Class 9 Maths Ganita Manjari Part 2 Chapter 9 Solutions

The complete NCERT-based solutions for Class 9 Maths, Ganita Manjari Part II, Chapter 1 (Propositions and their Converses, which is Chapter 9 of the complete book) are given below, covering all five worked examples, every "Think and Reflect" prompt and all 17 questions of Exercise Set 9.1, with a justification or counterexample for each statement. A proposition is a statement that is either true or false, and the converse of "if P then Q" is "if Q then P". This is the first chapter of Part II and the ninth chapter of the complete Class 9 book. Solutions for the earlier chapters are available in Class 9 Maths Ganita Manjari Solutions Part 1.

Key Ideas of the Chapter at a Glance

A proposition may be true while its converse is false, both may be true, or both may be false. A single counterexample is enough to prove a statement false, but a true statement needs a logical proof.

Think and Reflect (Page 1): Is the Converse of the Isosceles Triangle Theorem True?

Yes, Statement 2 is true: if two angles of a triangle are equal, then the sides opposite those angles are equal.

Proof. Let ABC be a triangle with angle B equal to angle C. Draw the altitude AD from A to BC. In triangles ABD and ACD, angle B equals angle C (given), angle ADB equals angle ADC (each 90 degrees) and AD is common to both. By the AAS criterion, triangle ABD is congruent to triangle ACD, so AB = AC. Hence the sides opposite the equal angles are equal.

Example 1: Rain and the Wet Road

Proposition P (if it rains, then the road is wet) is true, but its converse Q (if the road is wet, then it has rained) need not be true. The road may be wet because a tanker spilled water on it, and this situation is a counterexample to Q.

Example 2: Multiples of 6 and Multiples of 3

Why is Proposition P true?

If a number is a multiple of 6, it can be written as 6k for some integer k. Since 6k = 3 x (2k), the number is also a multiple of 3. Hence P is true.

Is the converse Q true?

No. Q says that every multiple of 3 is a multiple of 6. The number 9 is a multiple of 3 (3 x 3) but is not a multiple of 6, because 9 is not divisible by 2. So 9 is a counterexample and Q is false.

Example 3: Perfect Squares and the Number of Factors

Both statements are true. Proposition P (a perfect square has an odd number of factors) and its converse Q (a number with an odd number of factors is a perfect square) hold together, so "n is a perfect square" and "n has an odd number of factors" imply each other.

For example, 36 has nine factors (1, 2, 3, 4, 6, 9, 12, 18, 36) and is a perfect square, while 12 has six factors (an even number) and is not a perfect square.

Discussion: What Does the Factor-Partner Argument Prove?

The argument proves Statement Q, not Statement P. It starts from "n has an odd number of factors", moves to "a factor-partner pair (f, f) exists" and ends at "n = f x f". The direction of reasoning is from odd number of factors to perfect square, which is the converse statement.

The same argument does not prove P because the step from "a repeating pair exists" to "the number of factors is odd" is not automatic. If two repeating pairs (f, f) and (g, g) existed, the count of factors would be even. However, a number cannot have more than one such pair, because f x f = g x g forces f = g. Hence a square number has exactly one repeating pair, an odd number of factors, and P is also true.

Example 4: Same Area and Congruent Triangles

Proposition P: If two triangles have the same area, then they are congruent

This statement is false. Consider a right-angled triangle with perpendicular sides 6 and 2, and another right-angled triangle with sides 3, 4 and 5 (perpendicular sides 3 and 4). Each has area 6 square units, yet the triangles are not congruent because their sides differ. This is a counterexample.

Converse Q: If two triangles are congruent, then they have the same area

This statement is true. Congruent triangles have all corresponding sides and angles equal, so they can be placed exactly over each other. Hence their bases, heights and areas are the same.

Think and Reflect (Page 3): Both Proposition and Converse False

Take the proposition "If a number is prime, then it is odd". It is false because 2 is prime but even. Its converse, "If a number is odd, then it is prime", is also false because 9 is odd but not prime. Both the proposition and its converse are false.

