Argand diagram; algebra of complex numbers — practice questions

Question 1 · difficulty L2 · understanding

Let $$\alpha$$ and $$\beta$$ be the roots of the equation x 2 + (2i $$-$$ 1) = 0. Then, the value of |$$\alpha$$ 8 + $$\beta$$ 8 | is equal to :

• A. 50
• B. 250
• C. 1250
• D. 1500

Question 2 · difficulty L2 · understanding

If $$\alpha$$, $$\beta$$ $$\in$$ R are such that 1 $$-$$ 2i (here i 2 = $$-$$1) is a root of z 2 + $$\alpha$$z + $$\beta$$ = 0, then ($$\alpha$$ $$-$$ $$\beta$$) is equal to :

• A. $$-$$7
• B. 7
• C. 3
• D. $$-$$3

Question 3 · difficulty L3 · understanding

Let $$\alpha, \beta$$ be the roots of the equation $$x^2-x+2=0$$ with $$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Then $$\alpha^6+\alpha^4+\beta^4-5 \alpha^2$$ is equal to ___________.

Question 4 · difficulty L3 · understanding

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0$, then $5(3 A-2 B-C)$ is equal to ____________.

Question 5 · difficulty L3 · understanding

Let $$\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=1} \right\}$$ and $$B = \left\{ {z \in c:\left| {z + 3i} \right| = 4} \right\}$$. Then $$\sum\limits_{z \in A \cap B} {({\mathop{\rm Re}\nolimits} z - {\mathop{\rm Im}\nolimits} z)}$$ is equal to ____________.

Question 6 · difficulty L3 · understanding

If $${z^2} + z + 1 = 0$$, $$z \in C$$, then $$\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|$$ is equal to _________.

Question 7 · difficulty L3 · understanding

If the real part of the complex number $$z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\pi \over 2}} \right)$$ is zero, then the value of sin 2 3$$\theta$$ + cos 2 $$\theta$$ is equal to _______________.

Question 8 · difficulty L3 · understanding

Let \begin{aligned}S = \left\{ {n \in N\left| {{{\left( {\matrix{ 0 & i \cr 1 & 0 \cr } } \right)}^n}\left( {\matrix{ a & b \cr c & d \cr } } \right) = \left( {\matrix{ a & b \cr c & d \cr } } \right)\forall a,b,c,d \in R} \right.} \right\}\end{aligned}, where i = $$\sqrt { - 1}$$. Then the number of 2-digit numbers in the set S is _____________.

Question 9 · difficulty L3 · understanding

Let $S=\left\{z \in \mathbb{C}: z^2+\sqrt{6} i z-3=0\right\}$. Then $\sum\limits_{z \in S} z^8$ is equal to :

• A. 162
• B. 184
• C. 262
• D. 324

Question 10 · difficulty L3 · understanding

Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2+a x+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3 i \sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ is equal to:

• A. 160
• B. 176
• C. 194
• D. 187