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NDA 2 2024 Maths, question 45

Topic: Algebra

Passage · Q45 to Q46

Let (8+37)20=U+V(8+3\sqrt{7})^{20}=U+V and (8−37)20=W(8-3\sqrt{7})^{20}=W, where UU is an integer and 0<V<10<V<1.

Q45

What is V+WV+W equal to?

  1. (a) 88
  2. (b) 44
  3. (c) 22
  4. (d) 11Correct answer

Solution.

Answer: (d) 11

  1. Adding the conjugate binomial expansions cancels all odd powers of √7, so (U+V)+W is an integer.
  2. Also 0<8−3√7<1, so 0<W<1.
  3. Since 0<V<1, the integer V+W lies strictly between 0 and 2 and must equal 1.