# JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank

### done Conjugate, Modulus and Argument of complex number

• A) $x=n\pi$

B) $x=\left( n+\frac{1}{2} \right)\pi$

C) $x=0$

D) No value of x

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• A) 1

B) -1

C) 0

D) None of these

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• A) $z$is purely real

B) $z$is purely imaginary

C) Either $z$is purely real or purely imaginary

D) None of these

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• A) $z=0$

B) $\operatorname{Re}(z)=0$

C) $\operatorname{Im}\,(z)=0$

D) None of these

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• A) ${{A}^{2}}+{{B}^{2}}$

B) ${{A}^{2}}-{{B}^{2}}$

C) ${{A}^{2}}$

D) ${{B}^{2}}$

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• A) 1

B) 2

C) 3

D) 4

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• A) A real number

B) A imaginary number

C) Both are real numbers

D) Both are imaginary numbers

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• A) $(-2,-1)$or $(2,-1)$

B) $(-1,\text{ }2)$or $(-2,\text{ }1)$

C) $(1,\,2)$or $(-1,-2)$

D) None of these

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• A) $\frac{7-26i}{25}$

B) $\frac{-7-26i}{25}$

C) $\frac{-7+26i}{25}$

D) $\frac{7+26i}{25}$

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• A) $|z-a|$

B) ${{z}^{2}}+{{a}^{2}}$

C) $|z+a{{|}^{2}}$

D) None of these

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• A) 0

B) 1

C) 2

D) None of these

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• A) 1

B) $-1$

C) ${{c}^{2}}$

D) $-{{c}^{2}}$

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• A) $x=\frac{1}{5}$

B) $y=\frac{3}{5}$

C) $x+iy=\frac{1-i}{1-2i}$

D) $x-iy=\frac{1-i}{1+2i}$

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• A) $\frac{13}{2}+i\,\left( \frac{15}{2} \right)$

B) $\frac{13}{10}+i\left( \frac{-15}{2} \right)$

C) $\frac{13}{10}+i\,\left( \frac{-9}{10} \right)$

D) $\frac{13}{10}+i\,\left( \frac{9}{10} \right)$

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• A) $-3-5i$

B) $-3+5i$

C) $3+5i$

D) $3-5i$

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• A) $\frac{3i}{4}$

B) $\frac{11+10i}{17}$

C) $\frac{11-10i}{17}$

D) $\frac{2+3i}{4i}$

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• A) i

B) 1

C) 1 - i

D) 1 + i

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• A) $\operatorname{Re}(z)>0$

B) $\operatorname{Re}(z)<0$

C) $\operatorname{Re}(z)>2$

D) None of these

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• A) 3/2

B) 1

C) 2/3

D) 4/9

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• A) $2|{{z}_{1}}{{|}^{2}}\,|{{z}_{2}}{{|}^{2}}$

B) $2|{{z}_{1}}{{|}^{2}}+\,2\,\,|{{z}_{2}}{{|}^{2}}$

C) $|{{z}_{1}}{{|}^{2}}+\,|{{z}_{2}}{{|}^{2}}$

D) $2|{{z}_{1}}|\,\,|{{z}_{2}}|$

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• A) $|z|\,=0$

B) $|z|\,=1$

C) $|z|\,>1$

D) $|z|\,<1$

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• A) $|{{z}^{2}}|\,=\,|z{{|}^{2}}$

B) $|{{z}^{2}}|\,=\,|\bar{z}{{|}^{2}}$

C) $z=\bar{z}$

D) ${{\bar{z}}^{2}}={{\bar{z}}^{2}}$

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• A) $\sqrt{3}-1$

B) $\sqrt{3}+1$

C) $\sqrt{3}$

D) $\sqrt{2}+\sqrt{3}$

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• A) Positive real

B) Negative real

C) Zero or purely imaginary

D) None of these

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• A) $2-\frac{3}{2}i$

B) $\frac{3}{2}+2i$

C) $\frac{3}{2}-2i$

D) $-2+\frac{3}{2}i$

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• A) Purely imaginary

B) Real and positive

C) Real and negative

D) None of these

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• A) Less than unity

B) Greater than unity

C) Equal to unity

D) Any

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• A) Unit modulus

B) Less than unit modulus

C) Greater than unit modulus

D) None of these

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• A) $|z|\,<1$

B) $|z|\,=1$

C) $|z|\,>1$

D) None of these

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• A) 1

B) $|{{z}_{1}}|+|{{z}_{2}}|+.......+|{{z}_{n}}|$

C) $\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+.........+\frac{1}{{{z}_{n}}} \right|$

D) None of these

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• A) $z$ is a pure real number

B) $|z|=1$

C) $z$is a pure imaginary number

D) $z=1$

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• A) $\ge \,|{{z}_{1}}|-|{{z}_{2}}|$

