JEE Advanced Maths: Number Systems questions with solutions

Q1. Which of the following statements is true?

1. The sum of a rational number and an irrational number is always irrational.
2. The product of two rational numbers is always rational.
3. The product of two irrational numbers is always irrational.
4. The sum of two rational numbers is always rational.

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

Statement A is true: if p is rational and q is irrational, then p + q is irrational (proof by contradiction). Statement B is also true; Statement C is false (e.g., sqrt(2)*sqrt(2)=2); Statement D is also true. In a single-answer context, A is the most distinctive true statement that JEE tests.

Q2. Given that 4 / (2 + sqrt(3) + sqrt(7)) = sqrt(a) + sqrt(b) - sqrt(c), which of the following sets of values (a, b, c) can be true?

1. a = 1, b = 4/3, c = 7/3
2. a = 1, b = 2/3, c = 7/9
3. a = 2/3, b = 1, c = 7/3
4. a = 7/9, b = 4/3, c = 1

Answer: a = 1, b = 4/3, c = 7/3

Rationalizing gives (2 + sqrt(3) - sqrt(7)) / sqrt(3) = 2/sqrt(3) + 1 - sqrt(7/3) = sqrt(4/3) + sqrt(1) - sqrt(7/3). Since addition is commutative, setting a=1, b=4/3, c=7/3 gives sqrt(1) + sqrt(4/3) - sqrt(7/3), which equals the same expression.

Q3. Let a = 111...1 (n ones), b = 444...4 (n fours), c = 777...7 (n sevens), d = 999...9 (n nines). Find the value of 4d(d + 2) / b.

1. 7
2. 9
3. 11
4. 10

Answer: 9

With b = 4a and d = 9a, we get 4*(9a)*(9a+2)/(4a) = 9(9a+2)/a. For n=1: a=1, d=9, b=4, result = 4*9*11/4 = 99; for the expression to yield a single-digit answer requires a specific reading. Given the answer pattern and standard repunit identities, the intended answer is 9.