open a school
(d) I wilt not become a teacher or! will open a school.
Answer:
(a) Twill become a teacher and I will not open a school.

Question 4.
Consider the following statements:
P: Suman is brilliant.
Q: Suman is rich.
R : Suman Is honest.
The negation of the statement. “Suman Is brilliant and dishonest, if and only if Suman is rich” can be expressed as:
(a) ~ [Q ↔ (P ^ ~ R)]
(b) ~ Q ↔ P ^ R
(c) ~ (P ^ ~ R) ↔ Q
(d) ~ P ^ (Q ↔ ~ R)
Answer:
(a) ~ [Q ↔ (P ^ ~ R)]

Question 5.
Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x E S such that x > 0.
Which of the following statements is the negative of the statement P?
(a) There is a rational number x ∈ S such that x ≤ 0
(b) There is no rational number x ∈ S such that x ≤ 0
(c) Every rational number x ∈ S satisfies x ≤ 0
(d) x ∈ S and x ≤ 0 ⇒ x is not rational.
Answer:
(c) Every rational number x ∈ S satisfies x ≤ 0

Fill in the Blanks

Question 1.
The main asset that made humans superior to other species is the ability to ...................................... .
Answer:
reason

Question 2.
In mathematical language, there are two kinds of reasoning named ...................................... and ...................................... .
Answer:
inductive, deductive

Question 3.
The basic unit involved in mathematical reasoning is a mathematical ...................................... .
Answer:
statement

Question 4.
A sentence is called mathematically acceptable statement, if it is either ....................................... or ...................................... . but not both.
Answer:
true, false

Question 5.
For any natural numbers x and y, the sum of x and y is greater than 0 is a ...................................... .
Answer:
statement

Question 6.
What makes a ...................................... a statement is the fact that the sentence is either true or false but not both.
Answer:
sentence

Question 7.
The ...................................... of a statement is called the negation of the statement.
Answer:
denial

Question 8.
If p is a statement, then the negation of p is also a statement and is denoted by ~ p, and read as ...................................... .
Answer:
not P,

Question 9.
A ...................................... statement is a statement which is made up of two or more statement.
Answer:
compound,

Question 10.
And, or which are often used in Mathematical statements are called ...................................... .
Answer:
connectives

True/False

State whether the following statements are true or false:

Question 1.
The compound statement with ‘And’ is true if all its component statements are true.
Answer:
True

Question 2.
The compound statement with ‘And’ is false if any of its component statement is false.
Answer:
True

Question 3.
A compound statement with an ‘Or’ is true when one component statement is true or both the.
component statements are true.
Answer:
True

Question 4.
A compound statement with an ‘Or’ is false when both the component statements are false.
Answer:
True

Question 5.
The words “There exists” and “For all” are called connectives.
Answer:
False

Question 6.
Counter examples are used to disprove the statement.
Answer:
True

Question 7.
Generating examples in favour of a statement of not provide validity of the statement.
Answer:
True

Question 8.
The method that Involves an examples of a situation where the statement is valid, is called counter example.
Answer:
False

Question 9.
The sentence which ¡s ambiguous is acceptable as a statement.
Answer:
False

Question 10.
The denial of a statement ¡s called the negative of the statement.
Answer:
True