Let alpha and beta be the roots of x^2 + sqrt(3)x - 16 = 0, and gamma and delta be the roots of x^2 + 3x - 1 = 0. If P_n = alpha^n + beta^n and Q_n = gamma^n + delta^n, then (P_25 + sqrt(3)P_24) / (2P_23) + (Q_25 - Q_23) / Q_24 is equal to

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q3 – JEE-MAIN 2025

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Home Maths Complex & QuadraticQuestion #87

Let

α

and

β

be the roots of

x^{2} + \sqrt{3} x - 16 = 0

, and

γ

and

δ

be the roots of

x^{2} + 3 x - 1 = 0

. If

P_{n} = α^{n} + β^{n}

and

Q_{n} = γ^{n} + δ^{n}

, then

\frac{P_{25} + \sqrt{3} P_{24}}{2 P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}

is equal to

Clue 1 of 3: Recurrence Relations

For a quadratic equation $a x^{2} + b x + c = 0$ with roots $α, β$ , the sum of powers $P_{n} = α^{n} + β^{n}$ follows a linear recurrence relation.

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Step 1: Establish Recurrence for

P_{n}

✦ Active

For the quadratic equation $a x^{2} + b x + c = 0$ with roots $α, β$ , the sum of powers $P_{n} = α^{n} + β^{n}$ satisfies the recurrence relation $a P_{n} + b P_{n - 1} + c P_{n - 2} = 0$ . For the equation $x^{2} + \sqrt{3} x - 16 = 0$ , we have $a = 1, b = \sqrt{3}, c = - 16$ . Thus, the recurrence is:

P_{n} + \sqrt{3} P_{n - 1} - 16 P_{n - 2} = 0

Setting $n = 25$ , we get:

P_{25} + \sqrt{3} P_{24} - 16 P_{23} = 0

Rearranging, we find $P_{25} + \sqrt{3} P_{24} = 16 P_{23}$ . Therefore, the first part of the expression is:

\frac{P_{25} + \sqrt{3} P_{24}}{2 P_{23}} = \frac{16 P_{23}}{2 P_{23}} = 8

Step 2: Establish Recurrence for

Q_{n}

○ Expand

Similarly, for the equation $x^{2} + 3 x - 1 = 0$ with roots $γ, δ$ , the sum of powers $Q_{n} = γ^{n} + δ^{n}$ satisfies the recurrence relation $a Q_{n} + b Q_{n - 1} + c Q_{n - 2} = 0$ . Here, $a = 1, b = 3, c = - 1$ . Thus, the recurrence is:

Q_{n} + 3 Q_{n - 1} - Q_{n - 2} = 0

Setting $n = 25$ , we get:

Q_{25} + 3 Q_{24} - Q_{23} = 0

Rearranging, we find $Q_{25} - Q_{23} = - 3 Q_{24}$ . Therefore, the second part of the expression is:

\frac{Q_{25} - Q_{23}}{Q_{24}} = \frac{- 3 Q_{24}}{Q_{24}} = - 3

Step 3: Calculate the Final Value○ Expand

Adding the results from the two parts, the total expression is:

8 + (- 3) = 5

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Maths Question 3 – JEE-MAIN 2025

Clue 1 of 3: Recurrence Relations

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