Mathematical reasoning: Questions and Answers

30 mathematical reasoning questions solved step by step. Open a question for the full working, the marks for each step and exam practice.

• If a proposition is true, then is its converse always true?Answer: No. A true proposition can have a false converse. ‘If a figure is a square, then it has four sides’ is true, but its converse ‘If a figure has four sides, then it is a square’ is false: a 5 cm × 3 cm rectangle has four sides but is not a square.
• Statement 1: If two sides of a triangle are equal, then the angles opposite the equal sides are equal.
  Statement 2: If two angles of a triangle are equal, then the sides opposite the equal angles have equal lengths.
  We have proved the first statement in an earlier grade. Is the second statement true? Can you prove it?
  (Hint: Draw the altitude from the vertex containing the third angle.)Answer: Yes, Statement 2 is true. If ∠B = ∠C in △ABC, draw the altitude AD. Then △ABD ≅ △ACD (AAS), so AB = AC.
• Proposition P: If it rains, then the road is wet.
  Converse Q: If the road is wet, then it has rained.Answer: Proposition P is true, but its converse Q may not be true: the road may be wet because a tanker spilled water on it, though it has not rained. Such a case is a counterexample to Q.
• Proposition P: If a number is a multiple of 6, then it is a multiple of 3.
  Converse Q: If a number is a multiple of 3, then it is a multiple of 6.Answer: (i) P is true: a multiple of 6 is 6k = 3 × 2k, a multiple of 3. (ii) No. Q is false: 9 is a multiple of 3 but not a multiple of 6.
• In this example, n is any positive integer.
  Proposition P: If n is a perfect square, then it has an odd number of factors.
  Converse Q: If n has an odd number of factors, then it is a perfect square.
  You may recall that we came across these statements in the previous grade. Which of them are true?Answer: Both P and Q are true. ‘n is a perfect square’ and ‘n has an odd number of factors’ imply each other. For example, 36 = 6 × 6 has 9 factors, while 12 (not a square) has 6.
• Consider the following argument.
  Each factor of a number has a ‘partner’ factor such that their product yields the given number, e.g., 5 is a factor of 35, and 5 × 7 = 35. Here, 7 is the partner factor of 5, and vice-versa. Let us focus on factor-partner pairs of numbers, e.g., (1, 12), (2, 6), (3, 4) are the pairs for the number 12.
  If a number has an odd number of factors, there must be a factor-partner pair in which the same number repeats (e.g., the partner factor of 5 in 25). If not, the given number will have an even number of factors since each factor can be paired with its factor pair.
  Thus, if a number has an odd number of factors, then it is a perfect square.
  What does this argument prove? Statement P or Q?Answer: It proves Statement Q: if a number has an odd number of factors, then it is a perfect square. The argument starts from ‘odd number of factors’ and ends at ‘perfect square’. It does not prove P.
• This only proves Q. It doesn’t prove that every square number has an odd number of factors (Proposition P).
  Is P true?Answer: Yes, P is true. If n = f × f, then (f, f) is the only factor–partner pair that repeats a number; every other factor has a different partner. So the number of factors is (an even number) + 1, which is odd.
• (i) n has an odd number of factors
  which implies
  (ii) the existence of a factor-partner pair in which the same number repeats, say (f, f)
  which implies
  (iii) n is a square number: n = f × f
  We start with (iii). Clearly, (iii) implies (ii). Does (ii) imply (i)?Answer: Not by itself. (ii) only says there is at least one repeated pair (f, f); it does not say how many. With two such pairs the number of factors would be even. (ii) leads to (i) only once we add that a number has at most one repeated pair.
• For example, if the number of such pairs is two — say (f, f) and (g, g) — what can we say about the number of factors?Answer: The number of factors would be even: the other factors come in pairs of two different numbers (an even count), and f and g add 2 more. (This can never actually happen, since f × f = g × g forces f = g.)
• Proposition P: If two triangles have the same area, then they are congruent.
  Converse Q: If two triangles are congruent, then they have the same area.
  Determine if these statements are true or not. Justify the true statements and give a counterexample for each false statement.Answer: P is false: right triangles with legs 6 cm, 4 cm and legs 8 cm, 3 cm both have area 12 cm² but are not congruent. Q is true: congruent triangles fit exactly on each other, so they cover the same region and have the same area.
• It can also happen that both the proposition and converse are false! Can you give an example?Answer: Proposition: ‘If a number is a multiple of 4, then it is a multiple of 6.’ False: 8. Converse: ‘If a number is a multiple of 6, then it is a multiple of 4.’ False: 6. Both are false.
• Here is a statement of the Baudhāyana–Pythagoras theorem.
  Let a, b, c be the sidelengths of a triangle. If the triangle is right-angled, then a² + b² = c².
  What is its converse? Is this converse true?Answer: (a) Converse: if a² + b² = c², then the triangle is right-angled. (b) XY = c. (c) △ABC ≅ △XYZ (SSS), so ∠C = ∠Z = 90°. (d) Yes, the converse is true.
