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NDA 2 2025 Maths, question 8

Topic: Algebra

Q8

If

α=−1+−32\alpha=\frac{-1+\sqrt{-3}}{2}

then what is the value of (1+α19−α35)100−(1−3α25+α38)50(1+\alpha^{19}-\alpha^{35})^{100}-(1-3\alpha^{25}+\alpha^{38})^{50}?

  1. (a) −2-2
  2. (b) −1-1
  3. (c) 00Correct answer
  4. (d) 22

Solution.

Answer: (c) 00

  1. α³=1 and 1+α+α²=0.
  2. Reducing exponents modulo 3, the first base is 1+α−α²=−2α², and the second is 1−3α+α²=−4α.
  3. Their powers are 2¹⁰⁰α²⁰⁰ and 4⁵⁰α⁵⁰.
  4. Both equal 2¹⁰⁰α², so their difference is zero.