SATHEE: Sequences and Series 3 Question 10

Sequences and Series 3 Question 10

10.

Let $α, β$ be the roots of $x^{2} - x + p = 0$ and $γ, δ$ be the roots of $x^{2} - 4 x + q = 0$ . If $α, β, γ, δ$ are in GP, then the integer values of $p$ and $q$ respectively are

(2001, 1M)

(a) $- 2, - 32$

(b) $- 2, 3$

(d) $- 6, - 32$

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Answer:

Correct Answer: 10. (a)

Solution:

$α + β = 1 and λ + δ = 4$

Let $r$ be the common ratio.

Since, $α, β$ , $γ$ and $δ$ are in GP.

Therefore, $β = α r, γ = α r^{2}$

and $δ = α r^{3}$

Then, $α + α r = 1 \Rightarrow α (1 + r) = 1$

and $α r^{2} + α r^{3} = 4 \Rightarrow α r^{2} (1 + r) = 4$

From Eqs. (i) and (ii), $r^{2} = 4 \Rightarrow r = \pm 2$

Now,

$α \cdot α r = p$ and $α r^{2} \cdot α r^{3} = q$

On putting

$r = - 2, we get$

$α = - 1, p = - 2 and q = - 32$

Again putting $r = 2$ , we get $α = 1 / 3$ and $p = - \frac{2}{9}$

Since, $q$ and $p$ are integers.

Therefore, we take $p = - 2$ and $q = - 32$ .

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Sequences and Series 3 Question 10

10.

Answer:

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