Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the converse of the statement: "If n is divisible by 4, then n is even."

If n is , then n is by .

Correct! The converse switches the hypothesis and conclusion.

(2) Give a counterexample to show that the product of two even numbers is not always prime.

Example: 23 × 23 = (which is not prime)

Perfect! 4 is composite, not prime.

(3) State whether true or false: "If a number is divisible by 8, then it is divisible by 2."

Answer:

Excellent! Since 8 = 23, any multiple of 8 is also a multiple of 2.

(4) Write the contrapositive of: "If a figure is a rhombus, then it is a parallelogram."

If a figure is notis a , then it is notis a .

Correct! The contrapositive negates both parts and switches them.

(5) Which proof method is suitable to prove "5 is irrational"?

Method: Proof by

Great! We assume 5 is rational and derive a contradiction.

Short Answer Questions (2 Marks Each)

(1) Prove that the sum of three consecutive odd numbers is divisible by 3.

(2) Disprove the statement: "All multiples of 3 are prime."

(3) Write the converse and contrapositive of: "If a quadrilateral is a square, then it is a parallelogram." Which of these are true?

Converse: If a quadrilateral isis not a , then it isis not a .

Contrapositive: If a quadrilateral is notis a , then it is notis a .

So, we have: Original: TrueFalse, Converse: , Contrapositive:

(4) Prove by direct method that the square of any odd integer is always odd.

(5) Show by counterexample that the statement "Every rectangle is a square" is false.

Long Answer Questions (4 Marks Each)

(1) Prove by contradiction that 2 is irrational.

(2) Prove that if n is an even integer, then n2 is divisible by 4.

(3) Using contrapositive method, prove: "If a number is divisible by 6, then it is divisible by 3."

(4) Prove that the product of two consecutive integers is always even.

(5) Write the converse and contrapositive of: "If a number is divisible by 12, then it is divisible by 6." Check which are true and justify.

Converse: If a number isis not divisible by , then it isis not divisible by .

Contrapositive: If a number is notis divisible by , then it is notis divisible by .

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The contrapositive of "If x is a multiple of 9, then x is divisible by 3" is:

(a) If x is divisible by 3, then x is a multiple of 9

(b) If x is not divisible by 3, then x is not a multiple of 9

(c) If x is not a multiple of 9, then x is not divisible by 3

(d) None of these

Correct! Contrapositive negates both parts and switches them.

(2) Which of the following is a counterexample for the statement "All prime numbers are odd"?

(a) 1 (b) 2 (c) 3 (d) 5

Correct! 2 is the only even prime number.

(3) To prove that 11 is irrational, which method is used?

(a) Direct proof (b) Proof by contradiction (c) Counterexample (d) Converse

Correct! We assume 11 is rational and derive a contradiction.

(4) The converse of "If a figure is a square, then it has four equal sides" is:

(a) If a figure has four equal sides, then it is a square (b) If a figure has four sides, then it is a square (c) If a figure is not a square, then it does not have four equal sides (d) None of these

Correct! Converse switches hypothesis and conclusion.

(5) Which statement is false?

(a) Every square is a rectangle (b) Every rectangle is a square (c) Every square is a parallelogram (d) Every square is a quadrilateral

Correct! Not every rectangle is a square (counterexample: 4×2 rectangle).

(6) The contrapositive of "If a number is divisible by 15, then it is divisible by 5" is:

(a) If a number is not divisible by 5, then it is not divisible by 15

(b) If a number is divisible by 5, then it is divisible by 15

(c) If a number is not divisible by 15, then it is not divisible by 5

(d) None of these

Correct! Contrapositive: ¬q → ¬p.

(7) Which of the following disproves the statement "Every natural number is even"?

(a) 1 (b) 2 (c) 3 (d) Both (a) and (c)

Correct! Both 1 and 3 are odd natural numbers, disproving the statement.

(8) Which method is most suitable to prove "If n is an odd integer, then n2 is odd"?

(a) Direct proof (b) Proof by contradiction (c) Counterexample (d) Converse

Correct! Direct proof: n = 2k+1 ⟹ n2 = 4k2 + 4k + 1 = 22k2+2k + 1.

(9) The square of an even integer is always:

(a) Odd (b) Divisible by 4 (c) Prime (d) None of these

Correct! If n = 2k, then n2 = 4k2, which is divisible by 4.

(10) Which of the following is the contrapositive of "If n is divisible by 4, then n is divisible by 2"?

(a) If n is not divisible by 2, then n is not divisible by 4

(b) If n is divisible by 2, then n is divisible by 4

(c) If n is not divisible by 4, then n is not divisible by 2

(d) If n is divisible by 2, then n is not divisible by 4

Correct! Contrapositive: ¬(divisible by 2) → ¬(divisible by 4).

Mathematical Reasoning Challenge

Determine whether these statements are True or False:

Mathematical Reasoning Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Mathematical Reasoning" hard worksheet and learned:

(1) Logical Statements: Understanding conditional statements, converse, and contrapositive relationships

(2) Proof Techniques: Mastering direct proof, proof by contradiction, and contrapositive methods

(3) Counterexamples: Finding specific cases that disprove general statements effectively

(4) Logical Equivalence: Recognizing when statements are logically equivalent or independent

(5) Mathematical Arguments: Constructing rigorous proofs for number theory properties

(6) Truth Tables: Understanding when implications are true or false based on hypothesis and conclusion

(7) Irrationality Proofs: Applying proof by contradiction to show numbers like √2, √3 are irrational

(8) Divisibility Properties: Proving relationships between divisibility conditions using various methods

(9) Geometric Reasoning: Analyzing logical relationships between different types of quadrilaterals

(10) Critical Thinking: Evaluating mathematical statements for validity and finding flaws in reasoning

(11) Proof Strategy: Selecting appropriate proof methods based on the type of statement to prove

(12) Logical Connections: Understanding the relationship between a statement and its contrapositive

Outstanding work! You now have advanced skills in mathematical reasoning and proof techniques!