Complex Numbers And Quadratic Equations

The value of $(1 + i) (1 + i^{2}) (1 + i^{3}) (1 + i^{4})$ is:
1. 2
2. 0
3. 1
4. i
If $\sqrt{a + i b} = x + i y$ , then possible value of $\sqrt{a - i b}$ is:
1. $x^{2} + y^{2}$
2. $\sqrt{x^{2} + y^{2}}$
3. $x + i y$
4. $x - i y$
The polar form of ${(i^{25})}^{3}$ is:
1. $\cos (\frac{π}{2}) + i \sin (\frac{π}{2})$
2. $\cos (π) + i \sin (π)$
3. $\cos (π) - i \sin (π)$
4. $\cos (\frac{π}{2}) - i \sin (\frac{π}{2})$
If $z = \frac{- 2}{1 + i \sqrt{3}}$ , then the value of arg(z) is:
1. $π$
2. $\frac{π}{3}$
3. $\frac{2 π}{3}$
4. $\frac{π}{4}$
The principal value of the amplitude of (1 + i) is:
1. $\frac{π}{4}$
2. $\frac{π}{12}$
3. $\frac{3 π}{4}$
4. $π$
The complete set of values of k, for which the quadratic equation $x^{2} - k x + k + 2 = 0$ has equal roots, consists of:
1. $2 + \sqrt{12}$
2. $2 \pm \sqrt{12}$
3. $2 - \sqrt{12}$
4. $- 2 - \sqrt{2}$
For the equation ${|x|}^{2} + |x| - 6 = 0$ , the sum of real roots is:
1. 1
2. 0
3. 2
4. None of these
If a, b are roots of the equation $4 x^{2} + 3 x + 7 = 0$ , then $\frac{1}{a} + \frac{1}{b}$ is equal to:
1. $\frac{7}{3}$
2. $- \frac{7}{3}$
3. $\frac{3}{7}$
4. $- \frac{3}{7}$
The value of x satisfying $\log_{3} (x^{2} + 4 x + 12) = 2$ are:
1. 2, - 4
2. 1, - 3
3. - 1, 3
4. - 1, - 3
The number of real roots of the equation ${(x^{2} + 2 x)}^{2} - {(x + 1)}^{2} - 55 = 0$ is:
1. 2
2. 1
3. 4
4. None of these
If z is a non-zero complex number, then $|\frac{{|\bar{z}|}^{2}}{z \bar{z}}|$ is equal to:
1. $|\frac{\bar{z}}{z}|$
2. $|z|$
3. $|\bar{z}|$
4. None of these
If a = 1 + i, then $a^{2}$ equals:
1. 1 - i
2. 2i
3. (1 + i) (1 - i)
4. i - 1
The argument of $\frac{1 - i \sqrt{3}}{1 + i \sqrt{3}}$ is:
1. $60 °$
2. $120 °$
3. $210 °$
4. $240 °$
If $z = \frac{1}{{(2 + 3 i)}^{2}}$ , then $|z|$ equals:
1. $\frac{1}{13}$
2. $\frac{1}{5}$
3. $\frac{1}{12}$
4. None of these
If ${(x + i y)}^{\frac{1}{3}} = a + i b$ , then $\frac{x}{a} + \frac{y}{b} =$ ?
1. 0
2. 1
3. - 1
4. None of these
If $α$ , $β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then $\frac{1}{a α + b} + \frac{1}{a β + b} =$ ?
1. $\frac{c}{a b}$
2. $\frac{a}{b c}$
3. $\frac{b}{a c}$
4. None of these
If x is real and $k = \frac{x^{2} - x + 1}{x^{2} + x + 1}$ , then:
1. $k \in [\frac{1}{3}, 3]$
2. $k \geq 3$
3. $k \leq \frac{1}{3}$
4. None of these
The values of k for which the quadratic equation $k x^{2} + 1 = k x + 3 x - 11 x^{2}$ has real and equal roots are:
1. - 11, - 3
2. 5, 7
3. 5, - 7
4. None of these
If the equations $x^{2} + 2 x + 3 λ = 0$ and $2 x^{2} + 3 x + 5 λ = 0$ have a non-zero common roots, then $λ$ is:
1. 1
2. - 1
3. 3
4. 0
The set of all values of m for which both the roots of the equation $x^{2} - (m + 1) x + m + 4 = 0$ are real and negative, is:
1. $(- \infty, - 3] \cup [5, \infty)$
2. $[- 3, 5]$
3. $(- 4, - 3]$
4. $(- 3, - 1]$
If $x + i y = \frac{3 + 5 i}{7 - 6 i}$ , then y = ?
1. $\frac{9}{85}$
2. $- \frac{9}{85}$
3. $\frac{53}{85}$
4. None of these
The amplitude of $\frac{1}{i}$ is equal to:
1. 0
2. $\frac{π}{2}$
3. $- \frac{π}{2}$
4. $π$
The argument of $\frac{1 - i}{1 + i}$ is:
1. $- \frac{π}{2}$
2. $\frac{π}{2}$
3. $\frac{3 π}{2}$
4. $\frac{5 π}{2}$
The value of $\frac{(i^{5} + i^{6} + i^{7} + i^{8} + i^{9})}{(1 + i)}$ is:
1. $\frac{1}{2} (1 + i)$
2. $\frac{1}{2} (1 - i)$
3. 1
4. $\frac{1}{2}$
$\frac{1 + 2 i + 3 i^{2}}{1 - 2 i + 3 i^{2}}$ equals:
1. i
2. - 1
3. - i
4. 4
The number of solutions of $x^{2} + |x - 1| = 1$ is:
1. 0
2. 1
3. 2
4. 3
If one root of the equation $x^{2} + p x + 12 = 0$ is 4, while the equation $x^{2} + p x + q = 0$ has equal roots, the value of q is:
1. $\frac{49}{4}$
2. $\frac{4}{49}$
3. 4
4. None of these
If $α$ , $β$ are the roots of the equation $x^{2} - p (x + 1) - c = 0$ , then $(α + 1) (β + 1)$ is:
1. c
2. c - 1
3. 1 - c
4. None of these
The least value of k which makes the roots of the equation $x^{2} + 5 x + k = 0$ imaginary is:
1. 4
2. 5
3. 6
4. 7
The equation of the smallest degree with real coefficients having 1 + i as one of the roots is:
1. $x^{2} + x + 1 = 0$
2. $x^{2} - 2 x + 2 = 0$
3. $x^{2} + 2 x + 2 = 0$
4. $x^{2} + 2 x - 2 = 0$

Complex Numbers And Quadratic Equations

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