Complex Numbers & Quadratic Equations · JEE Main previous year questions

Let $\mathrm{Q}$ and $\mathrm{R}$ be the feet of perpendiculars from the point $\mathrm{P}(a,a,a)$ on the lines $x=y,z=1$ and $x=-y,z=-1$ respectively. If $\angle \mathrm{QPR}$ is a right angle, then $12a^2$ is equal to ____.

Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15i$ Then, $\left|z_1^4+z_2^4\right|$ equals -

1. A.$15\sqrt{15}$
2. B.$30\sqrt{3}$
3. C.$25\sqrt{3}$
4. D.75

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2x^2+54$ at points $(x,y)$ and $(-x,y)$, where $y>0$, is :

1. A.108
2. B.122
3. C.88
4. D.92

If $z=x+iy,xy\ne 0$, satisfies the equation $z^2+i\bar{z}=0$, then $|z^2|$ is equal to :

1. A.9
2. B.$\frac{1}{4}$
3. C.4
4. D.1

If $2{\sin }^{3}x+\sin 2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0,\frac{n\pi }{2}\right],n\in \mathbf{N}$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to :

1. A.$(0,\infty )$
2. B.$\mathbf{Z}$
3. C.$\left(-\frac{\sqrt{17}}{2},\frac{\sqrt{17}}{2}\right)$
4. D.$(-\infty ,0)$

Let $\alpha ,\beta \in \mathbf{N}$ be roots of the equation $x^2-70x+\lambda =0$, where $\frac{\lambda }{2},\frac{\lambda }{3}\notin \mathbf{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha -1}+\sqrt{\beta -1})(\lambda +35)}{|\alpha -\beta |}$ is equal to :

Let $\alpha ,\beta$ be the roots of the equation $x^2-x+2=0$ with $\text{Im}(\alpha )>\text{Im}(\beta )$. Then ${\alpha }^6+{\alpha }^4+{\beta }^4-5{\alpha }^2$ is equal to _______.

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\left(2\tan \frac{5\pi }{8}\right)$, then $(r,\theta )$ is equal to

1. A.$\left(2\sec \frac{11\pi }{8},\frac{11\pi }{8}\right)$
2. B.$\left(2\sec \frac{3\pi }{8},\frac{3\pi }{8}\right)$
3. C.$\left(2\sec \frac{5\pi }{8},\frac{3\pi }{8}\right)$
4. D.$\left(2\sec \frac{3\pi }{8},\frac{5\pi }{8}\right)$

Let $\alpha ,\beta$ be the roots of the equation $x^2-\sqrt{6}x+3=0$ such that $\mathrm{Im}(\alpha )>\mathrm{Im}(\beta )$. Let $a,b$ be integers not divisible by 3 and $n$ be a natural number such that $\frac{{\alpha }^{99}}{\beta }+{\alpha }^{98}=3^n(a+ib),i=\sqrt{-1}$. Then $n+a+b$ is equal to _________.

Let for any three distinct consecutive terms $a,b,c$ of an A.P, the lines $ax+by+c=0$ be concurrent at the point $P$ and $Q(\alpha ,\beta )$ be a point such that the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$ and $x+2y+3z=4$, has infinitely many solutions. Then $(PQ{)}^2$ is equal to _________.

If $S=\{z\in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is :

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

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha {)}^7=A+B\alpha +C{\alpha }^2$, $A,B,C\ge 0$, then $5(3A-2B-C)$ is equal to ________.

If $\alpha ,\beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023{\alpha }^n+2024{\beta }^n$, then :

1. A.$2S_{12}=S_{11}+S_{10}$
2. B.$S_{12}=S_{11}+S_{10}$
3. C.$S_{11}=S_{10}+S_{12}$
4. D.$2S_{11}=S_{12}+S_{10}$

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha }}$ lie on the circles $|z-z_0{|}^2=4$ and $|z-z_0{|}^2=16$ respectively, where $z_0=1+i$. Then, the value of $100|\alpha {|}^2$ is _______.

Let $\alpha ,\beta$ be the roots of the equation $x^2-\sqrt{2}x+2=0$. Then ${\alpha }^{14}+{\beta }^{14}$ is equal to

1. A.$-64$
2. B.$-64\sqrt{2}$
3. C.$-128\sqrt{2}$
4. D.$-128$

If the system of equations $$2x+y-z=5$$ $$2x-5y+\lambda z=\mu$$ $$x+2y-5z=7$$ has infinitely many solutions, then $(\lambda +\mu {)}^2+(\lambda -\mu {)}^2$ is equal to

1. A.$916$
2. B.$912$
3. C.$920$
4. D.$904$

Let $\alpha ,\beta$ be the roots of the quadratic equation $x^2+\sqrt{6}x+3=0$. Then $\frac{{\alpha }^{23}+{\beta }^{23}+{\alpha }^{14}+{\beta }^{14}}{{\alpha }^{15}+{\beta }^{15}+{\alpha }^{10}+{\beta }^{10}}$ is equal to :

1. A.$72$
2. B.$9$
3. C.$729$
4. D.$81$

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to :

1. A.$-\pi +{\tan }^{-1}\frac{8}{9}$
2. B.$-\pi +{\tan }^{-1}\frac{33}{5}$
3. C.$\pi -{\tan }^{-1}\frac{8}{9}$
4. D.$\pi -{\tan }^{-1}\frac{33}{5}$

The number of triplets $(x,y,z)$, where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is :

1. A.136
2. B.80
3. C.92
4. D.114

If $a$ and $b$ are the roots of the equation $x^2-7x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to _________.

$\left|\begin{array}{ccc}x+1& x& x\\ x& x+\lambda & x\\ x& x& x+{\lambda }^2\end{array}\right|=\frac{9}{8}(103x+81)$, then $\lambda ,\frac{\lambda }{3}$ are the roots of the equation :

1. A.$4x^2+24x-27=0$
2. B.$4x^2-24x+27=0$
3. C.$4x^2-24x-27=0$
4. D.$4x^2+24x+27=0$

For the system of linear equations $$2x-y+3z=5$$ $$3x+2y-z=7$$ $$4x+5y+\alpha z=\beta ,$$ which of the following is NOT correct?

1. A.The system has infinitely many solutions for $\alpha =-6$ and $\beta =9$
2. B.The system has a unique solution for $\alpha \ne -5$ and $\beta =8$
3. C.The system is inconsistent for $\alpha =-5$ and $\beta =8$
4. D.The system has infinitely many solutions for $\alpha =-5$ and $\beta =9$

Let the complex number $z=x+iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$ is equal to :

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

Let A be the point $(1,2)$ and B be any point on the curve $x^2+y^2=16$. If the centre of the locus of the point P, which divides the line segment AB in the ratio $3:2$ is the point $C(\alpha ,\beta )$, then the length of the line segment AC is :

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

Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the x-intercept of the normal to the curve at the point $(\alpha ,\alpha +1)$ in the first quadrant is _________.