Propositions and Their Converses: CBSE Class 9 Mathematics practice questions

Practise 50 questions on Propositions and Their Converses, free, with an explanation for each answer.

Sample questions

Which of these is a proposition? (A proposition is a statement that is definitely either true or false.)

Four cards showing a question mark, an exclamation mark, a triangle and a speech bubble

Please sit down quietly.
A triangle can have two right angles.
How tall is this triangle?
What a lovely drawing!

Show the answer

B. A triangle can have two right angles.

A proposition is a statement that is either true or false. "A triangle can have two right angles" is a proposition because it can be judged: it is false, since the angle sum would exceed $180^\circ$ . A request, a question and an exclamation cannot be called true or false, so they are not propositions.

Q: If a number is a multiple of 3, then it is a multiple of 6. How many numbers from 1 to 30 are counterexamples to Q?

A grid of the numbers 1 to 30 with the multiples of 3 shaded

Show the answer

C. 5

Multiples of 3 up to 30: $3, 6, 9, \dots, 30$ , which is 10 numbers. Multiples of 6: $6, 12, 18, 24, 30$ , which is 5 numbers. Counterexamples are multiples of 3 that are not multiples of 6: $3, 9, 15, 21, 27$ , so $10 - 5 = 5$ . The trap 10 counts every multiple of 3; 25 counts all numbers that are not multiples of 6.

For a number $n$ , pair each factor with its 'partner' (the factor that multiplies with it to give $n$ ). The argument says: if there is an odd number of factors, there must be a pair $(f, f)$ , so $n = f\times f$ . What does this argument prove?

The factors of a number on a line with arcs joining partner factors

Only Q: if $n$ has an odd number of factors, then $n$ is a perfect square.
Only P: if $n$ is a perfect square, then it has an odd number of factors.
Both P and its converse Q.
Neither P nor Q.

Show the answer

A. Only Q: if $n$ has an odd number of factors, then $n$ is a perfect square.

The argument starts with 'an odd number of factors' and ends with 'a perfect square', so it proves Q only. It does not show that every square has an odd number of factors, because a square could in principle have several repeated pairs. P needs the extra fact that a number has at most one pair $(f, f)$ .

Topics covered

Propositions
Converse of a proposition
Using a true proposition
Proof of the converse
The proof's altitude
Example 1: rain and the road
Counterexamples
Converse of a true statement
Counterexamples and Fermat
Example 2: multiples of 6 and 3
Finding a counterexample
Example 3: factor-partner argument
Factor pairs of 36
Odd number of factors
Proving Q in Example 3
Example 4: same area
Four possible outcomes
Exercise 8: divisibility
Example 5: converse of the Pythagoras theorem
Proof of the converse of Pythagoras
The right angle in the proof
Using the converse
Which angle is a right angle?
Understanding converses
Writing a converse
Exercise 1: parallel lines
Exercise 2: square and angles
Exercise 3: incentre figure
Reading Fig. 9.1
Converse of the incentre statement
Exercise 5: products of squares
Exercise 6: squares of numbers
Exercises 4 to 9: converses
Exercise 8: smallest counterexample
Exercise 10: exactly 3 factors
Exercise 11: four divisors
Exercise 11: converse
Exercise 12: converse
Exercise 13(ii): neat expressions
Exercises 13 and 14: counterexamples
Exercise 15: divisible by 8
Exercise 16: divisibility by 3
Exercise 17(i): equal diagonals
Exercise 17(ii): converse property
Squares and rectangles
Propositions and their converses
Key terms
Disproving a claim
False propositions and converses
Fermat and Euler

Start practising free

Propositions and Their Converses: CBSE Class 9 Mathematics practice questions

Sample questions

Topics covered

More CBSE Class 9 Mathematics chapters