Algebra of Complex Numbers

Q.

The least positive integer n such that ${\left(\frac{2i}{1+i}\right)}^n$ is a positive integer, is

1. 4

2. 2

3. 8

4. 16

Q.

Show that $1+i^{10}+i^{20}+i^{30}$ is a real number.

Q. Let ${(-2-\frac{1}{3}i)}^3=\frac{x+iy}{27}(i=\sqrt{-1})$, where $x$ and $y$ are real numbers, then $y-x$ equals :

1. $91$
2. $85$
3. $-91$
4. $-85$

Q. The number of distinct real roots of equation $3x^4+4x^3-12x^2+4=0$ is

Q. If $z^3+(3+2i)z+(-1+ia)=0$ has one real root, then the value of the $a$ lies in the interval $aϵR$

1. $(-2,-1)$
2. $(1,2)$
3. $(-1,0)$
4. $(0,1)$

Q. If $(\sqrt{3}+i{)}^{100}=2^{99}(p+iq),$ then $p$ and $q$ are roots of the equation

1. $x^2-(\sqrt{3}-1)x-\sqrt{3}=0$
2. $x^2+(\sqrt{3}-1)x-\sqrt{3}=0$
3. $x^2-(\sqrt{3}+1)x+\sqrt{3}=0$
4. $x^2+(\sqrt{3}+1)x+\sqrt{3}=0$

Q. Let $x$ be a complex number such that $\left|\frac{z-i}{z+2i}\right|$ and $|z|=\frac{5}{2}$. Then the value of $|z+3i|$ is :

1. $\sqrt{10}$
2. $2\sqrt{3}$
3. $\frac{7}{2}$
4. $\frac{15}{4}$

Q. If $x+\frac{1}{x}=8,$ then the value of $x^3+\frac{1}{x^3}$ is

1. $432$
2. $356$
3. $512$
4. $488$

Q. If $(a+ib{)}^5=\alpha +i\beta then(b+ia{)}^5$
is equal to

1. β - iα
2. β + iα
3. α+β
4. -α-iβ

Q.

For a positive integer n, find the value of $(1-i{)}^n{(1-\frac{1}{i})}^n.$

Q.

Find the number of solutions of $z^2+|z^2|=0.$

Q. the value of [997]^1/3 according to binomial theorem is

Q. If $\alpha ,\beta ,\gamma$ and $a,b,c$ are the complex numbers such that $\frac{\alpha }{a}+\frac{\beta }{b}+\frac{\gamma }{c}=1+i$ and $\frac{a}{\alpha }+\frac{b}{\beta }+\frac{c}{\gamma }=0$ such that $\frac{{\alpha }^2}{a^2}+\frac{{\beta }^2}{b^2}+\frac{{\gamma }^2}{c^2}=p+iq$ where $p,q\in R$ then the value of $p+q$ is

Q.

$\text{If}a=1+i,\text{then}{a}^2\text{equals}$

1. 2i

2. i - 1

3. (1 + i) (1 - i)

4. 1 - i

Q. Let Let $Z_k(k=0,1,2,...............6)$ be the roots of the equation $(z+1{)}^7+z^7=0,$ then ${\sum }_{k=0}^{6}Re\left(zk\right)$ is

1. 
2. 3 + 2i
3. 3 - 2i
4. 0

Q.

Find the values of the following expressions :

$\left(i\right)i^{49}+i^{68}+i^{89}+i^{110}\left(ii\right)i^{30}+i^{80}+i^{120}\left(iii\right)i+i^2+i^3+i^4\left(iv\right)i^5+i^{10}+i^{15}\left(v\right)\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}\left(vi\right)1+i^2+i^4+i^6+i^8+...+i^{20}$

Q.

If the extremities of the base of an isosceles triangle are the points $(2a,0)$ and $(0,a)$ and the equation of one of the sides is $x=2a$, then the area of the triangle is

1. $5a^2$

2. $\frac{5a^2}{2}$

3. $\frac{25a^2}{2}$

4. None of these

Q. If x= t^3/3 +6t-5t^2/2+ 1, then what is the value of d^2x/dt^2 when dx/dt is zero?

Q.

If the roots of $z^3+i{z}^2+2i$ = 0 represent the vertices of a triangle in the argand plane, then its area is

1. 2

2. 

3. 

