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NDA 2 2025 Maths, question 31

Topic: Algebra

Passage · Q31 to Q32

Consider the equation

abx2+bcx+ca=cax2+abx+bcabx^{2}+bcx+ca=cax^{2}+abx+bc

Q31

If the roots of the equation are equal, then which one of the following is correct?

  1. (a) ac=b2ac=b^{2}
  2. (b) a+c=2ba+c=2b
  3. (c) 1a+1c=12b\frac{1}{a}+\frac{1}{c}=\frac{1}{2b}
  4. (d) 1a+1c=2b\frac{1}{a}+\frac{1}{c}=\frac{2}{b}Correct answer

Solution.

Answer: (d) 1a+1c=2b\frac{1}{a}+\frac{1}{c}=\frac{2}{b}

  1. Bring all terms to one side: a(b−c)x²+b(c−a)x+c(a−b)=0.
  2. Substitution shows x=1 is a root.
  3. For equal roots of a genuine quadratic, the other root is also 1, so a(b−c)=c(a−b).
  4. Thus b(a+c)=2ac and 1/a+1/c=2/b.