CBSE Class 10 Mathematics Real Numbers MCQs Set 33

Answer: (d) \( \frac{2837}{99} \)

Explanation:
1. \( 23.\overline{43} = 23 + \frac{43}{99} = \frac{2320}{99} \).
2. \( 5.\overline{2} = 5 + \frac{2}{9} = \frac{47}{9} = \frac{517}{99} \).
3. Sum = (2320 + 517)/99 = 2837/99.

Teacher's Note:
1. A bar over two digits means two nines in the denominator; over one digit, one nine.
2. Convert both to the denominator 99 before adding.

Answer: (d) \( \frac{21}{17} \)

Explanation:
1. Work from the bottom: 3 + 1/4 = 13/4.
2. 1 + 4/13 = 17/13, so the whole fraction = 13/17.
3. x = 2 − 13/17 = 21/17.

Teacher's Note:
1. Simplify a continued fraction from the innermost part outward.
2. Then isolate x.

Answer: (a) R is true and T is false.

Explanation:
1. A fraction p/q (p, q positive integers) is a rational number, so R is true.
2. Rational numbers include negative numbers and zero, e.g. −3/4 and 0, which are not fractions in the sense of positive p/q.
3. So T is false.

Teacher's Note:
1. Every fraction is rational but not vice versa.
2. Rational numbers include negatives and zero.

Answer: (c) \( \frac{47}{9} \)

Explanation:
1. Let x = 5.2222… Then 10x = 52.222…
2. Subtract: 9x = 47, so x = 47/9.

Teacher's Note:
1. Subtract the original from ten times the number to remove the repeating part.
2. Check: 47 ÷ 9 = 5.222…

Answer: (c) \( \frac{22}{7} \)

Explanation:
1. x + y = [(3 + √2)² + (3 − √2)²] / [(3 − √2)(3 + √2)].
2. Numerator = 2(9 + 2) = 22; denominator = 9 − 2 = 7.
3. x + y = 22/7.

Teacher's Note:
1. Use the common denominator (3 − √2)(3 + √2) = 7.
2. (a + b)² + (a − b)² = 2(a² + b²).

Answer: (a) x is false and y is the correct explanation of x.

Explanation:
1. Whole numbers are 0, 1, 2, 3, … and natural numbers are 1, 2, 3, …
2. 0 is a whole number but not a natural number, so x is false.
3. y is true, and it is exactly the reason why x fails.

Teacher's Note:
1. One counter-example is enough to disprove a general statement.
2. 0 is the only difference between the two sets.

Answer: (a) 34

Explanation:
1. Rationalising: a = 3 + 2√2 and b = 3 − 2√2.
2. a + b = 6 and ab = 9 − 8 = 1.
3. a² + b² = (a + b)² − 2ab = 36 − 2 = 34.

Teacher's Note:
1. Rationalise first; then use symmetric functions of a and b.
2. Here ab = 1, so b = 1/a.

Answer: (c) 198

Explanation:
1. a + b = 6 and ab = 1.
2. a³ + b³ = (a + b)³ − 3ab(a + b) = 216 − 18 = 198.

Teacher's Note:
1. Use the identity a³ + b³ = (a + b)³ − 3ab(a + b).
2. This is quicker than cubing 3 + 2√2.

Answer: (b) 194

Explanation:
1. 1/x = 1/(7 + 4√3) = 7 − 4√3, so x + 1/x = 14.
2. x² + 1/x² = (x + 1/x)² − 2 = 196 − 2 = 194.

Teacher's Note:
1. (7 + 4√3)(7 − 4√3) = 49 − 48 = 1, so they are reciprocals.
2. Square the sum and subtract 2.

Answer: (d) 4

Explanation:
1. 7 + 4√3 = (2 + √3)², since (2 + √3)² = 4 + 3 + 4√3.
2. √x = 2 + √3 and 1/√x = 2 − √3.
3. Their sum = 4.

Teacher's Note:
1. Look for a perfect square of the form (a + b√3)².
2. Reciprocals of conjugate pairs are conjugates.

Answer: (c) 4

Explanation:
1. 2 1/5 − 1 2/7 = 11/5 − 9/7 = (77 − 45)/35 = 32/35.
2. 6 − 1/6 = 35/6, so 1/(35/6) = 6/35 and 2 − 6/35 = 64/35.
3. The fraction = (32/35) ÷ (64/35) = 1/2.
4. 8 − 8 × ½ = 4.

Teacher's Note:
1. Handle the numerator and denominator separately.
2. Keep the common denominator 35 to avoid errors.

Answer: (c) \( \frac{27161}{9900} \)

Explanation:
1. Two digits repeat (35) after two non-repeating digits (74).
2. Fraction = (27435 − 274)/9900 = 27161/9900.

Teacher's Note:
1. Rule: (whole digit string − non-repeating part) over as many 9s as repeating digits, followed by as many 0s as non-repeating digits.
2. Here the denominator is 99 followed by 00.

Answer: (a) a = 2, b = −1

Explanation:
1. Multiply top and bottom by (√3 − 1): (√3 − 1)²/(3 − 1) = (4 − 2√3)/2 = 2 − √3.
2. So a = 2 and b = −1.

Teacher's Note:
1. Rationalise by multiplying by the conjugate of the denominator.
2. (√3 − 1)² = 3 + 1 − 2√3.

Answer: (c) 0.466

Explanation:
1. √3 = √6/√2 = 2.4495/1.4142 = 1.7321.
2. √3 + √2 − 1 = 1.7321 + 1.4142 − 1 = 2.1463.
3. 1/2.1463 = 0.4659, which is 0.466 to 3 decimal places.

Teacher's Note:
1. Find the missing square root from the ones given.
2. Divide carefully and round at the end.

Answer: (b) irrational

Explanation:
1. π has a non-terminating, non-repeating decimal expansion.
2. So it is irrational. (22/7 is only an approximation.)

Teacher's Note:
1. 22/7 is rational but only approximately equals π.
2. Irrational numbers are real, not imaginary.

Answer: (b) a non-terminating and non-repeating decimal

Explanation:
1. Terminating and recurring decimals are rational.
2. Only a decimal that neither terminates nor repeats is irrational.

Teacher's Note:
1. Rational: terminating or non-terminating repeating.
2. Irrational: non-terminating non-repeating.

Answer: (c) 22/7 is a rational number.

Explanation:
1. Every real number, rational or irrational, corresponds to a point on the number line, so (a) and (b) are false.
2. 22/7 is a ratio of integers, so it is rational: (c) is true.

Teacher's Note:
1. The number line is complete: it contains all real numbers.
2. Irrational points such as √2 can be constructed with a compass.

Answer: (b) irrational

Explanation:
1. If r is rational and s is irrational and r + s were rational, then s = (r + s) − r would be rational, a contradiction.
2. So the sum is always irrational.

Teacher's Note:
1. Proof by contradiction.
2. Example: 2 + √3 is irrational.

Answer: (b) irrational

Explanation:
1. If r ≠ 0 is rational and rs were rational, then s = (rs)/r would be rational, a contradiction.
2. So the product is irrational.

Teacher's Note:
1. The zero case (0 × √2 = 0) must be excluded.
2. Example: 3√5 is irrational.

Answer: (c) 1.5

Explanation:
1. √2 ≈ 1.414 and √3 ≈ 1.732.
2. 1.5 lies between them and is rational.
3. (√2 + √3)/2 and √6/2 are irrational; 1.8 is greater than √3.

Teacher's Note:
1. Check each option numerically and for rationality.
2. Use decimals for √2 and √3.