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JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Sequences and Series

Let three real numbers a, b, c be in arithmetic progression and...

Let three real numbers a,b,ca, b, c be in arithmetic progression and a+1,b,c+3a+1, b, c+3 be in geometric progression. If a>10a>10 and the arithmetic mean of a,ba, b and cc is 8, then the cube of the geometric mean of a,ba, b and cc is

Options

  1. A128
  2. B316
  3. C120
  4. D312

Correct answer

C. 120

Step-by-step solution

2b=a+c,b2=(a+1)(c+3)a+b+c3=8→b=8,a+c=1664=(a+1)(19−a)=19+18a−a2a2−18a−45=0→(a−15)(a+3)=0,(a>10)a=15,c=1,b=8((abc)1/3)3=abc=120\begin{aligned} & 2 b=a+c, b^2=(a+1)(c+3) \\ & \frac{a+b+c}{3}=8 \rightarrow b=8, a+c=16 \\ & 64=(a+1)(19-a)=19+18 a-a^2 \\ & a^2-18 a-45=0 \rightarrow(a-15)(a+3)=0,(a>10) \\ & a=15, c=1, b=8 \\ & \left((a b c)^{1 / 3}\right)^3=a b c=120\end{aligned}
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