From the chapter Mathematical Reasoning some of the important questions are covered.

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Questions:

Generic 2

10 x 2 = 20

Given below are two statements
p: 25 is a multiple of 5
q: 25 is a multiple of 8
Write the compound statements connecting these two statements with "and" and "or". In both cases check the validity of the compound statement.

For each of the following compound statements first identify the connecting words and then break it into component statements.
(i) The sun is cold and the moon rotates around the earth.
(ii)The school works or a holiday is declared.
(iii) Ankita is rich or happy

Write the negation of the following statements:
(i) Every natural number is greater than 0.
(ii) The sum of 4 and 6 is 10.
(iii) A whole number is an integer.

Write the negation of the following statements.
If 2 + 3 = 5, then 5 is an odd number.

For the following compound statement, first identify the connecting words and then break into component statement.
Write the contrapositive and converse of the statement 'Something is cold implies that it has low temperature'.

Write down the converse of the following statements.
If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.

Find the component statement of the following compound statements and check whether they are true or false.
100 is divisible by 3,11 and 5.

If p and q are two statements given by
p :30 is a multiple of 5.
q :30 is a multiple of 8.
Write the compound statement by using the connective 'and' check its validity.

Rewrite the following statements in the form of conditional statements
A necessary condition for Indian team to win a cricket match is that the selection committee selects an all-rounder.

10.

Find the conditional statement of 'You will get a sweet dish after the dinner'.

Generic 4

2 x 4 = 8

11.

Write down the negation
\(\triangle ABC\) is isosceles, if and only if \(\angle B=\angle C\)

12.

Write the component statements of the following compound statement and check whether the compound statement is true or false
The sun is a star or Sun is plant

Generic 6

3 x 6 = 18

13.

Check the validity of the statements given below by the method given against it
(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method)
(ii) q: If n is a real number with n > 3, then n² > 9 (by contradiction method)

14.

State the converse and contrapositive of each of the following statements:
(i) p: A positive integer is prime only if it has
no divisors other then 1 and itself.
(ii) q: I goto a beach whenever it is a sunny day.
(iii) r: if it is not outside, then you feel thirsty.

15.

For each of the following compound statement first identify the connecting words and then break it into component statements.
(i) All rational numbers are real and all real numbers are not complex.
(ii) Square of an integer is positive or negative.
(iii) The sand heats up quickly in the sun and does not cool down fast at night.
(iv) x = 2 and x = 3 are the roots of the equation 3x² - x - 10 = 0.

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Answers Key:

Generic 2

10 x 2 = 20

The compound statement with "and " is
25 is a multiple of 5 and 8.
Since p is true and q is false, so the compound statement with "and" is not true.
Thus the statement" p and q" is not valid.
Now the compound statement with "or" is
25 is a multiple of 5 or 8. Since p is true and q is false, so the compound statement with "or" is true Thus the statement "p or q" is valid.

(i)The component statement has the connecting word 'and'. Component statements are
p: The sun is cold.
q: the moon rotates around the earth.
(ii)The component statement has the connecting word 'or'. Component statements are
p: The school works.
q: A holiday is declared.
(iii)The component statement has the connecting word 'or'. Component statements are.
p: Ankita is rich.
q: Ankita is happy

(i) Negation of statement is: There exists a natural number which is not greater than 0.
(ii) Negation of statement is: The sum of 4 and 6 is not equal to 10.
(iii) Negation of statement is: A whole number is not an integer

Let p : 2 + 3 =5
and q : 5 is an odd number.
The given proposition is P\(\Rightarrow\)q.
\(\sim (p\Rightarrow q)\equiv p\wedge (\sim q)\)
= 2 + 3 = 5 and 5 is an even number.

Contrapositive statement If the temperature of something is not low, then it is not cold.
Converse statement If something has low temperature, then it is cold.

If the opposite angles of a quadrilateral are supplementary, then S is cyclic.

The component statements are
p :100 is divisible by 3
q :100 is divisible by 11
r : 100 is divisible by 5
p,q are false and r is true. So \(p\vee q\) is true.

The compound statement is '30 is a multiple of 5 and 8'.
The statement 'p and q' is not a valid statement.

The given statement is of form "q is necessary for p" where
p : Indian team wins a cricket match
q : The selection committee selects an all-rounder
Which is an equivalent form of "if p then q". So the equivalent formulation of the given statement is "If the teams wins a cricket match then selection committee selects an all-rounder ".

10.

The conditional statement of given statement is 'If you take dinner, then you will get a sweet dish.'

Generic 4

2 x 4 = 8

11.

Either \(\triangle ABC\) is isosceles and \(\angle B\neq \angle C\) or \(\triangle ABC\) is not isosceles and \(\angle B=\angle C\)

12.

Compound statement is true and its component statements are
p: The Sun is a star
q: The Sun is a plant

Generic 6

3 x 6 = 18

13.

(i)Let us assume that p is not true.
\(\therefore \)Sum of an irrational and a rational number is not irrational.
\(\Rightarrow \)There exists an irrational number a and a rational number b such that a + b is not irrational.
\(\Rightarrow \)a + b = c (say) is a rational number.
\(\Rightarrow \)a=c-b
\(\Rightarrow \)a is rational.
But a is irrational, which is contradiction
So our supposition in wrong.
Thus p is true.
(ii) Let rand s be the statements given by
r: n is a real number with n > 3
s: n²> 9
If possible let q is not true then
\(\Rightarrow \)\(\sim \)q is true
\(\Rightarrow \)(r\(\Rightarrow \)s) is true
[\(\therefore \)q:rs]
\(\Rightarrow \)r and \(\sim \)s is true.
\(\Rightarrow \)n is a real number with n > 3 and n² < 9
which is contradiction So our supposition is wrong.
Thus q is true.

14.

(i) The contrapositive of the given statement is: If a positive integer has divisors other than 1 and itself then it is not prime. The converse of the given statement is: If a positive integer has no divisors other than 1 and itself, then it is a prime
(ii) The contrapositive of the given statement is: If! do not go to a beach then it is not a sunny day. The converse of the given statement is: If I go to beach, then it is a sunny day.
(iii) The contrapositive of the given statement IS: If you do not feel thirsty, then it is not hot outside. The converse of the given statement is: If you feel thirsty, then it is hot outside.

15.

(l) The component statement has the connecting word 'and' component statements are
p: All rational numbers are real.
q: All real numbers are not complex.
(ii) The component statement has the connecting words 'or' component statements are
p: Square of an integer is positive.
q: Square of an integer is negative.
(iii) The component statement has the connecting word 'and' component statements are
p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.
(iv) The component statement has the connecting word 'and' component statements are
p: x = 2 is a root of the equation 3\({ x }^{ 2 }\)- x - 10 = 0.
q: x = 3 is a root of the equation 3\({ x }^{ 2 }\)- x - 10 = 0.