Complex Numbers & Quadratic Equations · JEE Main previous year questions

Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg (z_1)=\frac{-\pi }{4}$, $\arg (z_2)=0$ and $\arg (z_3)=\frac{\pi }{4}$. If $|z_1{\bar{z}}_2+z_2{\bar{z}}_3+z_3{\bar{z}}_1{|}^2=\alpha +\beta \sqrt{2}$, $\alpha ,\beta \in Z$, then the value of ${\alpha }^2+{\beta }^2$ is :

1. A.41
2. B.29
3. C.24
4. D.31

Let $\alpha ,\beta$ be the roots of the equation $x^2+2\sqrt{2}x-1=0$. The quadratic equation, whose roots are ${\alpha }^4+{\beta }^4$ and $\frac{1}{10}({\alpha }^6+{\beta }^6)$, is:

1. A.$x^2-180x+9506=0$
2. B.$x^2-195x+9506=0$
3. C.$x^2-190x+9466=0$
4. D.$x^2-195x+9466=0$

The sum of the square of the modulus of the elements in the set $\{z=a+\mathrm{i}b:a,b\in \mathbf{Z},z\in \mathbf{C},|z-1|\le 1,|z-5|\le |z-5\mathrm{i}|\}$ is ______.

Let $z$ be a complex number such that the real part of $\frac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z-(6+8i)|$ is equal to

1. A.$8$
2. B.$12$
3. C.$10$
4. D.$\infty$

Let $\alpha ,\beta ;\alpha >\beta$, be the roots of the equation $x^2-\sqrt{2}x-\sqrt{3}=0$. Let ${\mathrm{P}}_n={\alpha }^n-{\beta }^n,n\in \mathbb{N}$. Then $(11\sqrt{3}-10\sqrt{2}){\mathrm{P}}_{10}+(11\sqrt{2}+10){\mathrm{P}}_{11}-11{\mathrm{P}}_{12}$ is equal to

1. A.$10\sqrt{3}{\mathrm{P}}_9$
2. B.$11\sqrt{3}{\mathrm{P}}_9$
3. C.$11\sqrt{2}{\mathrm{P}}_9$
4. D.$10\sqrt{2}{\mathrm{P}}_9$

Let $z$ be a complex number such that $|z+2|=1$ and $\text{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\text{Re}(\overline{z+2})|$ is

1. A.$\frac{2\sqrt{6}}{5}$
2. B.$\frac{24}{5}$
3. C.$\frac{\sqrt{6}}{5}$
4. D.$\frac{1+\sqrt{6}}{5}$

Let $\underset{\alpha }{\overset{{\log }_{e}4}{\int }}\frac{\mathrm{d}x}{\sqrt{e^x-1}}=\frac{\pi }{6}$. Then $e^{\alpha }$ and $e^{-\alpha }$ are the roots of the equation :

1. A.$2x^2-5x+2=0$
2. B.$x^2-2x-8=0$
3. C.$2x^2-5x-2=0$
4. D.$x^2+2x-8=0$

If the system of equations $x+4y-z=\lambda ,7x+9y+\mu z=-3,5x+y+2z=-1$ has infinitely many solutions, then $(2\mu +3\lambda )$ is equal to :

1. A.$-2$
2. B.2
3. C.3
4. D.$-3$

The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$, is ____

If $z_1,z_2$ are two distinct complex numbers such that $\left|\frac{z_1-2z_2}{\frac{1}{2}-z_1{\bar{z}}_2}\right|=2$, then

1. A.either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius 1.
2. B.$z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius 1.
3. C.either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac{1}{2}$.
4. D.both $z_1$ and $z_2$ lie on the same circle.

Let $\alpha ,\beta$ be roots of $x^2+\sqrt{2}x-8=0$. If ${\mathrm{U}}_n={\alpha }^n+{\beta }^n$, then $\frac{{\mathrm{U}}_{10}+\sqrt{2}{\mathrm{U}}_9}{2{\mathrm{U}}_8}$ is equal to ____.

