JEE Mathematics Number System MCQs Set 01

Answer: (b) \( \sqrt{7} \)

Explanation:
1. \( \sqrt{49} = 7 \), \( \sqrt{0.25} = 0.5 \) and \( \sqrt{1.44} = 1.2 \) are all rational.
2. 7 is not a perfect square, so \( \sqrt{7} \) is not a ratio of integers.

Teacher's Note:
1. A square root of a positive integer is rational only when the integer is a perfect square.
2. Decimals such as 0.25 and 1.44 are perfect squares of terminating decimals, so check these first.

Answer: (a) x = 6 and x = −4

Explanation:
1. Case 1: x − 1 = 5 gives x = 6.
2. Case 2: x − 1 = −5 gives x = −4.
3. Both values satisfy |x − 1| = 5.

Teacher's Note:
1. Split an absolute-value equation of the form |A| = k (k > 0) into A = k and A = −k.
2. Always check that k is not negative before solving.

Answer: (d) x = 2 and x = −5

Explanation:
1. 2x + 3 = 7 gives x = 2.
2. 2x + 3 = −7 gives 2x = −10, so x = −5.

Teacher's Note:
1. Write both cases of the modulus separately.
2. Substitute back to confirm: |4 + 3| = 7 and |−10 + 3| = 7.

Answer: (c) no real solution

Explanation:
1. A modulus is never negative, so |x − 2| ≥ 0 for every real x.
2. Hence |x − 2| can never equal −3.

Teacher's Note:
1. Before solving, note the sign of the right-hand side.
2. A negative right-hand side makes an equation |A| = k impossible.

Answer: (b) 6

Explanation:
1. 1/7 = 0.142857 142857 …
2. The block 142857 has 6 digits.
3. Remainders on dividing by 7 can only be 1 to 6, so the block cannot exceed 6 digits.

Teacher's Note:
1. The repeating block of 1/p has length at most p − 1.
2. For 7 the maximum length 6 is actually reached.

Answer: (a) \( \frac{7}{16} \)

Explanation:
1. A fraction in lowest terms terminates only if its denominator has no prime factors except 2 and 5.
2. 16 = 2⁴, so \( \frac{7}{16} = 0.4375 \) terminates.
3. 7, 12 = 2²·3 and 3 contain the prime 7 or 3, so the others repeat.

Teacher's Note:
1. Factorise the denominator of the reduced fraction.
2. Only primes 2 and 5 are allowed for a terminating decimal.

Answer: (d) \( \frac{4}{11} \)

Explanation:
1. Let x = 0.3636…; then 100x = 36.3636….
2. Subtract: 99x = 36, so x = 36/99.
3. Divide by 9: x = 4/11.

Teacher's Note:
1. For a two-digit repeating block, multiply by 100 and subtract.
2. Always reduce the fraction to lowest terms.

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

Explanation:
1. Let x = 0.1666…; then 10x = 1.666… and 100x = 16.666….
2. Subtract: 90x = 15, so x = 15/90 = 1/6.

Teacher's Note:
1. When some digits do not repeat, multiply to align the repeating part before subtracting.
2. Check: 1 ÷ 6 = 0.1666…

Answer: (b) 7

Explanation:
1. \( (a + b)(a - b) = a^2 - b^2 \) with a = 3 and \( b = \sqrt{2} \).
2. So the product is 9 − 2 = 7, a rational number.

Teacher's Note:
1. Conjugate surds multiply to a rational number.
2. This is the idea behind rationalising denominators.

Answer: (a) The sum of a rational number and an irrational number is irrational.

Explanation:
1. If r is rational and s irrational, assume r + s = t is rational; then s = t − r would be rational, a contradiction.
2. Counter-examples to the others: \( \sqrt{2} + (-\sqrt{2}) = 0 \) and \( \sqrt{2} \cdot \sqrt{2} = 2 \) (rational), but \( \sqrt{2} \cdot \sqrt{3} = \sqrt{6} \) is irrational.

Teacher's Note:
1. Use a short proof by contradiction for the true statement.
2. One counter-example is enough to disprove an 'always' claim.

Answer: (d) infinite

Explanation:
1. Between any two distinct rational numbers there is another, for instance their average.
2. Repeating the process gives \( \frac{5}{12}, \frac{9}{24}, … \), without end.

Teacher's Note:
1. Rationals are dense on the number line.
2. The same holds for irrationals.

Answer: (c) \( \frac{5}{48} \)

Explanation:
1. Common denominator 48: 30/48, 28/48, 27/48, 25/48.
2. Greatest = 30/48 and least = 25/48.
3. Difference = 5/48.

Teacher's Note:
1. Convert to the LCM of the denominators before comparing.
2. LCM of 8, 12, 16 and 48 is 48.

Answer: (b) 12.09 m

Explanation:
1. 6.045 km = 6045 m.
2. One rod = 6045 ÷ 500 = 12.09 m.

Teacher's Note:
1. Convert units first: 1 km = 1000 m.
2. Sanity check: 500 × 12.09 = 6045.

Answer: (a) \( \frac{10}{3} \)

Explanation:
1. a² + 1 = 5, b² + 1 = 10, c² − 1 = 15.
2. Value = 50/15 = 10/3.

Teacher's Note:
1. Substitute carefully and then simplify.
2. Do not confuse c² − 1 with (c − 1)².

Answer: (d) \( \sqrt{5} \)

Explanation:
1. Triangle 1: hypotenuse = √(1² + 1²) = √2.
2. Triangle 2: √(2 + 1) = √3. Triangle 3: √(3 + 1) = 2.
3. Triangle 4: √(4 + 1) = √5.

Teacher's Note:
1. Each step adds 1 to the square of the hypotenuse.
2. The n-th hypotenuse is \( \sqrt{n + 1} \).

Answer: (c) \( \sqrt{3} + \sqrt{2} \)

Explanation:
1. Multiply numerator and denominator by the conjugate \( \sqrt{3} + \sqrt{2} \).
2. Denominator becomes 3 − 2 = 1.
3. So the value is \( \sqrt{3} + \sqrt{2} \).

Teacher's Note:
1. The conjugate of a − b is a + b.
2. (√a − √b)(√a + √b) = a − b.

Answer: (b) −i

Explanation:
1. i has period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
2. 23 = 4 × 5 + 3, so i²³ = i³ = −i.

Teacher's Note:
1. Divide the power by 4 and use the remainder.
2. A remainder of 0 gives 1.

Answer: (a) 0 is an integer but not a natural number.

Explanation:
1. The natural numbers are 1, 2, 3, … so 0 is not natural.
2. 0 belongs to Z, W, Q and R.
3. 0 is neither positive nor negative, and it equals 0/1, so it is rational.

Teacher's Note:
1. N does not include 0 in this convention; W does.
2. 0 is rational because it can be written as 0/1.

Answer: (d) 3 + 2i

Explanation:
1. Add real parts: 2 + 1 = 3.
2. Add imaginary parts: 3i − i = 2i.
3. The sum is 3 + 2i.

Teacher's Note:
1. Complex numbers add part by part.
2. Keep track of the sign of the imaginary part.

Answer: (c) 2i

Explanation:
1. (1 + i)² = 1 + 2i + i².
2. Since i² = −1 this equals 2i.

Teacher's Note:
1. Expand using (a + b)².
2. Remember i² = −1.