Example 5: Converse of the Baudhayana-Pythagoras Theorem

The converse of the Baudhayana-Pythagoras theorem is true. The theorem states that if a triangle with sidelengths a, b, c is right-angled, then a² + b² = c². Its converse states that if a² + b² = c², then the triangle is right-angled.

Proof.

• Let triangle ABC have sidelengths BC = a, CA = b and AB = c with a² + b² = c².
• Construct a right-angled triangle XYZ with the right angle at Z, YZ = a and XZ = b.
• By the Baudhayana-Pythagoras theorem in triangle XYZ, XY² = a² + b² = c², so XY = c.
• In triangles ABC and XYZ, BC = YZ = a, CA = XZ = b and AB = XY = c. By the SSS criterion, triangle ABC is congruent to triangle XYZ.
• Therefore angle C equals angle Z, which is 90 degrees. Triangle ABC is right-angled at C.

A Note and Think and Reflect (Page 5)

The statements "If X then Y", "X implies Y" and "Y when X" all carry the same meaning. The two forms of the perfect-square proposition on page 5 are therefore equivalent.

Question 1: Real-life examples where a proposition is true but not its converse

• Proposition: If a person lives in Dehradun, then the person lives in Uttarakhand. Converse: If a person lives in Uttarakhand, then the person lives in Dehradun. The converse is false, as a resident of Haridwar lives in Uttarakhand but not in Dehradun.
• Proposition: If a student scores 100 per cent, then the student has passed. Converse: If a student has passed, then the student scored 100 per cent. A student with 60 per cent has passed but did not score 100 per cent.
• Proposition: If a day is Sunday, then it is a weekend day. Converse: If a day is a weekend day, then it is Sunday. Saturday is a counterexample.

Question 2: More examples from geometry and number theory

• Geometry: If a quadrilateral is a rhombus, then its diagonals are perpendicular. The converse is false, because a kite has perpendicular diagonals but is not a rhombus.
• Geometry: If a quadrilateral is a rectangle, then its diagonals are equal. The converse is false, because an isosceles trapezium has equal diagonals but is not a rectangle.
• Number theory: If a number is divisible by 10, then it is divisible by 5. The converse is false, because 15 is divisible by 5 but not by 10.
• Number theory: If a number is divisible by 9, then it is divisible by 3. The converse is false, because 12 is divisible by 3 but not by 9.

Exercise Set 9.1: Complete Solutions

Question 1

Proposition: If two lines are parallel, then the corresponding angles formed by a transversal are equal. This is true, as it is a standard property of parallel lines.

Converse: If the corresponding angles formed by a transversal are equal, then the two lines are parallel. This is true. If the lines were not parallel, they would meet at a point and form a triangle with the transversal. One angle of this triangle would then be an exterior angle equal to its interior opposite angle, which contradicts the fact that an exterior angle is larger than either interior opposite angle. Hence the lines must be parallel.

Question 2

Proposition: If a quadrilateral is a square, then all its angles are equal. This is true, as each angle of a square is 90 degrees.

Converse: If all angles of a quadrilateral are equal, then it is a square. This is false. A rectangle with sides 3 cm and 5 cm has all angles equal to 90 degrees but is not a square.

Question 3 (Starred): Angle Bisectors and the Incentre

Proposition: If AB = AC, then IE = IF. This is true.

Justification. If AB = AC, then angle B equals angle C. Their halves are also equal, so angle IBC equals angle ICB, which gives IB = IC. Now consider triangles BCF and CBE. BC is common, angle FBC (which is angle B) equals angle ECB (which is angle C), and angle FCB (half of C) equals angle EBC (half of B). By the ASA criterion the triangles are congruent, so CF = BE. Subtracting equal parts, IE = BE - IB = CF - IC = IF.

Converse: If IE = IF, then AB = AC. This is false.

