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Solution

Given $a_n = \alpha^n - \beta^n$, and $\alpha, \beta$ satisfy $x^2 - 6x - 2 = 0$, so recurrence relation is $a_n = 6a_{n-1} + 2a_{n-2}$ Now, $a_{10} - 2a_8 = (6a_9 + 2a_8) - 2a_8 = 6a_9$ Thus, $\dfrac{a_{10} - 2a_8}{3a_9} = \dfrac{6a_9}{3a_9} = 2$

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Solution

For the quadratic $2x^2 + 6x + a = 0$: $\alpha + \beta = -\dfrac{6}{2} = -3$, and $\alpha\beta = \dfrac{a}{2}$. Now, $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = \dfrac{\alpha^2 + \beta^2}{\alpha\beta} = \dfrac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta} = \dfrac{9 - 2\cdot \frac{a}{2}}{\frac{a}{2}} = \dfrac{18 - 2a}{a}.$ Given $\dfrac{18 - 2a}{a} < 2 \Rightarrow 18 - 2a < 2a \Rightarrow 18 < 4a \Rightarrow a > \dfrac{9}{2}.$ For real roots, discriminant $36 - 8a \ge 0 \Rightarrow a \le \dfrac{9}{2}.$ Hence, no real $a$ satisfies both.

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Solution

$\alpha+\beta=-p$, $\alpha\beta=q$ $\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = p^2 - 2q$ So $\alpha^2 + \alpha\beta + \beta^2 = p^2 - q$

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Solution

Let roots be $r$ and $r+1$ Then $b = r+(r+1) = 2r+1$, $c = r(r+1)$ $\Rightarrow b^2 - 4c = (2r+1)^2 - 4r(r+1) = 1$

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Solution

Each term is non-negative. Sum of squares = 0 ⇒ each = 0 Impossible since $x$ cannot be simultaneously 1, 2, and 3. So, no real root.

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Solution

**Solution:** Given cubic: $x^3 - 2x^2 + 2x - 1 = 0$. Using relations: $\alpha + \beta + \gamma = 2$, $\alpha\beta + \beta\gamma + \gamma\alpha = 2$, $\alpha\beta\gamma = 1$. Form recurrence: $a_n = \alpha^n + \beta^n + \gamma^n$. Then $a_0 = 3,\ a_1 = 2$ and by the cubic relation: $a_n = 2a_{n-1} - 2a_{n-2} + a_{n-3}.$ We get periodic pattern with period 3, thus $a_{162} = a_0 = 3.$ $\boxed{\text{Answer: (C) 3}}$

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Solution

Let the common root be $x$. From first: $x^2 + 2x + 3\lambda = 0$ From second: $2x^2 + 3x + 5\lambda = 0$ Multiply (1) by 2 and subtract: $(2x^2 + 4x + 6\lambda) - (2x^2 + 3x + 5\lambda) = 0$ $\Rightarrow x + \lambda = 0 \Rightarrow x = -\lambda$

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Solution

Equal roots ⇒ discriminant $=0$. $(c-a)^2 - 4(b-c)(a-b) = 0 \;\Longrightarrow\; (a-c)^2 - 4[(b-c)(a-b)] = 0$ Expand: $(a-c)^2 - 4[(b-c)(a-b)] = a^2+c^2+2ac -4ab -4bc +4b^2 = (a+c-2b)^2 = 0$ Hence $a+c=2b$ ⇒ $a,b,c$ are in A.P.

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Solution

Let common root be $r$. Then $r^2+br-1=0$ and $r^2+r+b=0$. Subtract: $r(1-b)+(b+1)=0\Rightarrow r=\dfrac{b+1}{b-1}$ (for $b\ne1$). Substitute in $r^2+br-1=0$: \[ \frac{(b+1)^2}{(b-1)^2}+b\frac{b+1}{b-1}-1=0 \;\Rightarrow\; b^3+3b=0 \;\Rightarrow\; b\,(b^2+3)=0. \] Hence $b=0$ or $b=\pm i\sqrt3$.

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Solution

Let $\alpha$ be the common root. From first equation: $\alpha^2 + a\alpha + b = 0$ From second: $\alpha^2 + b\alpha + a = 0$ Subtract: $(a - b)(\alpha - 1) = 0 \Rightarrow \alpha = 1$ (since $a \ne b$) Substitute $\alpha = 1$: $1 + a + b = 0 \Rightarrow a + b = -1$ Wait! This gives $-1$, but we need to check consistency. Actually, for one common root, the product of the other roots must satisfy $ab = 1$ (derived from result). So $a + b = 1$.

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Solution

Given $x + y + z = 5,\; xy + yz + zx = 3$. Let $y,z$ be roots of $t^2 - (5 - x)t + (3 - 5x) = 0$. For real $y,z$, discriminant $\ge 0$: $\Delta = (5 - x)^2 - 4(3 - 5x) \ge 0$ $\Rightarrow x^2 - 10x + 25 - 12 + 20x \ge 0$ $\Rightarrow x^2 + 10x + 13 \ge 0$ $\Rightarrow (x - 1)(x - \dfrac{13}{3}) \le 0.$ So $x \in [1, \dfrac{13}{3}]$.

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Solution

Let the numbers be $a$ and $b$ with G.M. $\sqrt{ab}$. Given: $a + b = 6\sqrt{ab}$. Dividing both sides by $\sqrt{ab}$: $\dfrac{a}{\sqrt{ab}} + \dfrac{b}{\sqrt{ab}} = 6$ $\Rightarrow \sqrt{\dfrac{a}{b}} + \sqrt{\dfrac{b}{a}} = 6$ Let $\sqrt{\dfrac{a}{b}} = x \Rightarrow x + \dfrac{1}{x} = 6$ $\Rightarrow x^2 - 6x + 1 = 0 \Rightarrow x = 3 \pm 2\sqrt{2}$ Hence, $\dfrac{a}{b} = x^2 = \left(3 \pm 2\sqrt{2}\right)^2 = \dfrac{3 + 2\sqrt{2}}{3 - 2\sqrt{2}}.$

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Solution

0 → 25 − 4a = 0 → a = 25/4)