CBSE Class 8 Maths Advanced Rational Numbers MCQs Set 01

Answer: (d) rational number

Explanation:
1. This is the definition of a rational number: a quotient of two integers with a non-zero denominator.
2. Natural numbers, whole numbers and integers need not have this form of definition (although they can be written so), so the correct choice is rational number.

Teacher's Note:
1. Every integer is rational, but the word that defines the p/q form is 'rational'.
2. Remember the condition q ≠ 0.

Answer: (b) p and q are integers and q ≠ 0.

Explanation:
1. For p/q to make sense the denominator must not be zero.
2. The numerator p may be 0 (0 is a rational number, 0/5 = 0).
3. So the condition is p and q integers and q ≠ 0.

Teacher's Note:
1. Only the denominator is restricted.
2. 0 = 0/1 is a rational number, so p = 0 is allowed.

Answer: (d) \( \frac{q}{p} \)

Explanation:
1. The reciprocal of a/b is b/a, because a/b × b/a = 1.
2. So the reciprocal of p/q is q/p (it exists when p ≠ 0).

Teacher's Note:
1. Reciprocal means the number that multiplies with it to give 1.
2. Zero has no reciprocal, which is why p must be non-zero.

Answer: (a) between x and y

Explanation:
1. The number (x + y)/2 is the average of x and y.
2. The average of two different numbers lies between them (say x = 1, y = 3 gives 2).
3. So it is between x and y.

Teacher's Note:
1. Test with simple numbers, such as 2 and 6, whose average is 4.
2. If x = y the average equals both, so the numbers must be different.

Answer: (b) 241

Explanation:
1. 141 = 3 × 47, so it is not prime.
2. 241: its square root is about 15.5 and none of 2, 3, 5, 7, 11, 13 divides it, so 241 is prime.
3. 341 = 11 × 31 and 441 = 21 × 21 are composite.
4. So the prime number is 241.

Teacher's Note:
1. Test divisibility by the primes up to the square root.
2. Digit sum of 141 and 441 is a multiple of 3, a quick clue.

Answer: (b) 1

Explanation:
1. 0 is not a natural number, and 2 is prime.
2. 1 has only one factor, itself, so it is neither prime nor composite.
3. The answer is 1.

Teacher's Note:
1. Prime needs exactly two factors; composite needs more than two.
2. 1 has exactly one factor.

Answer: (b) is a negative rational number

Explanation:
1. The product of a number and its reciprocal is 1, which is positive.
2. So the two numbers have the same sign.
3. The reciprocal of a negative number is negative (for example the reciprocal of −3/4 is −4/3).

Teacher's Note:
1. A positive product needs equal signs.
2. Try it with −5: reciprocal is −1/5.

Answer: (b) \( -\frac{3}{5} \)

Explanation:
1. 219 = 3 × 73 and 365 = 5 × 73, so HCF = 73.
2. Dividing by 73: −219/365 = −3/5.

Teacher's Note:
1. Factorise numerator and denominator to spot the common factor.
2. Keep the negative sign with the fraction.

Answer: (d) \( -105 \)

Explanation:
1. |−138| = 138 and |−243| = 243.
2. 138 − 243 = −105.

Teacher's Note:
1. Take the absolute values first, then subtract.
2. Do not subtract the numbers inside the bars.

Answer: (b) (i) and (ii) only

Explanation:
1. (i) 1/x = 11/9, which is greater than 9/11.
2. (ii) (x + 1)/x = (20/11) ÷ (9/11) = 20/9, greater than 9/11.
3. (iii) (x + 1)/(x − 1) = (20/11) ÷ (−2/11) = −10, which is less than x.
4. So (i) and (ii) only.

Teacher's Note:
1. Substitute the value and compare each result with 9/11.
2. A negative number cannot exceed a positive one.