Counterexample. Take a triangle with BC = 7, CA = 3 and AB = 8. The incentre divides the bisector BE in the ratio (BC + AB) : CA, so IE = CA x BE / (AB + BC + CA), and similarly IF = AB x CF / (AB + BC + CA). Here BE² = 53.76 and CF² = 7.56, which gives IE² = IF² = 112/75, so IE = IF = 4 root 21 / 15 (about 1.22). Yet AB = 8 and AC = 3 are not equal. Hence the converse is false.

Question 4

Proposition: If x = y, then a + x = a + y. This is true, because adding the same number to equal numbers gives equal results.

Converse: If a + x = a + y, then x = y. This is true, because subtracting a from both sides gives x = y. Both the proposition and its converse are used while solving equations.

Question 5

Proposition: If a and b are perfect squares, then ab is a perfect square. This is true. Let a = m² and b = n². Then ab = m²n² = (mn)².

Converse: If ab is a perfect square, then a and b are perfect squares. This is false. Take a = 2 and b = 8. Then ab = 16 is a perfect square, but neither 2 nor 8 is a perfect square.

Question 6

Proposition: If x = y, then x² = y². This is true, because equal numbers have equal squares.

Converse: If x² = y², then x = y. This is false. Take x = 2 and y = -2. Then x² = y² = 4, but x is not equal to y.

Question 7

Proposition: If x = y, then x³ = y³. This is true, because equal numbers have equal cubes.

Converse: If x³ = y³, then x = y. This is true for real numbers. From x³ - y³ = 0 we get (x - y)(x² + xy + y²) = 0. The second factor equals (x + y/2)² + 3y²/4, which is zero only when x = y = 0. So the second factor cannot vanish unless x = y, and in every case x = y.

Question 8

Proposition: If n is divisible by 24, then it is divisible by both 4 and 6. This is true. If n = 24k, then n = 4 x (6k) and n = 6 x (4k).

Converse: If n is divisible by both 4 and 6, then it is divisible by 24. This is false. The number 12 is divisible by both 4 and 6 but not by 24. The smallest number divisible by both 4 and 6 is 12, not 24.

Question 9

Proposition: If n is divisible by 60, then it is divisible by both 5 and 12. This is true. If n = 60k, then n = 5 x (12k) and n = 12 x (5k).

Converse: If n is divisible by both 5 and 12, then it is divisible by 60. This is true. Since n is divisible by 12, write n = 12m. As 5 is a prime that does not divide 12, and 5 divides 12m, it must divide m. So m = 5k and n = 60k.

Question 10

Proposition: If n is the square of a prime number, then it has exactly 3 factors. This is true. If n = p² with p prime, its factors are 1, p and p² only.

Converse: If n has exactly 3 factors, then it is the square of a prime number. This is true. A number with an odd number of factors is a perfect square, so let n = m². Here m is not 1, since 1 has only one factor. If m were composite, say m = r x s with 1 < r < m, then 1, r, m and m² would be four distinct factors, which is a contradiction. So m is prime, and n is the square of a prime.

Question 11

Proposition: If n is a product of two unequal prime numbers, then it has exactly 4 divisors. This is true. If n = pq with p and q distinct primes, its divisors are 1, p, q and pq.

Converse: If n has exactly 4 divisors, then it is a product of two unequal primes. This is false. The number 8 has exactly four divisors (1, 2, 4, 8), but 8 = 2 x 2 x 2 is not a product of two unequal primes. The number 27 (divisors 1, 3, 9, 27) is another counterexample.

Question 12

Proposition: If n and n + 3 have no factors in common (other than 1), then n is not a multiple of 3. This is true. If n were a multiple of 3, say n = 3k, then n + 3 = 3(k + 1) would also be a multiple of 3, and 3 would be a common factor.

Converse: If n is not a multiple of 3, then n and n + 3 have no common factor other than 1. This is true. Any common factor d of n and n + 3 divides their difference, which is 3, so d is 1 or 3. If d were 3, then n would be a multiple of 3, which is not the case. Hence d = 1.

