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Complex Numbers and Quadratic Equations

Multiple Choice Questions

71.

If $α and β$ B are the roots of the quadratic equation is, x² + ax + b = 0, (b $\neq$ 0), then the quadratic equation whose roots are $α - \frac{1}{β}, β - \frac{1}{α}$ , is

ax² + a(b - 1)x + (a - 1)² = 0
bx² + a(b - 1)x + (b - 1)² = 0
x² + ax + b = 0
abx² + bx + a = 0

72.
If $α and β$ are the roots of the quadratic equation ax² + bx + c = 0 and 3b² = 16ac, then

$α = 4 β or β = 4 α$

$α = - 4 β or β = - 4 α$

$α = 3 β or β = 3 α$

$α = - 3 β or β = - 3 α$

$α = 3 β or β = 3 α$

Given that, ( $α, β$ ) are roots of the quadratic equation

ax² + bx + c = 0

and 3b² = 16ac ...(i)

$∴ α + β = \frac{- b}{a} and αβ = \frac{c}{a}$ ...(ii)

From Eq. (i), we get

3b . b = 16a . c

$\Rightarrow 3 {(\frac{b}{a})}^{2} = 16 (\frac{c}{a}) [∵ divide by a^{2}] \Rightarrow 3 {(α + β)}^{2} = 16 αβ [∵ from Eq . (ii)] \Rightarrow 3 α^{2} + 3 β^{2} + 6 αβ = 16 αβ \Rightarrow 3 α^{2} + 3 β^{2} = 10 αβ \Rightarrow 3 (\frac{α}{β}) + 3 (\frac{β}{α}) = 10$

$Let \frac{α}{β} = x Then, 3 x + \frac{3}{x} = 10 \Rightarrow 3 x^{2} - 10 x + 3 = 0 \Rightarrow 3 x (x - 3) - 1 (x - 3) = 0 \Rightarrow (x - 3) (3 x - 1) = 0 \Rightarrow x = 3, \frac{1}{3} \Rightarrow \frac{α}{β} = \frac{1}{3}, 3 ∴ β = 3 α or α = 3 β$

73.

The number of solutions of the equation

$\frac{1}{2} \log_{\sqrt{3}} (\frac{x + 1}{x + 5}) + \log_{e} {(x + 5)}^{2}$ = 1

0
1
2
infinite

74.

If $\sin (α), \cos (α)$ be the roots of the equation x² - bx + c = 0, Then, which of the following statements is/are correct?

$c \leq \frac{1}{2}$
$b \leq \sqrt{2}$
$c > \frac{1}{2}$
b $> \sqrt{2}$

75.

If $(α + \sqrt{β}) and (α - \sqrt{β})$ are the roots of the equation x² + px + q = 0, where $α, β, p and q$ are real, then the roots of the equation (p² - 4q)(p²x² + 4px) - 16q = 0 are

$(\frac{1}{α} + \frac{1}{\sqrt{β}}) and (\frac{1}{α} - \frac{1}{\sqrt{β}})$
$(\frac{1}{\sqrt{α}} + \frac{1}{β}) and (\frac{1}{\sqrt{α}} - \frac{1}{β})$
$(\frac{1}{\sqrt{α}} + \frac{1}{\sqrt{β}}) and (\frac{1}{\sqrt{α}} - \frac{1}{\sqrt{β}})$
$(\sqrt{α} + \sqrt{β}) and (\sqrt{α} - \sqrt{β})$

76.

The number of solutions of the equation $\log_{2} (x^{2} + 2 x - 1) = 1$ is

77.

Let R be the set of real numbers and the functions f : R ➔ R and g : R ➔ R be defined by f(x) = x² + 2x - 3 and g(x) = x + 1. Then, the value of x for which f(g(x)) = g(f(x)) is

78.

The maximum value of $|z|$ , when the complex number z satisfies the condition $|z + \frac{2}{z}|$ = 2 is

$\sqrt{3}$
$\sqrt{3} + \sqrt{2}$
$\sqrt{3} + 1$
$\sqrt{3} - 1$

79.

If ${(\frac{3}{2} + i \frac{\sqrt{3}}{2})}^{50} = 3^{25} (x + iy)$ , where x and y are real, then the ordered pair (x, y) is

$(- 3, 0)$
$(0, 3)$
$(0, - 3)$
$(\frac{1}{2}, \frac{\sqrt{3}}{2})$

80.

If $\frac{z - 1}{z + 1}$ is pure imaginary, then

$|z| = \frac{1}{2}$
$|z| = 1$
$|z| = 2$
$|z| = 3$

Previous Year Papers

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Multiple Choice Questions

72. If α and β are the roots of the quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac, then α = 4β or β = 4α α = - 4β or β = - 4α α = 3β or β = 3α α = - 3β or β = - 3α

72.
If $α and β$ are the roots of the quadratic equation ax² + bx + c = 0 and 3b² = 16ac, then

$α = 4 β or β = 4 α$

$α = - 4 β or β = - 4 α$

$α = 3 β or β = 3 α$

$α = - 3 β or β = - 3 α$