4. 4

Q. For $A(1,-2,4),B(5,-1,7),C(3,6,-2),D(4,5,-1)$, the projection of $\overrightarrow{AB}$ on $\overrightarrow{CD}$ is

1. $(2\sqrt{3},-2\sqrt{3},2\sqrt{3})$
2. $\frac{3}{13}(4,1,3)$
3. $(1,-1,1)$
4. $(2,-2,2)$

Q. Match the statements in Column-I with those in Column-II.
[Note: Here $z$ takes values in the complex plane and $\text{Im}z$ and $\text{Re}z$ denote, respectively, the imaginary part and the real part of $z$.]
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{The set of points}z\text{satisfying}& \text{(p)}& \text{An ellipse with}\\ & |z-i|z||=|z+i|z||\text{is}& & \text{eccentricity}\frac{4}{5}\\ & \text{contained in or equal to}& & \\ \text{(B)}& \text{The set of points}z\text{satisfying}& \text{(q)}& \text{The set of points}z\\ & |z+4|+|z-4|=10\text{is}& & \text{satisfying Im}z=0\\ & \text{contained in or equal to}& & \\ \text{(C)}& \text{If}|w|=2,\text{then the set of}& \text{(r)}& \text{The set of points}z\\ & \text{points}z=w-\frac{1}{w}\text{is}& & \text{satisfying}|\text{Im}z|\\ & \text{contained in or equal to}& & \le 1\\ \text{(D)}& \text{If}|w|=1,\text{then the set of}& \text{(r)}& \text{The set of points}z\\ & \text{points}z=w-\frac{1}{w}\text{is}& & \text{satisfying}|\text{Re}z|\\ & \text{contained in or equal to}& & \le 2\\ & & \text{(t)}& \text{The set of points}z\\ & & & \text{satisfying}|z|\le 3\end{array}$

1. Option c
2. A-Q, R; B-P; C-P, S, T; D-Q, R, S, T
3. Option d
4. Option b

Q. \begin{vmatrix}2y+4 &5y+7 &8y+a 3y+5 &6y+8 &9y+b 4y+6& 7y+9& 10y+c\end{vmatrix}evaluate the determinant and find the values of a, b, c if they are in A.P

Q.

Evaluate the following :

$\left(i\right)i^{457}\left(ii\right)i^{528}\left(iii\right)\frac{1}{i^{58}}\left(iv\right)i^{37}+\frac{1}{i^{47}}\left(v\right){(i^{41}+\frac{1}{i^{257}})}^9\left(vi\right)(i^{77}+i^{70}+i^{87}+i^{414}{)}^3(vii)i^{30}+i^{40}+i^{60}(viii)i^{49}+i^{68}+i^{89}+i^{110}$

Q.

Say true or false and justify your answer:$10\times {10}^{11}={100}^{11}$

1. True
2. False

Q.

If $x,y,z$ are real numbers satisfying the expression $25(9x^2+y^2)+9z^2-15(5xy+yz+3xz)=0$, then $x,y,z$ are in

1. AP

2. GP

3. HP

4. AGP

Q. Let $P$ and $T$ be the subsets of $X-Y$ plane defined by
$P=\left\{\right(x,y):x>0,y>0\text{and}{x}^2+y^2=1\}$
$T=\left\{\right(x,y):x>0,y>0\text{and}{x}^8+y^8<1\}$
Then $P\cap T$ is

1. the void set $\Phi$
2. $P-T^C$
3. $P$
4. $T$

Q. A point P lies inside the circles $x^2+y^2-4=0$ and $x^2+y^2-8x+7=0$. The point P starts moving under the conditions that its path encloses greatest possible area and it is at a fixed distance from any arbitrarily chosen fixed point in its region. The locus of P is

Q. The value of $(1-i{)}^4$ is

1. $4$
2. $-4$
3. $4i$
4. $-4i$

Q.

If $-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$, then the value of $\log \left(secx\right)$ is

1. $2cot{h}^{-1}\left(\cos e{c}^2\left(\left(\frac{x}{2}\right)-1\right)\right)$

2. $2\mathrm{co}t{h}^{-1}\left(\cos e{c}^2\left(\left(\frac{x}{2}\right)-1\right)\right)$

3. $2\cos ec{h}^{-1}\left(\mathrm{co}{t}^2\left(\left(\frac{x}{2}\right)-1\right)\right)$

4. $2\cos ec{h}^{-1}\left(\mathrm{co}{t}^2\left(\left(\frac{x}{2}\right)+1\right)\right)$

Q. If $z=x+iy$ is a complex number, then the curve represented by $Im\left(z^2\right)=c^2$, where $c$ is non zero real constant, is a

1. circle
2. straight line
3. ellipse
4. rectangular hyperbola