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NCERT :GRADE-9: MATHS CHAPTER -1 WORKSHEETS
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Grade 9 Mathematics
Propositions and Their Converses
5 Professional Practice Worksheets
• All Questions Carry 1 Mark
π Practice Series
Test your understanding of propositions, converses,
counterexamples, factor-partner reasoning, divisibility,
geometry and the converse of the BaudhΔyana–Pythagoras theorem.
Important:
Every question carries 1 mark.
Attempt the questions independently and explain your reasoning
wherever required.
Worksheet 1 — Source-Based Questions
15 Questions × 1 Mark = 15 Marks
Read each source carefully and answer the questions that follow.
SOURCE A — PROPOSITIONS
A proposition is a statement that is either true or false.
A statement in the form “If X then Y” can also be written
as “X implies Y”.
1.
What is a proposition?
[1]
2.
Complete: “If X then Y” can also be written as
“X ______ Y”.
[1]
3.
Is a proposition necessarily true?
[1]
SOURCE B — CONVERSE
If a proposition is “If X then Y”, its converse is
“If Y then X”.
4.
Write the converse of:
“If a number is divisible by 6, then it is divisible by 3.”
[1]
5.
What two parts are interchanged when a converse is formed?
[1]
6.
Write the converse of:
“If x = y, then x² = y².”
[1]
SOURCE C — COUNTEREXAMPLE
A proposition may be true while its converse is false.
A counterexample is an example that contradicts a stated
proposition.
7.
What is a counterexample?
[1]
8.
Give one counterexample to:
“Every multiple of 3 is a multiple of 6.”
[1]
9.
Can one suitable counterexample disprove a universal claim?
[1]
SOURCE D — FACTOR PARTNERS
Factors of a number occur in partner pairs. If a number has
an odd number of factors, one factor-partner pair contains
the same number twice.
10.
Give the repeated factor pair of 25.
[1]
11.
Is 36 a perfect square?
[1]
12.
How many factors does 49 have?
[1]
SOURCE E — PYTHAGOREAN CONVERSE
If a triangle is right-angled, then
a² + b² = c². The converse states that if
a² + b² = c², then the triangle is right-angled.
13.
Write the converse in one sentence.
[1]
14.
Does 3² + 4² = 5²?
[1]
15.
Which congruence criterion is used in the chapter's
proof of the converse?
[1]
Worksheet 2 — Case-Based Questions
15 Questions × 1 Mark = 15 Marks
CASE 1 — PARALLEL LINES
A student writes: “If two lines are parallel, then the
corresponding angles formed by a transversal are equal.”
1.
What is the converse of this statement?
[1]
2.
What geometric object cuts the two lines?
[1]
3.
What type of angles are compared?
[1]
CASE 2 — PERFECT SQUARES
A number's factors can be arranged in partner pairs.
For a perfect square, the square root forms a repeated
factor pair.
4.
What is the square root of 81?
[1]
5.
Write the repeated factor pair of 81.
[1]
6.
Is 81's number of factors odd or even?
[1]
CASE 3 — DIVISIBILITY
Consider the statement:
“If n is divisible by 24, then n is divisible by both 4 and 6.”
7.
Is 48 divisible by 24?
[1]
8.
Is 48 divisible by 6?
[1]
9.
Is 12 a multiple of 24?
[1]
CASE 4 — PRIME CLAIMS
The chapter asks students to challenge claims about
expressions that are said to produce only prime numbers.
10.
Is 2047 prime?
[1]
11.
What is 2¹¹ − 1?
[1]
12.
Name the method used to disprove a universal prime claim.
[1]
CASE 5 — TRIANGLES
13.
What is 5² + 12²?
[1]
14.
What is the square root of 169?
[1]
15.
Are 5, 12 and 13 the sides of a right triangle?
[1]
Worksheet 3 — Assertion & Reason
15 Questions × 1 Mark = 15 Marks
For each question select the correct option:
A — Both Assertion and Reason are true,
and Reason explains Assertion.