B) $\le \,|{{z}_{1}}|-|{{z}_{2}}|$

C) $\ge \,|{{z}_{1}}|+|{{z}_{2}}|$

D) $\le \,|{{z}_{2}}|-|{{z}_{1}}|$

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• A) Any real number

B) Any complex number

C) Any natural number

D) None of these

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• A) $\sqrt{{{(x-5)}^{2}}+{{y}^{2}}}$

B) ${{x}^{2}}+\sqrt{{{(y-5)}^{2}}}$

C) $\sqrt{{{(x-y)}^{2}}+{{5}^{2}}}$

D) $\sqrt{{{x}^{2}}+{{(y-5)}^{2}}}$

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• A) $-\frac{1}{2}$

B) $\frac{1}{2}$

C) 1

D) $-1$

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• A) ${{\tan }^{-1}}\left( \frac{2k}{{{k}^{2}}+1} \right)$

B) ${{\tan }^{-1}}\left( \frac{2k}{1-{{k}^{2}}} \right)$

C) - $2{{\tan }^{-1}}k$

D) $2{{\tan }^{-1}}k$

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• A) $\operatorname{Im}(z)=0$

B) $\operatorname{Re}(z)=0$

C) $amp(z)=\pi$

D) None of these

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• A) $|{{z}_{1}}|$

B) $|{{z}_{2}}|$

C) $|{{z}_{1}}+{{z}_{2}}|$

D) $|{{z}_{1}}+{{z}_{2}}|+|{{z}_{1}}-{{z}_{2}}|$

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• A) $|{{z}_{1}}+{{z}_{2}}|$

B) $|{{z}_{1}}-{{z}_{2}}|$

C) $|{{z}_{1}}+{{z}_{2}}|$

D) $|{{z}_{1}}|-|{{z}_{2}}|$

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• A) 1

B) 1/2

C) 2

D) $\sqrt{2}$

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• A) Purely real

B) Purely imaginary

C) Zero

D) Undefined

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• A) 0

B) $1/2$

C) $1/3$

D) 2/3

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• A) $0$

B) $-\frac{1}{|z+1{{|}^{2}}}$

C) $\left| \frac{z}{z+1} \right|\,.\frac{1}{|z+1{{|}^{2}}}$

D) $\frac{\sqrt{2}}{|z+1{{|}^{2}}}$

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• A) ${{\alpha }^{2}}-{{\beta }^{2}}=-1$

B) ${{\alpha }^{2}}-{{\beta }^{2}}=1$

C) ${{\alpha }^{2}}+{{\beta }^{2}}=1$

D) ${{\alpha }^{2}}-{{\beta }^{2}}=2$

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• A) 0

B) 1

C) - 1

D) 2

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• A) $-\pi$

B) $-\frac{\pi }{2}$

C) $\frac{\pi }{2}$

D) 0

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• A) ${{\tan }^{-1}}\frac{5}{\sqrt{3}}$

B) ${{\tan }^{-1}}\left( -\,\frac{5}{\sqrt{3}} \right)$

C) ${{\tan }^{-1}}\frac{\sqrt{3}}{5}$

D) ${{\tan }^{-1}}\left( -\frac{\sqrt{3}}{5} \right)$

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• A) $\frac{-\pi }{2}$and 1

B) $\frac{\pi }{2}$and $\sqrt{2}$

C) 0 and $\sqrt{2}$

D) $\frac{\pi }{2}$and 1

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• A) $|z|\,=\,|\bar{z}|$

B) $z.\,\bar{z}=|\bar{z}{{|}^{2}}$

C) $\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$

D) $arg\,z=arg\,\bar{z}$

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• A) $2\sqrt{3}-2i$

B) $2\sqrt{3}+2i$

C) $-2\sqrt{3}+2i$

D) $-\sqrt{3}+i$

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• A) ${{60}^{o}}$

B) ${{120}^{o}}$

C) ${{240}^{o}}$

D) ${{300}^{o}}$

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• A) $\theta$

B) $-\theta$

C) $\pi -\theta$

D) $\theta -\pi$

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• A) $2\sin \frac{\alpha }{2}$

B) $\frac{\alpha }{2}$

C) $\alpha$

D) None of these

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• A) $\frac{\pi }{3}$

B) $-\frac{\pi }{3}$

C) $\frac{\pi }{6}$

D) $-\frac{\pi }{6}$

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• A) $-{{60}^{o}}$

B) ${{60}^{o}}$

C) ${{120}^{o}}$

D) $-{{120}^{o}}$

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• A) $\frac{\pi }{2}$

B) $-\frac{\pi }{2}$

C) 0

D) $\frac{\pi }{4}$

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• A) Multiple of $\pi$

B) Multiple of$\frac{\pi }{2}$

C) Greater than $\pi$

D) Less than $\pi$

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• A) $\pi$

B) $\frac{\pi }{2}$

C) 0

D) $-\frac{\pi }{2}$

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• A) $\pi$

B) $\frac{\pi }{2}$

C) 0

D) $-\frac{\pi }{2}$

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• A) $\pi$

B) $\frac{\pi }{2}$

C) 0

D) $-\frac{\pi }{2}$

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• A) $\pi$

B) 0

C) $-\frac{\pi }{2}$

D) None of these

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• A) $\operatorname{Re}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0$

