Maths Q17 – JEE-MAIN 2025

Home Maths Complex & QuadraticQuestion #84

Let

P_{n} = α^{n} + β^{n}

n \in N

. If

P_{10} = 123

P_{9} = 76

P_{8} = 47

and

P_{1} = 1

, then the quadratic equation having roots

\frac{1}{α}

and

\frac{1}{β}

is:

Clue 1 of 3: Recurrence Relation

For a sequence $P_{n} = α^{n} + β^{n}$ , where $α$ and $β$ are roots of a quadratic equation $x^{2} - A x + B = 0$ , the sequence satisfies a linear recurrence relation.

🥷

Ninja Strategy•Reciprocal Roots Property

Recognize that if $a x^{2} + b x + c = 0$ has roots $α, β$ , then $c x^{2} + b x + a = 0$ has roots $1 / α, 1 / β$ . First find the equation for $α, β$ and then apply this transformation.

🧠 Full Solution Path

● Video Walkthrough

Step 1: Determine the sum and product of

α

and

β

✦ Active

The sequence $P_{n} = α^{n} + β^{n}$ satisfies the recurrence relation $P_{n} = (α + β) P_{n - 1} - (α β) P_{n - 2}$ , where $α$ and $β$ are roots of $x^{2} - (α + β) x + α β = 0$ .

Given $P_{1} = α + β = 1$ .

Using the recurrence relation for $n = 10$ : $P_{10} = (α + β) P_{9} - (α β) P_{8}$ . Substituting $P_{1} = 1$ :

P_{10} = P_{9} - (α β) P_{8}

Substitute the given values $P_{10} = 123$ , $P_{9} = 76$ , $P_{8} = 47$ :

123 = 76 - (α β) (47)

47 = - 47 (α β)

α β = - 1

Thus, we have $α + β = 1$ and $α β = - 1$ .

Step 2: Form the quadratic equation with roots

\frac{1}{α}

and

\frac{1}{β}

○ Expand

Let the new roots be $r_{1} = \frac{1}{α}$ and $r_{2} = \frac{1}{β}$ .

The sum of the new roots is $r_{1} + r_{2} = \frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$ .

The product of the new roots is $r_{1} r_{2} = \frac{1}{α} \cdot \frac{1}{β} = \frac{1}{α β}$ .

Substitute the values of $α + β = 1$ and $α β = - 1$ :

Sum = \frac{1}{- 1} = - 1

Product = \frac{1}{- 1} = - 1

The quadratic equation with roots $r_{1}$ and $r_{2}$ is $x^{2} - (Sum of roots) x + (Product of roots) = 0$ .

x^{2} - (- 1) x + (- 1) = 0

x^{2} + x - 1 = 0

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 17 – JEE-MAIN 2025

Clue 1 of 3: Recurrence Relation

🧠 Full Solution Path