B — Both are true, but Reason does not explain
Assertion.
C — Assertion is true, but Reason is false.
D — Assertion is false, but Reason is true.
1.
Assertion: A proposition is either true or false.
Reason: A proposition is a mathematical statement.
2.
Assertion: The converse of “If X then Y” is “If Y then X”.
Reason: The two parts are interchanged.
3.
Assertion: A true proposition always has a true converse.
Reason: A proposition and its converse have different forms.
4.
Assertion: A counterexample can disprove a universal statement.
Reason: A counterexample contradicts the stated claim.
5.
Assertion: Every multiple of 6 is a multiple of 3.
Reason: 6 = 3 × 2.
6.
Assertion: Every multiple of 3 is a multiple of 6.
Reason: 9 is a multiple of 3.
7.
Assertion: A perfect square has an odd number of factors.
Reason: Its square root forms a repeated factor pair.
8.
Assertion: 25 has an odd number of factors.
Reason: Its repeated factor pair is (5,5).
9.
Assertion: Congruent triangles have equal areas.
Reason: Congruent triangles have the same size and shape.
10.
Assertion: Equal-area triangles must be congruent.
Reason: Different triangles can have the same area.
11.
Assertion: If a²+b²=c², the triangle is right-angled.
Reason: This is the converse of the Pythagorean theorem.
12.
Assertion: “Y when X” can express the meaning “If X then Y”.
Reason: The chapter gives both forms as equivalent expressions.
13.
Assertion: The converse of a proposition may be false.
Reason: The truth of a proposition does not automatically establish its converse.
14.
Assertion: 36 has an odd number of factors.
Reason: The factor pair (6,6) occurs for 36.
15.
Assertion: The converse of the Pythagorean theorem is true.
Reason: The chapter proves it using construction and SSS congruence.
Worksheet 4 — Multiple Choice Questions
15 Questions × 1 Mark = 15 Marks
Select the most appropriate option. Every question carries
one mark.
1.
Which is the converse of “If a number is divisible by 8,
then it is divisible by 4”?
2.
Which number is a counterexample to “Every multiple of 3
is a multiple of 6”?
3.
Which number is a perfect square?
4.
How many factors does 25 have?
5.
Which pair is a factor pair of 35?
6.
Which number has exactly three factors?
7.
Which expression means “X implies Y”?
8.
Which number disproves the claim “All even n make
2βΏ+1 prime”?
9.
What is 7²?
10.
Which is the converse of “If x=y, then x³=y³”?
11.
Which statement is true?
12.
Which pair satisfies the Pythagorean relation?
13.
Which criterion is used in the proof of the converse
of the Pythagorean theorem?
14.
Which statement describes a counterexample?
15.
Which number has an odd number of factors?
Worksheet 5 — Mixed Competency & HOTS
15 Questions × 1 Mark = 15 Marks
Answer briefly. Focus on mathematical reasoning rather
than lengthy calculations.
1.
Write the converse of:
“If a quadrilateral is a square, then all its angles are equal.”
[1]
2.
Give a counterexample to:
“Every multiple of 5 is a multiple of 10.”
[1]
3.
Is the converse of “If it rains, the road is wet”
necessarily true?
[1]
4.
Give one reason why a road may be wet without rain.
[1]
5.
Write one factor-partner pair of 64 other than (1,64).
[1]
6.
Is 64's number of factors odd or even?
[1]
7.
What is the repeated factor pair of 64?
[1]
8.
Write the converse of:
“If n is divisible by 60, then n is divisible by 5 and 12.”
[1]
9.
Is 60 divisible by both 5 and 12?
[1]
10.
Express “a number is divisible by 3” using the
digit-sum condition.
[1]
11.
What is the sum of the digits of 123?
[1]
12.
Is 123 divisible by 3?
[1]
13.
For sides 9, 40 and 41, calculate 9²+40².
[1]
14.
Does 9²+40² equal 41²?
[1]
15.
Based on the converse of the Pythagorean theorem,
what type of triangle has sides 9, 40 and 41?
[1]
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