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NDA 1 2024 Maths, question 106

Topic: Algebra

Q106Not scored: no correct option as printed

If a,b,ca,b,c are in HP, then what is 1b−a+1b−c\frac{1}{b-a}+\frac{1}{b-c} equal to ?

  1. 2b\frac{2}{b}

  2. 1a+1c\frac{1}{a}+\frac{1}{c}

  3. 12(1a+1b+1c)\frac{1}{2}\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right)

Select the correct answer using the code given below :

  1. (a) 1 only
  2. (b) 2 only
  3. (c) 3 only
  4. (d) 1, 2 and 3

Solution.

Not scored: no correct option as printed.

Answer: not scored. No option is correct as printed.

  1. For harmonic progression, a + c = 2ac/b. Then 1/(b − a) + 1/(b − c) = (2b − a − c)/[b² − b(a + c) + ac] = 2/b, wherever the denominators are nonzero.
  2. This also equals 1/a + 1/c.
  3. Statement 3 equals 3/(2b), so it is false.
  4. The correct combination is statements 1 and 2 only, which is absent from the options.
  5. Equal a, b, c would additionally make the requested expression undefined.