Sequences & Series Question - None

Question

Let $\alpha, \beta \in \mathbb{R}$. If $\alpha, \beta^2$ are the roots of quadratic equation $x^2 - px + 1 = 0$ and $\alpha^2, \beta$ is the roots of quadratic equation $x^2 - qx + 8 = 0$, then the value of $r$ if $\dfrac{r}{8}$ is the arithmetic mean of $p$ and $q$, is

Accepted Answer

Step 1: For equation x² − px + 1 = 0 with roots α, β²:

• Sum: α + β² = p

• Product: α · β² = 1, so αβ² = 1

Step 2: For equation x² − qx + 8 = 0 with roots α², β:

• Sum: α² + β = q

• Product: α² · β = 8, so α²β = 8

Step 3: Divide the second product equation by the first:

$$\frac{\alpha^2\beta}{\alpha\beta^2} = \frac{8}{1} \implies \frac{\alpha}{\beta} = 8$$

Therefore α = 8β

Step 4: Substitute into αβ² = 1:

$$8\beta \cdot \beta^2 = 1 \implies 8\beta^3 = 1 \implies \beta = \frac{1}{2}$$

Thus α = 4

Step 5: Calculate p and q:

$$p = \alpha + \beta^2 = 4 + \frac{1}{4} = \frac{17}{4}$$

$$q = \alpha^2 + \beta = 16 + \frac{1}{2} = \frac{33}{2}$$

Step 6: Find arithmetic mean and solve for r:

$$\frac{r}{8} = \frac{p + q}{2} = \frac{\frac{17}{4} + \frac{33}{2}}{2} = \frac{\frac{17 + 66}{4}}{2} = \frac{83}{8}$$

$$ herefore \mathbf{r = 83}$$

Question:

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