Q168 / 270 · NDA 2016 (Phase 2) · Mathematics

Paragraph: Let $\mathrm{a}, \mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{b}$ be in AP , where $\mathrm{x}+\mathrm{y}+\mathrm{z}=15$. Let $\mathrm{a}, \mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{b}$ be in HP , where $\mathrm{p}^{-1}+\mathrm{q}^{-1}+\mathrm{r}^{-1}=\frac{5}{3}$
Question: What is the value of pqr?

  • A
    $35/243$
  • B
    $81 / 35$
  • Correct answer
    C
    $243/35$
  • D
    Cannot be determined

Solution

Since a, p, q, r, b or $1, \mathrm{p}, \mathrm{q}, \mathrm{r}, 9$ are in H.P. $\begin{aligned} \Rightarrow & \frac{1}{1+4 \mathrm{~d}}=9 \Rightarrow d=-\frac{2}{9} \\ & \frac{1}{\mathrm{p}}=1-\frac{2}{9}=\frac{7}{9} \Rightarrow p=\frac{9}{7} \\ & \frac{1}{\mathrm{q}}=\frac{7}{9}-\frac{2}{9}=\frac{5}{9} \Rightarrow q=\frac{9}{5} \\ & \quad \& \frac{1}{\mathrm{r}}=\frac{5}{9}-\frac{2}{9}=\frac{3}{9} \Rightarrow r=\frac{9}{3} \\ \Rightarrow & \mathrm{p} \times \mathrm{q} \times \mathrm{r}=\frac{243}{35}\end{aligned}$

Hence, the correct answer is option: 'C'.

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