JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For three positive integers $p, q, r$ , $x^{p q^{2}} = y^{q r} = z^{p^{2} r}$ and $r = pq + 1$ such that $3, 3 lo g_{y} x, 3 lo g_{z} y, 7 lo g_{x} z$ are in A.P. with common difference $\frac{1}{2}$ . Then $r - p - q$ is equal to:

Choose the correct answer:

Explanation

Solution

A.P. Series: Di gayi series hai $3, 3.5, 4, 4.5$ .
- $3\log_y x = 3.5 \implies \log_y x = \frac{7}{6}$
- $3\log_z y = 4 \implies \log_z y = \frac{4}{3}$
- $7\log_x z = 4.5 \implies \log_x z = \frac{9}{14}$
Powers Relation: Maan lijiye $x^{pq^2} = y^{qr} = z^{p^2r} = K$ .
- $x = K^{1/pq^2}, y = K^{1/qr}, z = K^{1/p^2r}$ .
Substitution: $\log_y x = \frac{1/pq^2}{1/qr} = \frac{r}{pq} = \frac{7}{6}$ .
- Kyonki $r = pq + 1$ , isliye $\frac{pq+1}{pq} = \frac{7}{6} \implies 6pq + 6 = 7pq \implies \mathbf{pq = 6}$ aur $\mathbf{r = 7}$ .
$p$ aur $q$ nikalna: Doosre relation $\log_z y = \frac{1/qr}{1/p^2r} = \frac{p^2}{q} = \frac{4}{3}$ ka use karke:
- $3p^2 = 4q$ . $pq = 6$ se $q = 6/p$ rakhne par: $3p^2 = 4(6/p) \implies 3p^3 = 24 \implies p^3 = 8 \implies \mathbf{p = 2}$ .
- Agar $p = 2$ , to $q = 6/2 = \mathbf{3}$ .
Final Value: $r - p - q = 7 - 2 - 3 = \mathbf{2}$ . (Option D)

Choose the correct answer:

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