Easy Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) What is a proof in mathematics?
Correct! A proof is a logical argument that demonstrates why a mathematical statement is true using established facts and reasoning.
(2) State whether the statement is true or false: "The sum of two even numbers is even."
Perfect! The sum of two even numbers is always even. For example: 2 + 4 = 6.
(3) Give an example of a counterexample.
Statement: All birds can fly.
Counterexample:
Excellent! A counterexample disproves a false statement by providing a specific case where it fails.
(4) Write the converse of the statement: "If a number is divisible by 4, then it is even." If a number is
Great! The converse switches the hypothesis and conclusion of a conditional statement.
(5) Check whether the statement "All prime numbers are odd" is true or false.
Correct! This is false because 2 is a prime number and it is even, not odd.
Short Answer Questions (2 Marks Each)
Write down the answers in a sheet and submit to the subject teacher for all the subjective answers.
(1) Prove that the sum of two odd numbers is even.
(2) Disprove the statement "All rectangles are squares."
(3) Show that if a number is divisible by 2, then it is even.
(4) Using a counterexample, disprove the statement "All multiples of 3 are prime numbers."
(5) Write the contrapositive of the statement: "If a figure is a square, then it is a rectangle."
Contrapositive: If a figure
Long Answer Questions (4 Marks Each)
Note: Write down the answers in a sheet and submit to the subject teacher for all the subjective answers.
(1) Prove that the product of two even numbers is always even.
(2) Show by direct proof that the square of an even number is even.
(3) Verify that
(4) Prove that the square of an odd number is odd.
(5) Disprove the statement "Every quadrilateral is a parallelogram."
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) A proof in mathematics is based on:
(a) Assumptions (b) Logical reasoning (c) Guesswork (d) Examples only
Correct! Mathematical proofs are based on logical reasoning using established facts and axioms.
(2) The statement "The sum of two odd numbers is odd" is:
(a) True (b) False (c) Sometimes true (d) Cannot say
Correct! This statement is false. The sum of two odd numbers is always even.
(3) A counterexample is used to:
(a) Prove a statement true (b) Disprove a false statement (c) Guess an answer (d) Create a new theorem
Correct! A counterexample provides a specific case that shows a statement is false.
(4) The contrapositive of "If a number is divisible by 5, then it ends in 0 or 5" is:
(a) If a number does not end in 0 or 5, then it is not divisible by 5
(b) If a number ends in 0 or 5, then it is divisible by 5
(c) If a number is not divisible by 5, then it does not end in 0 or 5 (d) None of these
Correct! The contrapositive negates both parts and reverses their order.
(5) The converse of "If a figure is a square, then it is a rectangle" is:
(a) If a figure is a rectangle, then it is a square
(b) If a figure is not a square, then it is not a rectangle
(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.
(6) Which of the following is true?
(a) All prime numbers are even (b) Every square is a rectangle (c) Every rectangle is a square (d) The product of two odd numbers is even
Correct! Every square has four equal sides and right angles, making it a special type of rectangle.
(7) Proof by contradiction is used to prove:
(a) Simple statements (b) Irrationality of numbers like
Correct! Proof by contradiction is particularly useful for proving irrationality and other statements that are difficult to prove directly.
(8) The method used to show that a statement is false is:
(a) Direct proof (b) Counterexample (c) Contrapositive (d) Converse
Correct! A counterexample disproves a statement by showing a specific case where it fails.
(9) Which of these is a true statement?
(a) The square of any even number is odd (b) The sum of two odd numbers is even (c) Every rectangle is a square (d) Every quadrilateral is a parallelogram
Correct! When two odd numbers are added, the result is always even.
(10) Which of the following is NOT a method of proof?
(a) Direct proof (b) Proof by contradiction (c) Counterexample (d) Guesswork
Correct! Guesswork is not a valid method of mathematical proof. Proofs require logical reasoning.
Mathematical Reasoning Challenge
Determine whether these statements are True or False: