Complex Numbers & Quadratic Equations · JEE Main previous year questions

The sum of all the real solutions of the equation ${\log }_{(x+3)}\left(6x^2+28x+30\right)=5-2{\log }_{(6x+10)}\left(x^2+6x+9\right)$ is equal to :

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

The system of linear equations $$x+y+z=6$$ $$2x+5y+az=36$$ $$x+2y+3z=b$$ has :

1. A.unique solution for $a=8$ and $b=16$
2. B.infinitely many solutions for $a=8$ and $b=14$
3. C.infinitely many solutions for $a=8$ and $b=16$
4. D.unique solution for $a=8$ and $b=14$

The sum of all the roots of the equation $(x-1{)}^2-5|x-1|+6=0$, is :

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

Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2-2f^{\prime }(x)m+f^{\prime \prime }(x)=0$ in m , has two equal roots for every $x\in \mathbf{R}$. If $f(0)=1$, $f^{\prime }(0)=2$, and $(\alpha ,\beta )$ is the largest interval in which the function $f({\log }_{e}x-x)$ is increasing, then $\alpha +\beta$ is equal to ____ .

If the system of equations $3x+y+4z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$ has no solution, then the value of $\alpha$ is equal to:

1. A.13
2. B.4
3. C.19
4. D.23

Let $z$ be the complex number satisfying $|z-5|\le 3$ and having maximum positive principal argument. Then $34{\left|\frac{5z-12}{5iz+16}\right|}^2$ is equal to:

1. A.20
2. B.26
3. C.12
4. D.16

Let $A=\{x:|x^2-10|\le 6\}$ and $B=\{x:|x-2|>1\}$. Then

1. A.$A\cup B=(-\infty ,1]\cup (2,\infty )$
2. B.$B-A=(-\infty ,-4)\cup (-2,1)\cup (4,\infty )$
3. C.$A-B=[2,3)$
4. D.$A\cap B=[-4,-2]\cup [3,4]$

Let $\alpha$ be a solution of $x^2+x+1=0$, and for some $a$ and $b$ in $\mathbb{R}$, $\left[\begin{array}{ccc}4& a& b\end{array}\right]\left[\begin{array}{ccc}1& 16& 13\\ -1& -1& 2\\ -2& -14& -8\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$. If $\frac{4}{{\alpha }^4}+\frac{m}{{\alpha }^a}+\frac{n}{{\alpha }^b}=3$, then $m+n$ is equal to $\underline{}$

1. A.$11$
2. B.$3$
3. C.$8$
4. D.$7$

Let $A=\{z\in \mathrm{C}:|z-2-i|=3\},B=\{z\in \mathrm{C}:\mathrm{Re}(z-iz)=2\}$ and $S=A\cap B$. Then ${\sum }_{z\in S}|z{|}^2$ is equal to ____.

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3x-1=0$. If $P_n={\alpha }^n+{\beta }^n$ and $Q_n={\gamma }^n+{\delta }^n$, then $\frac{P_{25}+\sqrt{3}{P}_{24}}{2P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

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

Let $z$ be a complex number such that $|z|=1$. If $\frac{2+k^{2}z}{k+\bar{z}}=kz$, $k\in \mathbf{R}$, then the maximum distance of $k+i{k}^2$ from the circle $|z-(1+2i)|=1$ is :

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

Let $P_n={\alpha }^n+{\beta }^n,n\in \mathbf{N}$. If $P_{10}=123,P_9=76,P_8=47$ and $P_1=1$, then the quadratic equation having roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is :

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

If the system of linear equations $$3x+y+\beta z=3$$ $$2x+\alpha y-z=-3$$ $$x+2y+z=4$$ has infinitely many solutions, then the value of $22\beta -9\alpha$ is :

1. A.31
2. B.37
3. C.43
4. D.49

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

1. A.30
2. B.26
3. C.22
4. D.18

Let $\alpha ,\beta (\alpha \ne \beta )$ be the values of $m$, for which the equations $x+y+z=1$, $x+2y+4z=m$ and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of ${\sum }_{n=1}^{10}(n^{\alpha }+n^{\beta })$ is equal to :

1. A.$3410$
2. B.$560$
3. C.$3080$
4. D.$440$

The sum, of the squares of all the roots of the equation $x^2+|2x-3|-4=0$, is

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

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2\sqrt{2}i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi }{6}$. Then

1. A.area of triangle ABO is $\frac{11}{4}$
2. B.area of triangle ABO is $\frac{11}{\sqrt{3}}$
3. C.ABO is a scalene triangle
4. D.ABO is an obtuse angled isosceles triangle

The area (in sq. units) of the region $\left\{(x,y):0\le y\le 2|x|+1,0\le y\le x^2+1,|x|\le 3\right\}$ is

1. A.$\frac{32}{3}$
2. B.$\frac{64}{3}$
3. C.$\frac{17}{3}$
4. D.$\frac{80}{3}$

The product of all the rational roots of the equation ${\left(x^2-9x+11\right)}^2-(x-4)(x-5)=3$, is equal to

1. A.$7$
2. B.$21$
3. C.$28$
4. D.$14$

If $\alpha$ and $\beta$ are the roots of the equation $2z^2-3z-2i=0$, where $i=\sqrt{-1}$, then $16\cdot \mathrm{Re}\left(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}}\right)\cdot \mathrm{Im}\left(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}}\right)$ is equal to

1. A.$441$
2. B.$312$
3. C.$409$
4. D.$398$

The area of the region $\left\{(x,y):x^2+4x+2\le y\le |x+2|\right\}$ is equal to

1. A.$7$
2. B.$\frac{24}{5}$
3. C.$\frac{20}{3}$
4. D.$5$

If the system of equations $2x-y+z=4$ $5x+\lambda y+3z=12$ $100x-47y+\mu z=212$ has infinitely many solutions, then $\mu -2\lambda$ is equal to

1. A.$56$
2. B.$59$
3. C.$57$
4. D.$55$

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

1. A.8
2. B.10
3. C.4
4. D.6

The system of equations $x+y+z=6$, $x+2y+5z=9$, $x+5y+\lambda z=\mu$, has no solution if

1. A.$\lambda =17,\mu =18$
2. B.$\lambda =17,\mu \ne 18$
3. C.$\lambda =15,\mu \ne 17$
4. D.$\lambda \ne 17,\mu \ne 18$

Let $\alpha ,\beta$ be the roots of the equation $x^2-\mathrm{a}x-\mathrm{b}=0$ with $\mathrm{Im}(\alpha )<\mathrm{Im}(\beta )$. Let ${\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}-{\beta }^{\mathrm{n}}$. If ${\mathrm{P}}_3=-5\sqrt{7}i,{\mathrm{P}}_4=-3\sqrt{7}i,{\mathrm{P}}_5=11\sqrt{7}i$ and ${\mathrm{P}}_6=45\sqrt{7}i$, then $\left|{\alpha }^4+{\beta }^4\right|$ is equal to _________.