• 1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
  2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples.Answer: 1. ‘If a person lives in Manama, the capital of Bahrain, then the person lives in Bahrain’ (true); converse false — someone living in Muharraq. ‘If a vehicle is a car, then it has wheels’; converse false — a bicycle. 2. ‘If a quadrilateral is a rectangle, its diagonals are equal’; converse false — an isosceles trapezium. ‘If a and b are both even, then a + b is even’; converse false — 3 + 5 = 8.
• If two lines are parallel, then the corresponding angles formed by a transversal are equal.Answer: Converse: If the corresponding angles formed by a transversal are equal, then the lines are parallel. Both are true.
• If a quadrilateral is a square, then all its angles are equal.Answer: Converse: If all the angles of a quadrilateral are equal, then it is a square. The proposition is true; the converse is false — a 6 cm × 4 cm rectangle has four equal angles but is not a square.
• Given any ∆ABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle.
  Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown.
  Proposition: If AB = AC, then IE = IF.Answer: Converse: If IE = IF, then AB = AC. The proposition is true (△IBF ≅ △ICE by ASA). The converse is false: in a triangle with ∠A = 60°, ∠B = 90°, ∠C = 30° (AB = 4 cm, AC = 8 cm), IE = IF ≈ 1.52 cm but AB ≠ AC.
• 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.Answer: Converse: If a + x = a + y, then x = y. Both are true.
• If a and b are perfect squares, then ab is a perfect square.Answer: Converse: If ab is a perfect square, then a and b are perfect squares. The proposition is true (a = m², b = k² ⇒ ab = (mk)²). The converse is false: a = 2, b = 8 gives ab = 16 = 4², but 2 and 8 are not perfect squares.
• If x = y, then x² = y².Answer: Converse: If x² = y², then x = y. The proposition is true; the converse is false: x = 3, y = −3 give x² = y² = 9 but x ≠ y.
• If x = y, then x³ = y³.Answer: Converse: If x³ = y³, then x = y. Both are true for real numbers (unlike squares, cubes keep the sign: (−2)³ = −8 ≠ 8).
• If n is divisible by 24, then it is divisible by both 4 and 6.Answer: Converse: If n is divisible by both 4 and 6, then it is divisible by 24. The proposition is true; the converse is false: 12 is divisible by 4 and by 6 but not by 24.
• If n is divisible by 60, then it is divisible by both 5 and 12.Answer: Converse: If n is divisible by both 5 and 12, then it is divisible by 60. Both are true (5 and 12 have no common factor other than 1).
• If n is the square of a prime number, then it has exactly 3 factors.Answer: Converse: If n has exactly 3 factors, then it is the square of a prime number. Both are true. (e.g. 49 = 7² has factors 1, 7, 49.)
• If n is a product of two unequal prime numbers, then it has exactly 4 divisors.Answer: Converse: If n has exactly 4 divisors, then it is a product of two unequal prime numbers. The proposition is true; the converse is false: 8 has exactly 4 divisors (1, 2, 4, 8) but 8 = 2 × 2 × 2.
• If n and n + 3 have no factors in common, then n is not a multiple of 3.Answer: Converse: If n is not a multiple of 3, then n and n + 3 have no factors in common. Both are true (‘no factors in common’ means no common factor other than 1).
• There are no known ‘neat’ expressions that generate only primes! Find counterexamples to the following claims.Answer: (i) n = 4: 4 × 16 + 1 = 65 = 5 × 13. (ii) n = 10: 100 + 10 + 11 = 121 = 11 × 11. (iii) n = 4: 4⁴ + 3 = 259 = 7 × 37.
• Find counterexamples to the following statements.Answer: (i) n = 11 (prime): 2¹¹ − 1 = 2047 = 23 × 89, not prime. (ii) n = 6 (even): 2⁶ + 1 = 65 = 5 × 13, not prime.
• Consider the statement: ‘If a number is divisible by 8, then it is divisible by both 2 and 4’.Answer: (i) If n = 8k, then n = 2 × 4k = 4 × 2k, so it is divisible by 2 and 4. (ii) No. The converse is false: 12 is divisible by 2 and by 4 but not by 8. To test for 8, check whether the number formed by the last three digits is divisible by 8.
• 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.Answer: If a number is divisible by 3, then the sum of its digits is a multiple of 3. If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3. Both are true: each statement is the converse of the other.
• We have identified different types of quadrilaterals — squares, rectangles, parallelograms, rhombi, kites and trapezia. One can identify more types (e.g., we could create a category of quadrilaterals that have equal-length opposite sides).
  Suppose we have identified a category of quadrilaterals called Q, and we have to construct a quadrilateral of this type. For this, we are to use two thin sticks, put them together as diagonals so that the quadrilateral obtained by joining their endpoints is of type Q (see the figure).Answer: (i)(a) Yes: every type-Q quadrilateral has equal diagonals, so unequal sticks can never give type Q. (i)(b) It may matter: equal diagonals are needed but may not be enough. (ii) (a) Equal sticks are a sure way, but not necessarily the only way. (b) No: equal sticks crossed in any way give equal diagonals, which is enough for type Q.

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