JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $a_1, a_2, a_3, \dots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to

Choose the correct answer:

Explanation

Solution:

GP Property: $a_4 \cdot a_6 = a_5^2$ . Isliye $a_5^2 = 9 \implies a_5 = 3$ (kyuki numbers positive hain).
Second Condition: $a_5 + a_7 = 24 \implies 3 + a_7 = 24 \implies a_7 = 21$ .
Common Ratio ( $r$ ): $\frac{a_7}{a_5} = r^2 \implies \frac{21}{3} = 7$ , yani $r^2 = 7$ .
Expression Solve Karna:
- $a_1 a_9 = a_5^2 = 3^2 = 9$ .
- $a_2 a_4 a_9 = (a_1 r) (a_1 r^3) (a_1 r^8) = a_1^3 r^{12} = (a_1 r^4)^3 = a_5^3 = 3^3 = 27$ .
- $a_5 + a_7 = 24$ (pehle se diya hai).
Final Value: $9 + 27 + 24 = \mathbf{60}$ .