Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the contrapositive of the statement: "If a number is divisible by 9, then it is divisible by 3." If a number is notis divisible by , then it is notis divisible by .

Correct! The contrapositive negates both parts and reverses their order.

(2) Give a counterexample for the statement: "The product of two prime numbers is always prime." 2 × 3 = 63 × 6 = 18

Perfect! 6 has factors other than 1 and itself, so it's not prime.

(3) State whether true or false: "If a number is divisible by 10, then it is divisible by 2 and 5."

Excellent! If a number is divisible by 10, it must be divisible by both factors of 10.

(4) Write the converse of: "If a polygon is a square, then it has four sides." If a polygon has four sidesis a squareisn't a square, then it is a squarehas four sidesdoesn't have four sides.

Great! The converse switches the hypothesis and conclusion.

(5) What method is generally used to prove irrationality of square roots like 2 or 3? Proof by

Correct! We assume the square root is rational and derive a contradiction.

Short Answer Questions (2 Marks Each)

(1) Prove that the sum of two consecutive integers is always odd.

(2) Disprove the statement "Every natural number is prime."

(3) Write the converse and contrapositive of: "If a number is divisible by 6, then it is divisible by 2."

Converse: If a number isisn't divisible by , then it isisn't divisible by .

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

(4) Prove that the product of two odd numbers is always odd.

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

Counterexample: A rhombussquare is a but isn'tis a rectangle, as it doesn't have angles.

Long Answer Questions (4 Marks Each)

(1) Prove that 3 is irrational using proof by contradiction.

(2) Using direct proof, show that the square of any even integer is divisible by 4.

(3) If a number is divisible by 12, prove that it is divisible by both 3 and 4.

(4) Write the converse and contrapositive of: "If a number is a multiple of 15, then it is a multiple of 5." Also state which are true.

Original: If a number is a multiple of 15, then it is a multiple of 5.

Converse: If a number isis not a multiple of , then it isis not a multiple of .

Contrapositive: If a number is notis a multiple of , then it is notis a multiple of .

(5) Show that if a number is divisible by 2 and 3, then it is divisible by 6.

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 an even number, then x2 is even" is:

(a) If x2 is not even, then x is not even (b) If x2 is even, then x is even (c) If x2 is odd, then x is odd (d) None of these

Correct! Both (a) and (c) are equivalent ways to state the contrapositive.

(2) To disprove the statement "All numbers are prime", which number is used as counterexample?

(a) 1 (b) 2 (c) 4 (d) 9

Correct! Both 4 and 9 are composite numbers that disprove the statement.

(3) The converse of "If a figure is a square, then it is a rectangle" is:

(a) If a figure is not a square, then it is not a rectangle

(b) If a figure is a rectangle, then it is a square

(c) If a figure is not a rectangle, then it is not a square

(d) None of these

Correct! The converse switches the hypothesis and conclusion.

(4) Which method is used to prove 7 is irrational?

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

Correct! Irrationality proofs typically use proof by contradiction.

(5) Which of the following is a true statement?

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

Correct! A square has all properties of a rectangle plus equal sides.

(6) The contrapositive of "If a number is divisible by 5, then it ends with 0 or 5" is:

(a) If a number is not divisible by 5, then it does not end in 0 or 5

(b) If a number does not end in 0 or 5, then it is not divisible by 5

(c) If a number ends in 0 or 5, then it is divisible by 5

(d) None of these

Correct! The contrapositive negates both parts and reverses order.

(7) The square of an odd number is always:

(a) Odd (b) Even (c) Prime (d) Composite

Correct! The square of an odd number is always odd.

(8) Which of the following disproves "Every prime number is odd"?

(a) 2 (b) 3 (c) 5 (d) 7

Correct! 2 is prime but even, not odd.

(9) If a number is divisible by both 2 and 3, then it is divisible by:

(a) 4 (b) 5 (c) 6 (d) 12

Correct! If divisible by both 2 and 3, it must be divisible by their LCM, which is 6.

(10) Which of the following is not a valid method of proof?

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

Correct! Guesswork is not a valid mathematical proof method.

Advanced Reasoning Challenge

Determine whether these statements are True or False:

Advanced Mathematical Reasoning Quiz