CBSE Class 9 Mathematics Proofs in Mathematics MCQs Set 20

Answer: (a) always false

Explanation:
1. A year has 12 months, so the statement is false without exception.
2. A statement that is always false is a valid mathematical statement.

Teacher's Note:
1. Check whether it is true or false in every case.

Answer: (b) ambiguous

Explanation:
1. The day on which Diwali falls changes from year to year.
2. So the statement is sometimes true and sometimes false: it is ambiguous.

Teacher's Note:
1. An ambiguous statement is not valid in mathematics.

Answer: (c) ambiguous

Explanation:
1. The temperature changes with time, so truth depends on when it is checked.

Teacher's Note:
1. Valid statements must be either always true or always false.

Answer: (d) always false

Explanation:
1. Dogs cannot fly, so it is false in every case.

Teacher's Note:
1. Always false statements are still valid statements.

Answer: (a) false, because February has 29 days in a leap year

Explanation:
1. In a leap year February has 29 days, so the statement is not always true.
2. Taken as a general statement it is false.

Teacher's Note:
1. A general statement is false if one case fails.

Answer: (b) false, because the sum is 360°

Explanation:
1. The angle sum of any quadrilateral is (4 − 2) × 180° = 360°.

Teacher's Note:
1. Split the quadrilateral into two triangles.

Answer: (c) always true

Explanation:
1. The square of a positive or negative number is positive and 0² = 0.
2. So x² ≥ 0 for every real x.

Teacher's Note:
1. Example: (−2)² = 4.

Answer: (d) always false

Explanation:
1. Odd + odd = even, for example 3 + 5 = 8 and 3 + 9 = 12.
2. So the statement is false.

Teacher's Note:
1. Write odd numbers as 2m + 1 and 2n + 1.

Answer: (a) always true

Explanation:
1. 2m + 2n = 2(m + n), which is even for all integers m and n.

Teacher's Note:
1. Examples 2 + 2 = 4 and 6 + 4 = 10 are not a proof; the algebra is.

Answer: (b) Every odd number greater than 1 is prime

Explanation:
1. 9 is odd and greater than 1 but not prime, so the first statement is false.
2. The other three are true.

Teacher's Note:
1. One counter-example disproves a statement.

Answer: (c) x³ ≥ x

Explanation:
1. For x = 1/2, x³ = 1/8 < 1/2 = x, so x³ ≥ x is not true for every real x.
2. The other three hold for all x.

Teacher's Note:
1. Try a fraction between 0 and 1.

Answer: (d) the number is greater than 1

Explanation:
1. For x > 1, x² − x = x(x − 1) > 0.

Teacher's Note:
1. Test x = 2: 4 > 2.

Answer: (a) rectangle

Explanation:
1. The diagonals of a rectangle are equal.
2. The diagonals of a rhombus are not equal in general.

Teacher's Note:
1. A rectangle is an equiangular parallelogram.

Answer: (b) There are three and only three angles in a triangle

Explanation:
1. A triangle has exactly three angles.

Teacher's Note:
1. Correct the number, not the figure.

Answer: (c) All prime numbers greater than 2 are odd

Explanation:
1. 2 is an even prime; every other prime is odd.

Teacher's Note:
1. Exclude the counter-example 2.

Answer: (d) the real number is an integer

Explanation:
1. Even numbers are integers; 2 × (1.5) = 3 is not even.
2. If the number is an integer n, then 2n is even.

Teacher's Note:
1. Evenness applies to integers only.

Answer: (a) x > 1

Explanation:
1. 3x + 1 > 4 means 3x > 3, so x > 1.

Teacher's Note:
1. Solve the inequality.

Answer: (b) x ≥ 0

Explanation:
1. A negative number cubed is negative; for x ≥ 0, x³ ≥ 0.

Teacher's Note:
1. The sign of x³ is the sign of x.

Answer: (c) alternate interior angles

Explanation:
1. ∠DAB = ∠B and ∠EAC = ∠C as alternate interior angles.
2. Then the angles at A on the straight line DE add up to 180°.

Teacher's Note:
1. DE ∥ BC with AB and AC as transversals.

Answer: (d) n even and n odd

Explanation:
1. If n = 2m then n(n + 1) = 2m(2m + 1). If n = 2r − 1 then n(n + 1) = (2r − 1)2r. Both are multiples of 2.

Teacher's Note:
1. Every integer is either even or odd.

Answer: (a) √2 is an irrational number

Explanation:
1. √2 is irrational. Jaipur is in Rajasthan, a prime has exactly two factors, and odd + odd is even.

Teacher's Note:
1. Evaluate each statement separately.

Answer: (b) 4 is not a multiple of 20

Explanation:
1. The negation simply denies the statement: 4 is not a multiple of 20.

Teacher's Note:
1. Negation reverses the truth value.

Answer: (c) The roots of the equation are not all equal

Explanation:
1. The negation of 'all roots are equal' is 'not all roots are equal'.

Teacher's Note:
1. The equation is (x + 1)⁴ = 0, so the original statement is true.

Answer: (d) false

Explanation:
1. February has 28 or 29 days, never 30.

Teacher's Note:
1. A false statement can still be a valid statement.

Answer: (a) 385

Explanation:
1. 10 × 11 × 21 ÷ 6 = 385.

Teacher's Note:
1. Check by adding the first few squares.

Answer: (b) 144

Explanation:
1. The sum of the first 12 odd numbers is 12² = 144.

Teacher's Note:
1. Sum of first n odd numbers is n².

Answer: (c) only the one about 3

Explanation:
1. For n = 1, 4 − 1 = 3 is not divisible by 8, so the first fails.
2. 4ⁿ − 1 = (4 − 1)(4ⁿ⁻¹ + ... + 1) is divisible by 3 for every n.

Teacher's Note:
1. One counter-example is enough to disprove.

Answer: (d) Yes, it is 11⁷

Explanation:
1. Digits: 1, 9, 4, 8, 7, 1, 7, 1. Sum of alternate digits: 1 + 4 + 7 + 7 = 19 and 9 + 8 + 1 + 1 = 19; the difference is 0.
2. The pattern 11¹, 11², ... confirms 11⁷ = 19487171.

Teacher's Note:
1. Powers of 11 have alternating digit sums that agree modulo 11.

Answer: (a) false, since 3² + 4² = 25 is not prime

Explanation:
1. With a = 3 and b = 4, a² + b² = 25 = 5 × 5.

Teacher's Note:
1. A single counter-example disproves a general statement.

Answer: (b) Yes, since the sum of the digits 36 is divisible by 3

Explanation:
1. Digits: 9 + 8 + 7 + 5 + 3 + 4 + 0 = 36 and 36 is divisible by 3.

Teacher's Note:
1. Apply the given rule directly.