Complex Numbers & Quadratic Equations · JEE Main previous year questions

The number of distinct real roots of $x^4-4x+1=0$ is :

1. A.4
2. B.2
3. C.1
4. D.0

If the sum of all the roots of the equation $$e^{2x}-11e^x-45e^{-x}+\frac{81}{2}=0$$ is ${\log }_{e}p$, then $p$ is equal to ____.

The number of points of intersection of $|z-(4+3i)|=2$ and $|z|+|z-4|=6$, $z\in \mathrm{C}$, is :

1. A.0
2. B.1
3. C.2
4. D.3

Let $\alpha ,\beta$ be the roots of the equation $x^2-4\lambda x+5=0$ and $\alpha ,\gamma$ be the roots of the equation $x^2-\left(3\sqrt{2}+2\sqrt{3}\right)x+7+3\lambda \sqrt{3}=0$, $\lambda >0$. If $\beta +\gamma =3\sqrt{2}$, then $(\alpha +2\beta +\gamma {)}^2$ is equal to ________.

The ordered pair (a, b), for which the system of linear equations $3x-2y+z=b$ $5x-8y+9z=3$ $2x+y+az=-1$ has no solution, is :

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

The area bounded by the curve $y=|x^2-9|$ and the line $y=3$ is :

1. A.$4(2\sqrt{3}+\sqrt{6}-4)$
2. B.$4(4\sqrt{3}+\sqrt{6}-4)$
3. C.$8(4\sqrt{3}+3\sqrt{6}-9)$
4. D.$8(4\sqrt{3}+2\sqrt{6}-9)$

The sum of the cubes of all the roots of the equation $$x^4-3x^3-2x^2+3x+1=0$$ is ____.

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

1. A.$(1,-3)$
2. B.$(-1,3)$
3. C.$(1,3)$
4. D.$(-1,-3)$

Let $X=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$, $Y=\alpha I+\beta X+\gamma X^2$ and $Z={\alpha }^{2}I-\alpha \beta X+({\beta }^2-\alpha \gamma )X^2$, $\alpha ,\beta ,\gamma \in \mathrm{R}$. If $Y^{-1}=\left[\begin{array}{ccc}\frac{1}{5}& \frac{-2}{5}& \frac{1}{5}\\ 0& \frac{1}{5}& \frac{-2}{5}\\ 0& 0& \frac{1}{5}\end{array}\right]$, then $(\alpha -\beta +\gamma {)}^2$ is equal to ________.

Let a circle $\mathrm{C}$ in complex plane pass through the points $z_1=3+4i$, $z_2=4+3i$ and $z_3=5i$. If $z(\ne z_1)$ is a point on $\mathrm{C}$ such that the line through $z$ and $z_1$ is perpendicular to the line through $z_2$ and $z_3$, then $\arg (z)$ is equal to :

1. A.${\tan }^{-1}\left(\frac{2}{\sqrt{5}}\right)-\pi$
2. B.${\tan }^{-1}\left(\frac{24}{7}\right)-\pi$
3. C.${\tan }^{-1}(3)-\pi$
4. D.${\tan }^{-1}\left(\frac{3}{4}\right)-\pi$

Let $a,b\in \mathbb{R}$ be such that the equation $a{x}^2-2bx+15=0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation $x^2-2bx+21=0$, then ${\alpha }^2+{\beta }^2$ is equal to :

1. A.$37$
2. B.$58$
3. C.$68$
4. D.$92$

Let $z_1$ and $z_2$ be two complex numbers such that ${\bar{z}}_1=i{\bar{z}}_2$ and $\arg \left(\frac{z_1}{z_2}\right)=\pi$. Then :

1. A.$\arg z_2=\frac{\pi }{4}$
2. B.$\arg z_2=-\frac{3\pi }{4}$
3. C.$\arg z_1=\frac{\pi }{4}$
4. D.$\arg z_1=-\frac{3\pi }{4}$

Let $x\ast y=x^2+y^3$ and $(x\ast 1)\ast 1=x\ast (1\ast 1)$. Then a value of $2{\sin }^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)$ is :

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{2}$
4. D.$\frac{\pi }{6}$

The sum of all the real roots of the equation $(e^{2x}-4)(6e^{2x}-5e^x+1)=0$ is

1. A.${\log }_{e}3$
2. B.$-{\log }_{e}3$
3. C.${\log }_{e}6$
4. D.$-{\log }_{e}6$

Let the system of linear equations $x+y+\alpha z=2$ $3x+y+z=4$ $x+2z=1$ have a unique solution $(x^{\ast },y^{\ast },z^{\ast })$. If $(\alpha ,x^{\ast })$, $(y^{\ast },\alpha )$ and $(x^{\ast },-y^{\ast })$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

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

The number of distinct real roots of the equation $x^7-7x-2=0$ is

1. A.5
2. B.7
3. C.1
4. D.3

Let $S=\{z\in \mathrm{C}:|z-3|\le 1\text{ and }z(4+3i)+\overline{z}(4-3i)\le 24\}$. If $\alpha +i\beta$ is the point in S which is closest to $4i$, then $25(\alpha +\beta )$ is equal to ____.

The number of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation $x^2+ax+b=0$, ${\alpha }^2-2$ is also a root of this equation, is :

1. A.6
2. B.2
3. C.4
4. D.8

If for the complex numbers z satisfying $|z-2-2i|\le 1$, the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to ___________.

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is ___________.

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

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

Let $\lambda \ne 0$ be in $\mathbb{R}$. If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+2\lambda =0$, and $\alpha$ and $\gamma$ are the roots of equation $3x^2-10x+27\lambda =0$, then $\frac{\beta \gamma }{\lambda }$ is equal to ________.

Let C be the set of all complex numbers. Let $S_1=\{z\in C:|z-2|\le 1\}$ and $S_2=\{z\in C:z(1+i)+\bar{z}(1-i)\ge 4\}$. Then, the maximum value of ${\left|z-\frac{5}{2}\right|}^2$ for $z\in S_1\cap S_2$ is equal to :

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

Let $\alpha =\underset{x\in R}{\max }\{8^{2\sin 3x}.4^{4\cos 3x}\}$ and $\beta =\underset{x\in R}{\min }\{8^{2\sin 3x}.4^{4\cos 3x}\}$. If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ is equal to :

1. A.42
2. B.47
3. C.43
4. D.50

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