CBSE Class 8 Mathematics Number System MCQs Set 33

Answer: (b) Their difference is a positive rational number

Explanation:
1. Sum, product and quotient of positive rational numbers are always positive.
2. The difference can be negative or zero, for example 3/5 - 9/2 is negative.

Teacher's Note:
1. Test each statement with an example.
2. One counter-example disproves a statement.

Answer: (b) \( -\frac{7}{4} \)

Explanation:
1. The denominators are equal, so compare numerators: -7 < -5.
2. Hence -7/4 < -5/4.

Teacher's Note:
1. Of two negative numbers the one with the larger absolute value is smaller.
2. On the number line -7/4 lies to the left of -5/4.

Answer: (a) \( -2,\ -\frac{1}{4},\ 0,\ 1,\ \frac{7}{4} \)

Explanation:
1. Negative numbers come first: -2 < -1/4.
2. Then 0, then the positives 1 < 7/4.
3. Order: -2, -1/4, 0, 1, 7/4.

Teacher's Note:
1. Place the numbers on a number line from left to right.
2. -2 is farther from zero than -1/4, so it is smaller.

Answer: (a) \( \frac{-7}{10}<\frac{2}{-3}<\frac{5}{-8} \)

Explanation:
1. Over the LCM 120 the numbers are -84/120, -75/120 and -80/120.
2. Ascending order of numerators: -84 < -80 < -75.
3. So -7/10 < 2/(-3) < 5/(-8).

Teacher's Note:
1. Write all fractions with the same positive denominator.
2. Then compare the numerators.

Answer: (d) a positive rational number

Explanation:
1. Two negative signs cancel, so the product is positive.
2. For example (-1/2) × (-2/3) = 1/3, which is not an integer.

Teacher's Note:
1. Like signs give a positive product.
2. The product need not be an integer.

Answer: (c) a negative rational number

Explanation:
1. A number and its reciprocal have the same sign because their product is 1.
2. So the reciprocal of a negative rational is negative.

Teacher's Note:
1. Product of a number and its reciprocal is 1, a positive number.
2. Same sign is needed for a positive product.

Answer: (b) no

Explanation:
1. The reciprocal of a would be 1/a, and 1/0 is not defined.
2. So 0 has no reciprocal.

Teacher's Note:
1. Division by zero is not defined.
2. Every non-zero rational number has a reciprocal.

Answer: (a) 15 and 8

Explanation:
1. (x + y)² = x² + y² + 2xy = 289 + 240 = 529, so x + y = 23.
2. (x - y)² = 289 - 240 = 49, so x - y = 7.
3. x = 15 and y = 8 (check: 15 × 8 = 120).

Teacher's Note:
1. Use (x + y)² = x² + y² + 2xy and (x - y)² = x² + y² - 2xy.
2. The sum and difference give both numbers.

Answer: (b) rational

Explanation:
1. An integer n can be written as n/1, so it is a rational number.
2. Negative integers are not natural, whole or positive.

Teacher's Note:
1. Rational numbers have the form p/q with q not zero.
2. Integers are a subset of rational numbers.

Answer: (b) negative

Explanation:
1. Negative numbers lie to the left of 0 on the number line.
2. Positive and natural numbers are greater than 0, and rational numbers may be either sign.

Teacher's Note:
1. Smaller than 0 means to the left of zero.
2. Only negative numbers satisfy this.

Answer: (b) \( -\frac{1}{4} \)

Explanation:
1. Same denominator: (-1 + -1)/8 = -2/8.
2. -2/8 = -1/4.

Teacher's Note:
1. Add numerators when denominators are equal.
2. Reduce the result to lowest terms.

Answer: (c) \( \frac{1}{5} \)

Explanation:
1. Two negative signs cancel in a fraction.
2. So -1/-5 = 1/5.

Teacher's Note:
1. A negative divided by a negative is positive.
2. Standard form has a positive denominator.

Answer: (a) 9

Explanation:
1. Denominator 5 becomes -15 when multiplied by -3.
2. Multiply the numerator the same way: -3 × (-3) = 9.

Teacher's Note:
1. Multiply numerator and denominator by the same number.
2. Check: 9/(-15) = -3/5.

Answer: (b) -1

Explanation:
1. 1/2 + 1/2 = 1.
2. The minus sign outside makes it -1.

Teacher's Note:
1. Evaluate the bracket first.
2. The sign outside applies to the whole bracket.

Answer: (b) -1

Explanation:
1. (1/8) × (-8) = -8/8 = -1.

Teacher's Note:
1. A number times its reciprocal is 1.
2. The negative sign makes it -1.

Answer: (b) one

Explanation:
1. Among a, a+2, a+4 one is a multiple of 3 (they leave remainders 0, 2, 1 or similar).
2. The only prime multiple of 3 is 3, which gives a = 3: 3, 5, 7.
3. So only one value of a is possible.

Teacher's Note:
1. Check the remainders modulo 3.
2. Try small primes: 5, 7, 9 fails because 9 is not prime.

Answer: (a) II only

Explanation:
1. The sum of two negative numbers is negative, so (I) is false.
2. The product of two negatives is positive, so (II) is true.
3. x - y may be negative or positive (e.g., -5 - (-2) = -3), so (III) is not always true.

Teacher's Note:
1. Test with numbers like -2 and -3.
2. Only statements true for all negative numbers count.

Answer: (a) Integers

Explanation:
1. The difference of two integers is always an integer.
2. 3 - 5 = -2 is not a whole number and 1/2 - 3/4 is not a non-negative fraction; -2 - (-5) = 3 is not a negative integer.

Teacher's Note:
1. Closure means the result stays inside the set.
2. One counter-example breaks closure.

Answer: (d) 2

Explanation:
1. Bases 2 and 5 are different, so the powers are equal only when the exponents are 0.
2. x - 2 = 0 gives x = 2 (then 2⁰ = 5⁰ = 1).

Teacher's Note:
1. Two different bases give equal powers only for exponent 0.
2. Check: 2⁰ = 1 = 5⁰.

Answer: (a) 4

Explanation:
1. LCM of 4, 5, 6 is 60.
2. Multiples of 60 between 300 and 600: 360, 420, 480, 540.
3. There are 4 such numbers.

Teacher's Note:
1. A number divisible by all three is a multiple of their LCM.
2. 300 and 600 themselves are excluded.

Answer: (a) 3

Explanation:
1. The numbers are 14 and 11 (14 × 11 = 154, 14 + 11 = 25).
2. Difference = 14 - 11 = 3.

Teacher's Note:
1. Find the pair with the given sum and product.
2. (x - y)² = 625 - 616 = 9.