CBSE Class 8 Mathematics Number System MCQs Set 44

Answer: (a) \( \frac{7}{2} \)

Explanation:
1. 7/2 = 3.5, and 3 < 3.5 < 4.
2. The others are 2.5, 4.5 and 0.75.

Teacher's Note:
1. The mean (a + b)/2 of two numbers lies between them.
2. (3 + 4)/2 = 7/2.

Answer: (c) 0 and 1

Explanation:
1. b × b = b gives b(b - 1) = 0.
2. So b = 0 or b = 1.

Teacher's Note:
1. Factorise b² - b = 0.
2. -1 gives (-1)(-1) = 1, which is not -1.

Answer: (c) 5565

Explanation:
1. Sum of first n natural numbers = n(n + 1)/2.
2. 105 × 106 / 2 = 5565.

Teacher's Note:
1. Use the formula instead of adding term by term.
2. Double the sum would be 105 × 106 = 11130.

Answer: (b) 400

Explanation:
1. The sum of the first n odd numbers is n².
2. Here n = 20, so the sum = 400.

Teacher's Note:
1. The 20th odd number is 2 × 20 - 1 = 39.
2. The sum of first n even numbers is n(n + 1) = 420 for n = 20.

Answer: (b) 2550

Explanation:
1. There are 50 even numbers; their sum is n(n + 1) = 50 × 51.
2. = 2550.

Teacher's Note:
1. Sum of first n even numbers = n(n + 1).
2. 5050 is the sum of all the numbers from 1 to 100, not just the even ones.

Answer: (a) 385

Explanation:
1. Sum of squares of first n natural numbers = n(n + 1)(2n + 1)/6.
2. 10 × 11 × 21 / 6 = 385.

Teacher's Note:
1. Remember the formula n(n + 1)(2n + 1)/6.
2. 3025 is the sum of the cubes 1³ + ... + 10³.

Answer: (c) 44100

Explanation:
1. Sum of cubes = [n(n + 1)/2]².
2. [20 × 21 / 2]² = 210² = 44100.

Teacher's Note:
1. The sum of cubes equals the square of the sum of the numbers.
2. 1 + 2 + ... + 20 = 210.

Answer: (a) 4

Explanation:
1. Inside the bracket multiply first: 2 × 17 = 34.
2. 32 + 34 - 6 = 60.
3. 60 ÷ 15 = 4.

Teacher's Note:
1. Follow BODMAS: multiplication before addition.
2. Then divide the bracket value by 15.

Answer: (b) \( \frac{2}{7} \)

Explanation:
1. Multiply numerator and denominator by -1 to make the denominator positive: 16/56.
2. HCF of 16 and 56 is 8, so 16/56 = 2/7.

Teacher's Note:
1. Standard form has a positive denominator and no common factor.
2. Divide by the HCF to reach lowest terms.

Answer: (b) \( \frac{8}{-12} \) and \( \frac{-50}{75} \) are equal

Explanation:
1. 8/(-12) = -8/12 = -2/3.
2. -50/75 = -2/3 as well, so the two are equal.
3. -7/21 = -1/3 and 3/9 = +1/3 are not equal; 3/5 is positive, so it is greater than -2/3.

Teacher's Note:
1. Reduce both numbers to standard form and compare.
2. Or check the cross products: 8 × 75 = (-12)(-50) = 600.

Answer: (c) equal

Explanation:
1. -7/21 = -1/3.
2. 3/(-9) = -1/3.
3. Cross check: (-7)(-9) = 63 = 21 × 3.

Teacher's Note:
1. Use a × d = b × c to test equality of a/b and c/d.
2. Both reduce to -1/3.

Answer: (b) \( \sqrt{3} \)

Explanation:
1. √25 = 5, 22/7 is a ratio of integers and 0.333... = 1/3 are all rational.
2. √3 = 1.732... is non-terminating and non-recurring, so it is irrational.

Teacher's Note:
1. Rational numbers have terminating or repeating decimals.
2. The square root of a non-square natural number is irrational.

Answer: (d) irrational

Explanation:
1. If r + s were rational, then s = (r + s) - r would also be rational, a contradiction.
2. So r + s is irrational.

Teacher's Note:
1. Rational numbers are closed under subtraction.
2. Same reasoning shows rs is irrational for r not zero.

Answer: (c) Non-zero rational numbers

Explanation:
1. The quotient of two non-zero rationals p/q ÷ r/s = ps/(qr) is again a non-zero rational.
2. Integers, natural and whole numbers fail: 1 ÷ 2 is not an integer.

Teacher's Note:
1. Test with 1 ÷ 2.
2. Closure requires the answer to stay inside the set.

Answer: (d) the difference of the sums of its digits in odd and even places is 0 or a multiple of 11

Explanation:
1. The test compares the sum of digits at odd places with that at even places.
2. For 3245682 both sums are 15, so the number is divisible by 11.

Teacher's Note:
1. Alternate digit sum difference.
2. Example: 283712 gives sums 6 and 17 with difference 11.

Answer: (c) 2961

Explanation:
1. 296 - 2 × 1 = 294; 29 - 2 × 4 = 21, which is a multiple of 7.
2. So 2961 = 7 × 423 is divisible by 7; the others are not (2963 leaves remainder 2, 2962 leaves 1, 2965 leaves 4).

Teacher's Note:
1. Repeat the process until a small number is obtained.
2. 7 × 423 = 2961.

Answer: (b) 63924

Explanation:
1. 63924 is even and 6 + 3 + 9 + 2 + 4 = 24 is divisible by 3, so it is divisible by 6.
2. 154 is even but 1 + 5 + 4 = 10 is not a multiple of 3; 5673 is odd; 89612 has digit sum 26.

Teacher's Note:
1. A number divisible by 2 and 3 is divisible by 6.
2. Check each condition separately.

Answer: (a) 456138

Explanation:
1. 4 + 5 + 6 + 1 + 3 + 8 = 27, a multiple of 9.
2. Digit sums of the others: 26, 21, 23, none a multiple of 9.

Teacher's Note:
1. Divisible by 9 means the digit sum is divisible by 9.
2. 27 = 3 × 9.

Answer: (c) 134522

Explanation:
1. The last two digits decide: 22 is not divisible by 4.
2. 28, 00 and 00 are divisible by 4.

Teacher's Note:
1. Check only the last two digits.
2. Numbers ending in two zeros are divisible by 4.

Answer: (a) \( \frac{1}{13} \)

Explanation:
1. The numerators lie between -3 and 9 for a number between the two.
2. 1/13 is between -3/13 and 9/13; the others fall outside.

Teacher's Note:
1. Same denominator: compare numerators.
2. Infinitely many rational numbers lie between any two rational numbers.

Answer: (a) \( \frac{9}{2} \)

Explanation:
1. (4 + 5)/2 = 9/2 = 4.5.
2. It lies strictly between 4 and 5.

Teacher's Note:
1. The mean of two numbers lies between them.
2. Repeat the process to get more numbers.