Question 13: Counterexamples to Prime-Generating Claims

For claim (iii), n = 0 also works, since 4⁰ + 3 = 4 is composite, but n = 4 is the usual counterexample for positive n.

Question 14: Counterexamples to Statements

• (i) If n is a prime number, then 2ⁿ - 1 is a prime number. Take n = 11, which is prime. Then 2¹¹ - 1 = 2047 = 23 x 89, which is composite.
• (ii) If n is an even number, then 2ⁿ + 1 is a prime number. Take n = 6. Then 2⁶ + 1 = 65 = 5 x 13, which is composite.

Question 15

(i) Justification. If a number is divisible by 8, write it as 8k. Then 8k = 2 x (4k) and 8k = 4 x (2k), so it is divisible by both 2 and 4.

(ii) Is checking divisibility by 2 and 4 enough to test divisibility by 8? No. The number 12 is divisible by both 2 and 4 but not by 8. In fact, any number divisible by 4 is automatically divisible by 2, so the two checks add nothing beyond divisibility by 4. The converse of the given statement is false. The correct shortcut is to check whether the number formed by the last three digits is divisible by 8.

Question 16

The two statements are as follows.

• 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 statements are true, so each is the converse of the other. This works because every power of 10 leaves remainder 1 on division by 3 (10^k - 1 is a multiple of 9). A number therefore differs from its digit sum by a multiple of 9, and hence by a multiple of 3. For example, 123 has digit sum 6, and both 123 and 6 are multiples of 3.

Question 17: Constructing a Quadrilateral Using Two Sticks as Diagonals

(i) Suppose the property is: if a quadrilateral is of type Q, then it has equal-length diagonals.

• (a) Yes, the two sticks must be of equal length. Every quadrilateral of type Q has equal diagonals, so unequal sticks can never give a quadrilateral of type Q.
• (b) Yes, it can matter how the sticks are put together. The property says only that type Q implies equal diagonals. It does not say that every quadrilateral with equal diagonals is of type Q. Equal length is therefore necessary but not sufficient, and the position and angle at which the sticks cross must also be chosen suitably (for example, they must bisect each other if Q is the category of rectangles).

(ii) Suppose the property is: if a quadrilateral has equal diagonals, then it is of type Q.

• (a) Equal sticks are sufficient, but not necessarily required. Any quadrilateral with equal diagonals is of type Q, so equal sticks will work. The property does not rule out type Q quadrilaterals with unequal diagonals, so unequal sticks may or may not work.
• (b) No, it does not matter how the sticks are put together. As long as the sticks are of equal length, the quadrilateral formed has equal diagonals and is therefore of type Q, whatever the crossing point or angle.

Exercise Set 9.1 at a Glance

Frequently Asked Questions

What is a proposition in Class 9 Maths?

A proposition is a statement that is either true or false, but not both. For example, "If two sides of a triangle are equal, then the angles opposite them are equal" is a proposition.

What is the converse of a statement?

The converse of the proposition "if P then Q" is the proposition "if Q then P". The hypothesis and the conclusion are interchanged.

Is the converse of a true statement always true?

No. A true proposition may have a false converse, as in "if a number is a multiple of 6, then it is a multiple of 3". Sometimes both are true, as with the Baudhayana-Pythagoras theorem, and sometimes both are false.

What is a counterexample?

A counterexample is a single instance that contradicts a stated proposition and shows it to be false. For instance, 9 is a multiple of 3 but not of 6, so it disproves "every multiple of 3 is a multiple of 6".

Is the converse of the Baudhayana-Pythagoras theorem true?

Yes. If the sidelengths a, b, c of a triangle satisfy a² + b² = c², the triangle is right-angled. It is proved by constructing a right-angled triangle with sides a and b and applying the SSS congruence criterion.

Which chapter of Ganita Manjari Part II is Propositions and their Converses?

It is Chapter 1 of Part II of the Class 9 Ganita Manjari textbook, and it is numbered Chapter 9 in the complete book because Part I covers Chapters 1 to 8. It contains Exercise Set 9.1 with 17 questions.