JIPMAT 2023 (QA) - The number of integral solutions of the equation 7(y+1/y)-2(y^2+1/y^2)=9 is | PYQs + Solutions

JIPMAT 2023

Number System

Integral Solutions

Medium

The number of integral solutions of the equation $7\left(y+\frac{1}{y}\right)-2\left(y^2+\frac{1}{y^2}\right)=9 \text { is }$

Correct Option: 2

Let me solve to understand why the answer is 1:

7(y + \frac{1}{y}) - 2(y^2 + \frac{1}{y^2}) = 9

Let's let

y + \frac{1}{y} = t

Then

(y + \frac{1}{y})^2 = (t)^2

y^2 + \frac{1}{y^2} = t^2 - 2

Substituting:

\newline

7t - 2(t^2 - 2) = 9

7t - 2t^2 + 4 = 9

2t^2 - 7t + 5 = 0

Using quadratic formula:

\newline

t = \frac{7 \pm \sqrt{49-40}}{4} = \frac{7 \pm 3}{4}

t = \frac{10}{4}

\frac{4}{4}

t = \frac{5}{2}

1

When

t = 1

y+\frac{1}{y}=1

and

y^{2}+\frac{1}{y^2}=-1

On solving using these values, we get

7\times1-2\times(-1) = 9

Therefore, the equation has only 1 integral solution.

The answer is 1.