SATHEE: Theory of Equations 1 Question 37

Theory of Equations 1 Question 37

38. Let $a, b, c, p, q$ be the real numbers. Suppose $α, β$ are the roots of the equation $x^{2} + 2 p x + q = 0$ . and $α, \frac{1}{β}$ are the roots of the equation $a x^{2} + 2 b x + c = 0$ , where $β^{2} \notin - 1, 0, 1$ .

Statement I $(p^{2} - q) (b^{2} - a c) \geq 0$

Statement II $b \notin p a$ or $c \notin q a$ .

$(2008, 3 M)$

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Answer:

Correct Answer: 38. (b)

Solution:

Given, $x^{2} + 2 p x + q = 0$

$\begin{aligned} ∴ α + β = - 2 p \\ α β = q \\ And a x^{2} + 2 b x + c = 0 \\ ∴ α + \frac{1}{β} = - \frac{2 b}{a} \\ and \frac{α}{β} = \frac{c}{a} \end{aligned}$

Now, $(p^{2} - q) (b^{2} - a c)$

$\begin{aligned} = {\frac{α + β^{2}}{- 2} - α β}^{2} \frac{α + {\frac{1}{β}}^{2}}{2} - \frac{α}{β} a^{2} \\ = \frac{(α - β)^{2}}{16} α - {\frac{1}{β}}^{2} \cdot a^{2} \geq 0 \end{aligned}$

$∴$ Statement I is true.

Again, now $p a = - \frac{α + β}{2} a = - \frac{a}{2} (α + β)$

and $b = - \frac{a}{2} α + \frac{1}{β}$

Since,

$p a \neq b \Rightarrow α + \frac{1}{β} \neq α + β$

$\Rightarrow β^{2} \neq 1, β \neq - 1, 0, 1$ , which is correct.

Similarly, if $c \neq q a$

$\begin{array}{cc} \Rightarrow & a \frac{α}{β} \neq a α β \Rightarrow α β - \frac{1}{β} \neq 0 \\ \Rightarrow & α \neq 0 and β - \frac{1}{β} \neq 0 \\ \Rightarrow & β \neq - 1, 0, 1 \end{array}$

Statement II is true.

Both Statement I and Statement II are true. But Statement II does not explain Statement I.

Theory of Equations 1 Question 37

38. Let $a, b, c, p, q$ be the real numbers. Suppose $α, β$ are the roots of the equation $x^{2} + 2 p x + q = 0$ . and $α, \frac{1}{β}$ are the roots of the equation $a x^{2} + 2 b x + c = 0$ , where $β^{2} \notin - 1, 0, 1$ .

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Theory of Equations 1 Question 37

38. Let a,b,c,p,q be the real numbers. Suppose α,β are the roots of the equation x2+2px+q=0. and α,1β are the roots of the equation ax2+2bx+c=0, where β2∉−1,0,1.

Answer:

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38. Let $a, b, c, p, q$ be the real numbers. Suppose $α, β$ are the roots of the equation $x^{2} + 2 p x + q = 0$ . and $α, \frac{1}{β}$ are the roots of the equation $a x^{2} + 2 b x + c = 0$ , where $β^{2} \notin - 1, 0, 1$ .