Answer: (b) \( \frac{3}{7}, \frac{1}{2}, \frac{7}{9}, \frac{4}{5} \)

Explanation:
1. Decimal values: 3/7 ≈ 0.43, 4/5 = 0.80, 7/9 ≈ 0.78, 1/2 = 0.50.
2. Ascending order: 0.43 < 0.50 < 0.78 < 0.80.
3. That is 3/7, 1/2, 7/9, 4/5.

Teacher's Note:
1. Converting to decimals is quick for four fractions.
2. Ascending means smallest first.

Answer: (d) \( \frac{5}{3} \)

Explanation:
1. The multiplicative inverse of a/b is b/a.
2. For 3/5 it is 5/3, since 3/5 × 5/3 = 1.

Teacher's Note:
1. Just flip the fraction.
2. The sign stays the same for a multiplicative inverse.

Answer: (a) \( \frac{-53}{9} \)

Explanation:
1. Let the number be x: −5 − x = 8/9.
2. x = −5 − 8/9 = −45/9 − 8/9 = −53/9.

Teacher's Note:
1. Set up an equation and solve for x.
2. Write −5 as −45/9 to subtract.

Answer: (d) \( \frac{61}{11} \)

Explanation:
1. Take x = 6k and y = 5k.
2. x² + y² = 36k² + 25k² = 61k² and x² − y² = 36k² − 25k² = 11k².
3. The ratio is 61/11.

Teacher's Note:
1. Replace x and y by 6 and 5 (any common multiple).
2. Squares change the ratio to 36 : 25.

Answer: (a) Irrational number

Explanation:
1. If a non-zero rational r times an irrational s were rational, then s = (r × s)/r would also be rational, which is false.
2. So the product is an irrational number.

Teacher's Note:
1. Example: 2 × √2 = 2√2 is irrational.
2. The rational number must be non-zero; 0 × √2 = 0.

Answer: (d) 44

Explanation:
1. 6/11 is twice 3/11.
2. So 6/11 of the number = 2 × 22 = 44.

Teacher's Note:
1. Notice 6/11 = 2 × (3/11), so no need to find the number.
2. Check: the number is 242/3 and 6/11 of it is 44.

Answer: (d) Infinite

Explanation:
1. The mean of two rational numbers lies between them, and we can take means again and again.
2. So there are infinitely many rational numbers between any two.

Teacher's Note:
1. There is no 'next' rational number.
2. Repeated averaging never stops.

Answer: (b) \( \frac{6}{5} \)

Explanation:
1. 1 = 5/5 and 2 = 10/5.
2. 6/5 = 1.2 lies between them; 5/6 < 1, 3/7 < 1 and 7/3 ≈ 2.33 > 2.
3. The answer is 6/5.

Teacher's Note:
1. Convert each option to a decimal and compare with 1 and 2.
2. Only 1.2 lies in (1, 2).

Answer: (b) 13

Explanation:
1. 385 = 5 × 7 × 11, so the first three primes are 5, 7, 11.
2. 1001 = 7 × 11 × 13, so the last three primes are 7, 11, 13.
3. The four primes are 5, 7, 11, 13 and the last is 13.

Teacher's Note:
1. Prime-factorise both products.
2. The two middle primes appear in both products.

Answer: (a) \( (x-z)^2 y \) is even

Explanation:
1. x − z = odd − even = odd, so (x − z)² is odd, and odd × y (odd) = odd.
2. So (x − z)²y is odd and the statement 'is even' cannot be true.
3. The other three are possible: (x − z)y² is odd, (x − z)y is odd and (x − y)²z is even.

Teacher's Note:
1. Use odd/even rules: odd − even = odd; odd × odd = odd.
2. Even × anything is even.

Answer: (d) 50/75

Explanation:
1. 3/5 = 45/75 and 2/3 = 50/75.
2. Numbers strictly between them have numerators 46, 47, 48, 49 over 75.
3. 50/75 equals 2/3 itself, so it does not lie between them.

Teacher's Note:
1. Use the common denominator 75.
2. 'Between' excludes both end numbers.

