NIMCET 2023 — Mathematics PYQ

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Question

NIMCET 2023 — Mathematics PYQ

NIMCET | Mathematics | 2023

If a, b, c, d are in HP and arithmetic mean of ab, bc, cd is 9 then which of the following number is the value of ad?

Choose the correct answer:

Explanation

Solution

Step 1: Properties of HP

Since $a, b, c, d$ are in HP, their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in Arithmetic Progression (AP).

Let the common difference of this AP be $d'$ .

\frac{1}{b} - \frac{1}{a} = d' \implies \frac{a-b}{ab} = d' \implies ab = \frac{a-b}{d'}

\frac{1}{c} - \frac{1}{b} = d' \implies \frac{b-c}{bc} = d' \implies bc = \frac{b-c}{d'}

\frac{1}{d} - \frac{1}{c} = d' \implies \frac{c-d}{cd} = d' \implies cd = \frac{c-d}{d'}

Step 2: Use the Arithmetic Mean condition

The arithmetic mean of $ab, bc, cd$ is given as $9$ :

\frac{ab + bc + cd}{3} = 9

ab + bc + cd = 27

Step 3: Substitute the HP relations

Substitute the expressions from Step 1 into the equation:

\frac{a-b}{d'} + \frac{b-c}{d'} + \frac{c-d}{d'} = 27

\frac{1}{d'} (a - b + b - c + c - d) = 27

\frac{a - d}{d'} = 27

Step 4: Relate $a$ and $d$ using the AP property

We know that $\frac{1}{d}$ is the 4th term of the AP starting with $\frac{1}{a}$ :

\frac{1}{d} = \frac{1}{a} + (4-1)d'

\frac{1}{d} - \frac{1}{a} = 3d'

\frac{a - d}{ad} = 3d'

\frac{a - d}{d'} = 3ad

Step 5: Solve for $ad$

From Step 3, we have $\frac{a - d}{d'} = 27$ . Substitute this into the equation from Step 4:

27 = 3ad

ad = \frac{27}{3}

ad = 9

Final Answer:

The value of $ad$ is 9.