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Question Bank No: 1

1. $\frac{{(\cos θ + i \sin θ)}^{4}}{{(\sin θ + i \cos θ)}^{5}}$ = ?

a)		cos $θ$ - i sin $θ$
b)		sin $θ$ - i cos $θ$
c)		cos 9 $θ$ - i sin 9 $θ$
d)		sin 9 $θ$ - i cos 9 $θ$

2. $[\frac{1 + Sin θ + i \cos θ}{1 + \sin θ - i \cos θ}] n$ =

a)		Cos $(\frac{n π}{2} - n θ)$ + i sin $(\frac{n π}{2} - n θ)$
b)		Cos $(\frac{n π}{2} + n θ)$ + i sin $(\frac{n π}{2} + n θ)$
c)		Sin $(\frac{n π}{2} - n θ)$ + i cos $(\frac{n π}{2} - n θ)$
d)		Cos n $(\frac{n π}{2} - n θ)$ + i sin n $(\frac{n π}{2} - n θ)$

3. If $| \begin{matrix} 2 i & - 3 i & 1 \\ 3 & 3 i & - 1 \\ 4 & 3 & i \end{matrix} |$ = x+iy then

a)		x = 3, y = 1
b)		x = 2, y = 5
c)		x = 0, y = 0
d)		x = 1, y = 1

4. If z= r $e^{i θ}$
then | $e^{iz}$ | =

a)		$e^{2 rsin θ}$
b)		$e^{rsin θ}$
c)		${re}^{\sin θ}$
d)		1

5. ${(i)}^{2}$ + ${(i)}^{4}$ + ${(i)}^{6}$ + ......... to (2k + 1) terms=

a)		-1
b)		1
c)		0
d)		k

6. The Value of $(\cos \frac{π}{2} + i \sin \frac{π}{2})$ . $(\cos \frac{π}{2^{2}} + i \sin \frac{π}{2^{2}})$ . $(\cos \frac{π}{2^{3}} + i \sin \frac{π}{2^{3}})$ .......... $\infty$

a)		-1
b)		1
c)		0
d)		2

7. If $\sqrt{a + ib}$ = x+iy, then a possible valve of $\sqrt{a - ib}$ ,

a)		$x^{2}$ = $y^{2}$
b)		$\sqrt{x^{2} + y^{2}}$
c)		x - iy
d)		None

8. The equation of smallest degree with real coefficients have 1+i as one of the roots is

a)		$x^{2}$ +x+1= 0
b)		$x^{2}$ -2x+2= 0
c)		$x^{2}$ +2x+2= 0
d)		$x^{2}$ +2x-2= 0

9. If 2 +i is a root of the equation $x^{3}$ -5 $x^{2}$ + 9x - 5=0, then the other roots are

a)		1 and 2-i
b)		-1 and 3+i
c)		0 and 1
d)		-1 and i-2

10. If $i^{2}$ = -1, then the sum i+ $i^{2}$ + $i^{3}$ +......To 1000 terms is equal to

a)		1
b)		-1
c)		i
d)		0

11. The value of $\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}$ + 1 is equal to

a)		0
b)		-1
c)		1
d)		2

12. If ${(i)}^{2}$ = -1, then ${(i)}^{2}$ + (i)4 + ${(i)}^{6}$ +........... to (2n + 1) terms=

a)		-1
b)		1
c)		0
d)		-2

13. Let w be an imaginary root of $x^{n}$ = 1. Then (5-w) (5- $ω^{2}$ )..............(5- $ω^{n - 1}$ )

a)		1
b)		$\frac{5^{n} + 1}{4}$
c)		$4^{n - 1}$
d)		$\frac{5^{n} - 1}{4}$

14. If i = $\sqrt{- 1}$ , then 4+5 ${(\frac{- 1 + \sqrt{- 3}}{2})}^{334}$ + 3 ${(\frac{- 1 + \sqrt{- 3}}{2})}^{365}$ is

a)		1-i $\sqrt{3}$
b)		-1-i $\sqrt{3}$
c)		i $\sqrt{3}$
d)		-i $\sqrt{3}$

15. (2 - w) (2 - $ω^{2}$ ) ( 2 - $ω^{10}$ ) ( 2 - $ω^{11}$ ) =........., where w is the complex cube root of unity

a)		49
b)		50
c)		48
d)		47

16. If $ω$ is a complex cube root of unit, then $\frac{a + b ω + {c ω}^{2}}{c + a ω + {b ω}^{2}}$ + $\frac{a + b ω + {c ω}^{2}}{b + c ω + {a ω}^{2}}$ =

a)		1
b)		-1
c)		0
d)		2

17. If w is a complex cube root of unity then $\frac{a + b ω + {c ω}^{2}}{c + a ω + {b ω}^{2}}$ + $\frac{c + a ω + {b ω}^{2}}{a + b ω + {c ω}^{2}}$ + $\frac{b + c ω + {a ω}^{2}}{b + {c ω}^{4} + {a ω}^{5}}$ =

a)		1
b)		w
c)		$w^{2}$
d)		0

18. If $ω$ is a non real cube root of unity, then (a+b) (a+ b $ω$ )(a+b $ω^{2}$ )= (K - 2002)

a)		$a^{3}$ + $b^{3}$
b)		$a^{3}$ - $b^{3}$
c)		$a^{2}$ + $b^{2}$
d)		$a^{2}$ - $b^{2}$

19. The valve of ${(\frac{- 1 + \sqrt{- 3}}{2})}^{26}$ + ${(\frac{- 1 - \sqrt{- 3}}{2})}^{26}$ is

a)		-1
b)		1
c)		0
d)		None of these

20. If $ω$ is an imaginary cube root of 1, $(1 + ω$ - $ω^{2}$ $)^{5}$ + $(1 - ω + ω^{2})^{5}$ =

a)		16
b)		8
c)		32
d)		27