Answer: (d) 5

Explanation:
1. Numerator: −3 + 2 = −1; 5 − (−1) = 6; 6 ÷ 2 = 3; 8 − 3 = 5.
2. Denominator: |5 − 3| = 2, |5 − 8| = 3, 3 ÷ 3 = 1; 2 − 1 = 1.
3. Value = 5/1 = 5.

Teacher's Note:
1. Apply brackets first, then division, then subtraction (BODMAS).
2. Absolute values are positive.

Answer: (a) \( \frac{4}{5} \)

Explanation:
1. Let the number be x. Correct result = 3x/2; wrong result = x ÷ 3/2 = 2x/3.
2. Difference: 3x/2 − 2x/3 = 5x/6 = 2/3.
3. x = (2/3) × (6/5) = 4/5.

Teacher's Note:
1. Write the correct and the wrong results as expressions in x.
2. Subtract and solve the simple equation.

Answer: (b) \( \frac{-13}{38} \)

Explanation:
1. −8/28: HCF 4, reduces to −2/7. 14/(−49): HCF 7, reduces to −2/7. 16/56: HCF 8, reduces to 2/7.
2. 13 and 38 have no common factor other than 1, so −13/38 is already in simplest form.

Teacher's Note:
1. Look for a common factor of numerator and denominator.
2. Also, a simplest form is written with a positive denominator.

Answer: (a) a + b is rational

Explanation:
1. The sum of two integers is an integer, which is rational.
2. a − b can be zero or negative; a/b is a rational number, not irrational.
3. So a + b is rational.

Teacher's Note:
1. Test with a = 2, b = 5.
2. Every integer is a rational number.

Answer: (a) \( 1\frac{69}{80} \)

Explanation:
1. 9/4 × 3/5 ÷ 12/5 = 27/20 × 5/12 = 9/16.
2. 7/8 ÷ 5/4 = 7/8 × 4/5 = 7/10.
3. Sum = 9/16 + 7/10 + 3/5 = 45/80 + 56/80 + 48/80 = 149/80 = 1 69/80.

Teacher's Note:
1. Do multiplication and division first, then add.
2. LCM of 16, 10, 5 is 80.

Answer: (c) \( \frac{101}{3} \)

Explanation:
1. Each factor is 2 − (2k−1)/(2k+1) = (2k+3)/(2k+1).
2. So the product is 5/3 × 7/5 × 9/7 × ... × 101/99, and everything cancels except the first denominator and last numerator.
3. Product = 101/3.

Teacher's Note:
1. Write the first few factors as 5/3, 7/5, 9/7.
2. Telescoping products cancel in pairs.

Answer: (b) 50

Explanation:
1. The k-th fraction is [1/(2k) − 1/(2k+1)] ÷ [1/(2k+1) − 1/(2k+2)] = [1/(2k(2k+1))] ÷ [1/((2k+1)(2k+2))] = (2k+2)/(2k) = (k+1)/k.
2. There are 49 fractions (k = 1 to 49): 2/1 × 3/2 × 4/3 × ... × 50/49.
3. The product is 50.

Teacher's Note:
1. Simplify one fraction first, then spot the pattern.
2. The product telescopes just like the previous question.

Answer: (d) \( \frac{18}{999} \)

Explanation:
1. Three digits (018) repeat, so write the repeating block over 999.
2. \( 0.\overline{018} = \frac{18}{999} \).
3. The answer is 18/999.

Teacher's Note:
1. The number of 9s equals the number of repeating digits.
2. Here the whole block 018 repeats from the decimal point.

Answer: (c) 365737

Explanation:
1. Dividend = divisor × quotient + remainder.
2. = 999 × 366 + 103 = 365634 + 103 = 365737.

Teacher's Note:
1. Use the division algorithm.
2. 999 × 366 = 366000 − 366.

Answer: (d) 349

Explanation:
1. Let x = 2.525252... Then 100x = 252.5252..., so 99x = 250.
2. x = 250/99, which is already in lowest form (250 = 2·5³, 99 = 3²·11).
3. Sum = 250 + 99 = 349.

