Chapter 9 Solutions — Propositions and their Converses

Exercise Set 9.1

1. 1.
   If two lines are parallel, then the corresponding angles formed by a transversal are equal. Frame the converse and determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.
2. 2.
   If a quadrilateral is a square, then all its angles are equal. Frame the converse and determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.
3. 3.
   Given any △ABC, bisect the angles at B and C. The bisectors meet at the incentre I of the triangle. Extend the bisectors beyond I till they meet the opposite sides at E and F respectively (BE extended meets AC at E; CF extended meets AB at F). Proposition: If AB = AC, then IE = IF. Frame the converse and determine if each statement is true or not.
4. 4.
   If x = y, then a + x = a + y, where x, y and a are any three numbers. This proposition and its converse are routinely used while solving equations. Frame the converse and determine if each statement is true or not.
5. 5.
   If a and b are perfect squares, then ab is a perfect square. Frame the converse and determine if each statement is true or not.
6. 6.
   If x = y, then x² = y², where x and y are real numbers. Frame the converse and determine if each statement is true or not.
7. 7.
   If x = y, then x³ = y³, where x and y are real numbers. Frame the converse and determine if each statement is true or not.
8. 8.
   If n is divisible by 24, then it is divisible by both 4 and 6. (n is a positive integer.) Frame the converse and determine if each statement is true or not.
9. 9.
   If n is divisible by 60, then it is divisible by both 5 and 12. (n is a positive integer.) Frame the converse and determine if each statement is true or not.
10. 10.
    If n is the square of a prime number, then it has exactly 3 factors. (n is a positive integer.) Frame the converse and determine if each statement is true or not.
11. 11.
    If n is a product of two unequal prime numbers, then it has exactly 4 divisors. (n is a positive integer.) Frame the converse and determine if each statement is true or not.
12. 12.
    If n and n + 3 have no factors in common, then n is not a multiple of 3. (n is a positive integer.) Frame the converse and determine if each statement is true or not.
13. 13.
    There are no known 'neat' expressions that generate only primes! Find counterexamples to the following claims.

    • (i) All numbers of the form 4n² + 1 are prime.
    • (ii) All numbers of the form n² + n + 11 are prime.
    • (iii) All numbers of the form 4ⁿ + 3 are prime.
14. 14.
    Find counterexamples to the following statements.

    • (i) If n is a prime number, then 2ⁿ − 1 is a prime number.
    • (ii) If n is an even number, then 2ⁿ + 1 is a prime number.
15. 15.
    Consider the statement: 'If a number is divisible by 8, then it is divisible by both 2 and 4'.

    • (i) Justify the statement.
    • (ii) To check whether a given number is divisible by 8, is it enough to check whether it is divisible by 2 and 4? Why or why not?
16. 16.
    Recall that a shortcut to check whether a given number is divisible by 3 is to add the digits of the number and check if the sum is a multiple of 3. Express the relationship between 'a number is divisible by 3' and 'sum of the digits is a multiple of 3' using 'If-then' sentences.
17. 17.
    Suppose we have a category of quadrilaterals Q, to be constructed using two thin sticks as diagonals (endpoints joined to form the quadrilateral).

    • (i) Suppose Q satisfies: 'If a quadrilateral is of type Q, then it has equal-length diagonals.'
    • (a) Should the two sticks be of equal length?
    • (b) Will it matter how the two sticks are put together?
    • (ii) Instead, suppose Q satisfies: 'If a…