Mathematical Reasoning — Chapter Test | Class 11 IMO

Q1

The statement 'p only if q' is logically represented as:

The phrase 'p only if q' means that if p happens, q must have happened. It translates to p → q.

Q2

The contrapositive of 'If the power goes out, the AC stops working' is:

Swap and negate: 'AC does not stop working' implies 'power did not go out'.

Q3

Which of the following is a statement?

A statement is a sentence that is definitely true or false. '2 + 3 = 5' is true and hence a statement.

Q4

How many rows are there in the truth table of three propositions?

There are 2³ = 8 possible combinations.

Q5

If p is true, q is false and r is true, then (p ∧ r) ∨ q is:

p ∧ r is true, and true ∨ false is true.

Q6

Find the negation of: 'For every real number x, x² + 1 > 0'.

The negation of 'For every x, P(x)' is 'There exists an x such that ~P(x)'. The negation of '>' is '≤'.

Q7

The contrapositive of 'If a number is divisible by 5, then it ends in 0 or 5' is:

Reverse and negate both parts.

Q8

Which of the following is an example of an EXCLUSIVE 'OR'?

An exclusive OR means one or the other, but not both. A coin cannot land on both heads and tails simultaneously, making it strictly exclusive.

Q9

In the statement 'If it rains, then the match will be cancelled', the cancellation of the match is the:

In p → q, q is the necessary condition for p. If the match is not cancelled, it necessarily means it did not rain.

Q10

A truth table for a compound statement involving 3 distinct logical variables (p, q, r) will have how many rows?

Each variable can be True or False (2 possibilities). For n variables, the number of rows is 2ⁿ. For 3 variables, 2³ = 8 rows.

Q11

Identify the truth value of: '3 is a prime number and 4 is a prime number.'

The first part (3 is prime) is True, but the second part (4 is prime) is False. True AND False = False.

Q12

Which of the following compound statements is a Tautology?

If (p ∧ q) is true, then p must be true. Therefore, the implication is T → T, which is True. If (p ∧ q) is false, an implication starting with F is vacuously True. Thus, it's a tautology.

Q13

The converse of 'If a number is divisible by 4, then it is even' is:

Converse interchanges hypothesis and conclusion.

Q14

The negation of the statement 'All equilateral triangles are isosceles' is:

The negation of a universal statement 'For all x, P(x)' is an existential statement 'There exists an x such that not P(x)'.

Q15

If p is false and q is true, then p → q is:

Any implication with a false antecedent is true.