The values of $m,n$, for which the system of equations $x+y+z=4$, $2x+5y+5z=17$, $x+2y+mz=n$ has infinitely many solutions, satisfy the equation :

1. A.${\text{m}}^2+{\text{n}}^2-\text{m}-\text{n}=46$
2. B.${\text{m}}^2+{\text{n}}^2+\text{mn}=68$
3. C.${\text{m}}^2+{\text{n}}^2-\text{mn}=39$
4. D.${\text{m}}^2+{\text{n}}^2+\text{m}+\text{n}=64$

The coefficients $\text{a},\text{b},\text{c}$ in the quadratic equation $a{x}^2+bx+c=0$ are from the set $\{1,2,3,4,5,6\}$. If the probability of this equation having one real root bigger than the other is p, then $216\text{p}$ equals :

1. A.38
2. B.76
3. C.57
4. D.19

If 2 and 6 are the roots of the equation $a{x}^2+bx+1=0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is :

1. A.$x^2+8x+12=0$
2. B.$2x^2+11x+12=0$
3. C.$4x^2+14x+12=0$
4. D.$x^2+10x+16=0$

The value of $\frac{1\times 2^2+2\times 3^2+\dots +100\times (101{)}^2}{1^2\times 2+2^2\times 3+\dots +100^2\times 101}$ is

1. A.$\frac{305}{301}$
2. B.$\frac{306}{305}$
3. C.$\frac{32}{31}$
4. D.$\frac{31}{30}$

The area (in sq. units) of the region $S=\{z\in \mathbb{C}:|z-1|\le 2;(z+\bar{z})+i(z-\bar{z})\le 2,\mathrm{Im}(z)\ge 0\}$ is

1. A.$\frac{7\pi }{4}$
2. B.$\frac{3\pi }{2}$
3. C.$\frac{7\pi }{3}$
4. D.$\frac{17\pi }{8}$

Let $S=\left\{{\sin }^{2}2\theta :({\sin }^4\theta +{\cos }^4\theta )x^2+(\sin 2\theta )x+({\sin }^6\theta +{\cos }^6\theta )=0\text{ has real roots}\right\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3\left((\alpha -2{)}^2+(\beta -1{)}^2\right)$ equals _________.

Let $\mathrm{S}=\{\mathrm{z}\in \mathrm{C}:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}\}$. Let $z_1,z_2\in \mathrm{S}$ be such that $\left|z_1\right|={\max }_{z\in S}|z|$ and $\left|z_2\right|={\min }_{z\in S}|z|$. Then ${\left|\sqrt{2}{z}_1-z_2\right|}^2$ equals :

1. A.1
2. B.4
3. C.3
4. D.2

If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha \beta$ is equal to :

1. A.1110
2. B.1120
3. C.1210
4. D.1220

If $5f(x)+4f\left(\frac{1}{x}\right)=x^2-2,\forall x\ne 0$ and $y=9x^{2}f(x)$, then $y$ is strictly increasing in :

1. A.$\left(0,\frac{1}{\sqrt{5}}\right)\cup \left(\frac{1}{\sqrt{5}},\infty \right)$
2. B.$\left(-\frac{1}{\sqrt{5}},0\right)\cup \left(\frac{1}{\sqrt{5}},\infty \right)$
3. C.$\left(-\frac{1}{\sqrt{5}},0\right)\cup \left(0,\frac{1}{\sqrt{5}}\right)$
4. D.$\left(-\infty ,\frac{1}{\sqrt{5}}\right)\cup \left(0,\frac{1}{\sqrt{5}}\right)$

If $z$ is a complex number such that $|z|⩽1$, then the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$ is :

1. A.2
2. B.$\frac{5}{2}$
3. C.$\frac{3}{2}$
4. D.3

Let the system of equations $x+2y+3z=5$, $2x+3y+z=9$, $4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda +2\mu$ is equal to :

1. A.22
2. B.17
3. C.15
4. D.28

If the system of linear equations $x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha +13\beta$ is equal to

1. A.60
2. B.54
3. C.64
4. D.58

If $f(x)=\left|\begin{array}{ccc}{x}^3& 2x^2+1& 1+3x\\ 3x^2+2& 2x& x^3+6\\ x^3-x& 4& x^2-2\end{array}\right|$ for all $x\in \mathbb{R}$, then $2f(0)+f^{\prime }(0)$ is equal to

1. A.24
2. B.18
3. C.42
4. D.48

If $\alpha$ denotes the number of solutions of $|1-i{|}^x=2^x$ and $\beta =\left(\frac{|z|}{\arg (z)}\right)$, where $z=\frac{\pi }{4}(1+i{)}^4\left[\frac{1-\sqrt{\pi }i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }i}\right],i=\sqrt{-1}$, then the distance of the point $(\alpha ,\beta )$ from the line $4x-3y=7$ is ____.