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Complex Numbers and Quadratic Equations

Multiple Choice Questions

81.

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

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

82.

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

83.

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

84.

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

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

85.
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

$(- 3, 0)$

$(0, 3)$

$(0, - 3)$

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

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

$Let z = \frac{3}{2} + i \frac{\sqrt{3}}{2} r = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{\frac{12}{4}} = \sqrt{3} θ = \tan^{- 1} (\frac{\frac{\sqrt{3}}{2}}{\frac{3}{2}}) = \tan^{- 1} (\frac{1}{\sqrt{3}}) = \frac{π}{6} ∴ \frac{3}{2} + i \frac{\sqrt{3}}{2} = \sqrt{3} e^{\frac{iπ}{6}} ∴ {(\frac{3}{2} + i \frac{\sqrt{3}}{2})}^{50} = {(\sqrt{3} e^{\frac{iπ}{6}})}^{50} = {(\sqrt{3})}^{50} {(e^{\frac{iπ}{6}})}^{50} = 3^{25} e^{\frac{i 50 π}{6}} \Rightarrow {(\frac{3}{2} + i \frac{\sqrt{3}}{2})}^{50} = 3^{25} e^{\frac{i 25 π}{3}} = 3^{25} (\cos (\frac{25 π}{3}) + isin (\frac{25 π}{3})) = 3^{25} (\cos (1500) + isin (1500)) = 3^{25} [\cos (360 \times 4 + 60) + isin (360 \times 4 + 60)] = 3^{25} (\cos (60) + isin (60)) \Rightarrow {(\frac{3}{2} + i \frac{\sqrt{3}}{2})}^{50} = 3^{25} (\frac{1}{2} + i \frac{\sqrt{3}}{2}) . . . (i)$

According to question,

${(\frac{3}{2} + i \frac{\sqrt{3}}{2})}^{50} = 3^{25} (x + iy) 3^{25} (\frac{1}{2} + i \frac{\sqrt{3}}{2}) = 3^{25} (x + iy) which is true only when x = \frac{1}{2}, y = \frac{\sqrt{3}}{2}$

86.

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

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

87.

Let f(x) = ax² + bx + c, g(x) = px² + qx + r such that f(1) = g(1), f(2) = g(2) and f(3) - g(3) = 2. Then, f(4) - g(4) is

88.

The equations x² + x + a= 0 and x² + ax + 1 = 0 have a common real root

for no value of a
for exactly one value of a
for exactly two value of a
for exactly three value of a

89.

The points representing the complex number z for which arg $(\frac{z - 2}{z + 2}) = \frac{π}{3}$ lie on

a circle
a straight line
an ellipse
a parabola

90.

The quadratic equation 2x² - (a³ + 8a - 1)x + a² - 4a = 0 posses roots of opposite sign. Then,

a $\leq$ 0
0 < a < 4
$4 \leq a < 8$
$a \geq 8$

Previous Year Papers

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Test Yourself

Multiple Choice Questions

85. If 32 + i3250 = 325x + iy, where x and y are real, then the ordered pair (x, y) is - 3, 0 0, 3 0, - 3 12, 32

85.
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

$(- 3, 0)$

$(0, 3)$

$(0, - 3)$

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