CUET PG 2022 — Mathematics PYQ

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Question

CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

The a, b, c and d are in GP and are in ascending order such that a + d = 112 and b + c = 48. If the GP is continued with a as the first term, then the sum of the first six terms is

Choose the correct answer:

Explanation

Step 1: Equations banana

Di gayi conditions ke mutabiq:

$a + d = 112 \implies a + ar^3 = 112 \implies a(1 + r^3) = 112$ ---(Equation 1)
$b + c = 48 \implies ar + ar^2 = 48 \implies ar(1 + r) = 48$ ---(Equation 2)

Step 2: $r$ ki value nikaalna

Equation 1 ko Equation 2 se divide karte hain:

\frac{a(1 + r^3)}{ar(1 + r)} = \frac{112}{48}

Humein pata hai ki $1 + r^3 = (1 + r)(1 - r + r^2)$ , toh:

\frac{(1 + r)(1 - r + r^2)}{r(1 + r)} = \frac{7}{3}

\frac{1 - r + r^2}{r} = \frac{7}{3}

Ab cross-multiplication karte hain:

3(1 - r + r^2) = 7r

3 - 3r + 3r^2 = 7r

3r^2 - 10r + 3 = 0

Is quadratic equation ko solve karne par:

$3r^2 - 9r - r + 3 = 0$

$3r(r - 3) - 1(r - 3) = 0$

$(3r - 1)(r - 3) = 0$

Toh $r = 3$ ya $r = 1/3$ . Kyunki terms ascending order mein hain, isliye hum $r = 3$ lenge.

Step 3: $a$ ki value nikaalna

$r = 3$ ko Equation 2 mein rakhte hain:

$a(3)(1 + 3) = 48$

$a(3)(4) = 48$

$12a = 48$

$a = 4$

Step 4: Pehle 6 terms ka sum ( $S_6$ )

GP ke sum ka formula hota hai: $S_n = \frac{a(r^n - 1)}{r - 1}$

Yahan $a = 4, r = 3, n = 6$ :

S_6 = \frac{4(3^6 - 1)}{3 - 1}

S_6 = \frac{4(729 - 1)}{2}

S_6 = 2 \times 728

$S_6 = 1456$