Teacher's Note:
1. Multiply by 100 because two digits repeat.
2. Check the fraction is in lowest terms before adding.

Answer: (a) \( \frac{14}{9} \)

Explanation:
1. \( 0.\overline{2} = \frac{2}{9} \), \( 0.\overline{3} = \frac{3}{9} \), \( 0.\overline{4} = \frac{4}{9} \) and \( 0.\overline{5} = \frac{5}{9} \).
2. Sum = (2 + 3 + 4 + 5)/9 = 14/9.

Teacher's Note:
1. A single repeating digit d gives d/9.
2. Add the numerators over the common denominator 9.

Answer: (a) 2

Explanation:
1. The number is 45k + 32.
2. 45k is a multiple of 15, so the remainder depends only on 32.
3. 32 = 2 × 15 + 2, so the remainder is 2.

Teacher's Note:
1. 45 is a multiple of 15, so the remainder 32 can be reduced mod 15.
2. Try k = 1: 77 = 5 × 15 + 2.

Answer: (a) 2 or 5

Explanation:
1. A terminating decimal needs the denominator (in lowest form) to be of the form 2^m × 5^n.
2. So its prime factors are only 2 and/or 5 (no other prime).
3. The best choice is 2 or 5.

Teacher's Note:
1. Think of 1/2, 1/5, 1/20 which all terminate and 1/3 which does not.
2. Denominator in lowest form matters.

Answer: (d) \( \frac{1}{x} \)

Explanation:
1. Product = x²y × x/y = x³.
2. We need x² ÷ d = x³, so d = x²/x³ = 1/x.

Teacher's Note:
1. Simplify the product first.
2. Then solve x² ÷ d = x³ for d.

Answer: (d) any number except '0'

Explanation:
1. 1 + 1/x = x/x + 1/x = (x + 1)/x.
2. This is true for every x for which the expressions exist, that is every x ≠ 0.
3. So x can be any number except 0.

Teacher's Note:
1. It is an identity, not an equation to solve.
2. x = 0 is excluded because we cannot divide by 0.

Answer: (c) \( \frac{17}{30} \)

Explanation:
1. 1/3 ≈ 0.33 and 4/5 = 0.80.
2. 1/4 = 0.25 (too small), 9/10 = 0.9 (too big), 17/30 ≈ 0.57 (between), 1 7/10 = 1.7 (too big).
3. The answer is 17/30.

Teacher's Note:
1. Convert to decimals for quick comparison.
2. Check each option against both ends.

Answer: (b) There are 2n non-perfect square numbers between the squares

Explanation:
1. Between n² and (n + 1)² = n² + 2n + 1 there are 2n numbers (n² + 1 up to n² + 2n), and none of them is a perfect square.
2. (C) is false: 3² + 4² = 25 = 5². (D) is false: the difference of consecutive numbers is 1.
3. (A) is false since the count is 2n, not 2n + 1.
4. The true statement is (B).

Teacher's Note:
1. Test with n = 3: numbers between 9 and 16 are 10 to 15, six of them.
2. A single counter-example kills a statement.

Answer: (d) (i) and (ii) only

Explanation:
1. (i) 1/m = 11/9 > 9/11.
2. (ii) (m + 1)/m = 20/9 > 9/11.
3. (iii) (m + 1)/(m − 1) = (20/11)/(−2/11) = −10 < 9/11.
4. So (i) and (ii) are greater than m.

Teacher's Note:
1. Put m = 9/11 into each expression.
2. A negative value is always less than a positive m.

Answer: (c) \( \frac{-1}{6} \)

Explanation:
1. Decimal values: −1/6 ≈ −0.17, 4/9 ≈ 0.44, 6/(−7) ≈ −0.86, 2/5 = 0.40, −3/4 = −0.75.
2. Descending: 4/9, 2/5, −1/6, −3/4, 6/(−7).
3. The middle (third) number is −1/6.

Teacher's Note:
1. With five numbers the middle one is the third.
2. Convert everything to decimals.