CDS 2026 (I) Elementary Mathematics, question 85

Topic: Algebra

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Passage · Q85 to Q86

Let x=6+56−5x=\frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}-\sqrt{5}} and y=6−56+5y=\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}+\sqrt{5}}

Q85

What is (x2−y2)\left( x^{2}-y^{2} \right) equal to ?

  1. (a) 223022\sqrt{30}
  2. (b) 443044\sqrt{30}
  3. (c) 663066\sqrt{30}
  4. (d) 883088\sqrt{30}Correct answer

Solution.

Answer: (d) 883088\sqrt{30}

  1. Rationalising gives x=(6+5)2=11+230x = \left( \sqrt{6} + \sqrt{5} \right)^{2} = 11 + 2\sqrt{30} and y=(6−5)2=11−230y = \left( \sqrt{6} - \sqrt{5} \right)^{2} = 11 - 2\sqrt{30}.
  2. Thus x+y=22x + y = 22 and x−y=430x - y = 4\sqrt{30}, so x2−y2=(x+y)(x−y)=8830x^{2} - y^{2} = (x + y)(x - y) = 88\sqrt{30}.