B) $\operatorname{Im}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0$

C) $\operatorname{Re}({{z}_{1}}{{z}_{2}})=0$

D) $\operatorname{Im}({{z}_{1}}{{z}_{2}})=0$

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• A) $|{{z}_{1}}|+|{{z}_{2}}|$

B) $|{{z}_{1}}|-|{{z}_{2}}|$

C) $||{{z}_{1}}|-|{{z}_{2}}||$

D) 0

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• A) $\frac{\pi }{4}$

B) $\frac{\pi }{3}$

C) $\frac{\pi }{2}$

D) 0

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• A) 0

B) Purely imaginary

C) Purely real

D) None of these

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• A) 0

B) $2\,amp\text{ }(z)$

C) $\pi$

D) $-\pi$

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• A) $\frac{\alpha }{2}$

B) $-\frac{\alpha }{2}$

C) $\frac{\pi }{2}+\frac{\alpha }{2}$

D) $\frac{\pi }{2}-\frac{\alpha }{2}$

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• A) $amp\,({{z}_{1}}{{\overline{z}}_{2}})$

B) $amp\,({{\overline{z}}_{1}}{{z}_{2}})$

C) $amp\,\left( \frac{{{z}_{2}}}{{{{\bar{z}}}_{1}}} \right)$

D) $amp\,\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)$

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• A) $\frac{\pi }{3}$

B) $\frac{\pi }{4}$

C) $\frac{\pi }{5}$

D) $\frac{\pi }{6}$

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• A) ${{z}_{1}}={{z}_{2}}$

B) ${{\bar{z}}_{1}}={{z}_{2}}$

C) ${{z}_{1}}+{{z}_{2}}=0$

D) ${{\bar{z}}_{1}}={{\bar{z}}_{2}}$

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• A) ${{z}_{2}}={{\overline{z}}_{1}}$

B) ${{z}_{2}}=\frac{1}{{{z}_{1}}}$

C) $arg\,({{z}_{1}})=$arg $({{z}_{2}})$

D) $|{{z}_{1}}|\,=\,|{{z}_{2}}|$

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• A) -p/2

B) p/2

C) p/4

D) p/6

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• A) $|{{z}_{1}}{{z}_{2}}|\,=\,|{{z}_{1}}||{{z}_{2}}|$

B) $arg\,\,({{z}_{1}}{{z}_{2}})=(arg\,{{z}_{1}})(arg\,{{z}_{2}})$

C) $|{{z}_{1}}+{{z}_{2}}|\,=\,|{{z}_{1}}|+|{{z}_{2}}|$

D) $|{{z}_{1}}-{{z}_{2}}|\,\ge \,|{{z}_{1}}|-|{{z}_{2}}|$

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• A) $\frac{\pi }{6}$

B) $-\frac{\pi }{6}$

C) $\frac{\pi }{3}$

D) None of these

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• A) 0

B) $\pi /2$

C) $\pi$

D) None of these

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• A) $\pi$

B) $-\pi$

C) $-\frac{\pi }{2}$

D) $\frac{\pi }{2}$

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• A) 0

B) $\pi /6$

C) $\pi /3$

D) $\pi /2$

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• A) $\pi$

B) $\pi /3$

C) $2\pi /3$

D) $\pi /4$

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• A) $|z|\,=1,\,\,\,\,arg\,z=\frac{\pi }{4}$

B) $|z|\,=1,arg\,z=\frac{\pi }{6}$

C) $|z|\,=\frac{\sqrt{3}}{2},\,arg\,z=\frac{5\pi }{24}$

D) $|z|\,=\frac{\sqrt{3}}{2},\,\,arg\,z={{\tan }^{-1}}\frac{1}{\sqrt{2}}$

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• A) $\pi /5$

B) $2\pi /5$

C) $\pi /10$

D) $\pi /15$

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• A) $\frac{2\pi }{3}$

B) $\frac{\pi }{3}$

C) $-\frac{\pi }{3}$

D) $-\frac{2\pi }{3}$

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• A) 1

B) - 1

C) i

D) - i

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• A) $\bar{z}$

B) $-\overline{z}$

C) z

D) $-z$

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• A) $\sqrt{2}\text{ and }\frac{\pi }{6}$

B) 1 and 0

C) 1 and $\frac{\pi }{3}$

D) 1 and $\frac{\pi }{4}$

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• A) $\frac{-31}{17}$

B) $\frac{17}{22}$

C) $\frac{-17}{31}$

D) $\frac{22}{17}$

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• A) $x\,(3-i),\,x\in R$

B) $\frac{x}{3+i},\,x\in R$

C) $x(3+i),\,x\in R$

D) $x(-3+i),\,x\in R$

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• A) 3

B) 8

C) 9

D) $\sqrt{8}$

E) 4

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