Complex Numbers & Quadratic Equations · JEE Main previous year questions

The number of real roots of the equation $e^{4x}-e^{3x}-4e^{2x}-e^x+1=0$ is equal to ____.

The number of real roots of the equation $e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^x+1=0$ is :

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

If $\alpha ,\beta$ are roots of the equation $x^2+5(\sqrt{2})x+10=0,\alpha >\beta$ and $P_n={\alpha }^n-{\beta }^n$ for each positive integer n, then the value of $\left(\frac{P_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}\right)$ is equal to ____.

The number of real solutions of the equation, $x^2-|x|-12=0$ is :

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

The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6$, $3x+5y+5z=26$, $x+2y+\lambda z=\mu$ has no solution, are :

1. A.$\lambda =3,\mu =5$
2. B.$\lambda =3,\mu \ne 10$
3. C.$\lambda \ne 2,\mu =10$
4. D.$\lambda =2,\mu \ne 10$

Let n denote the number of solutions of the equation $z^2+3\bar{z}=0$, where z is a complex number. Then the value of ${\sum }_{k=0}^{\infty }\frac{1}{n^k}$ is equal to :

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

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+(3{)}^{1/4}x+3^{1/2}=0$, then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to :

1. A.$56\times 3^{25}$
2. B.$56\times 3^{24}$
3. C.$52\times 3^{24}$
4. D.$28\times 3^{25}$

If z and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\arg (z)-\arg (\omega )=\frac{3\pi }{2}$, then $\arg \left(\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega }\right)$ is : (Here $\arg (z)$ denotes the principal argument of complex number z)

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

Let $\alpha ,\beta ,\gamma$ be the real roots of the equation, $x^3+a{x}^2+bx+c=0$, $(a,b,c\in \mathrm{R}\text{ and }a,b\ne 0)$. If the system of equations (in $u,v,w$) given by $\alpha u+\beta v+\gamma w=0$; $\beta u+\gamma v+\alpha w=0$; $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the value of $\frac{a^2}{b}$ is

1. A.5
2. B.3
3. C.1
4. D.0

Let $(1+x+2x^2{)}^{20}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{40}{x}^{40}$. Then $a_1+a_3+a_5+\cdots +a_{37}$ is equal to

1. A.$2^{20}(2^{20}-21)$
2. B.$2^{19}(2^{20}-21)$
3. C.$2^{19}(2^{20}+21)$
4. D.$2^{20}(2^{20}+21)$

Let $z_1,z_2$ be the roots of the equation $z^2+az+12=0$ and $z_1,z_2$ form an equilateral triangle with origin. Then, the value of $|a|$ is :

Let the system of linear equations $4x+\lambda y+2z=0$ $2x-y+z=0$ $\mu x+2y+3z=0,\lambda ,\mu \in \mathrm{R}$. has a non-trivial solution. Then which of the following is true?

1. A.$\mu =6,\lambda \in \mathrm{R}$
2. B.$\lambda =3,\mu \in \mathrm{R}$
3. C.$\mu =-6,\lambda \in \mathrm{R}$
4. D.$\lambda =2,\mu \in \mathrm{R}$

Let a complex number be $w=1-\sqrt{3}i$. Let another complex number z be such that $|zw|=1$ and $\arg (z)-\arg (w)=\frac{\pi }{2}$. Then the area of the triangle with vertices origin, z and w is equal to :

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

If f(x) and g(x) are two polynomials such that the polynomial $\mathrm{P}(x)=f(x^3)+xg(x^3)$ is divisible by $x^2+x+1$, then P(1) is equal to ____.

Let a complex number z, $|z|\ne 1$, satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1{)}^2}\right)\le 2$. Then, the largest value of $|z|$ is equal to _________.

1. A.5
2. B.8
3. C.6
4. D.7

Let z and $\omega$ be two complex numbers such that $\omega =z\bar{z}-2z+2$, $\left|\frac{z+i}{z-3i}\right|=1$ and $\mathrm{Re}(\omega )$ has minimum value. Then, the minimum value of $n\in \mathbb{N}$ for which ${\omega }^n$ is real, is equal to _________.

The sum of $162^{\text{th}}$ power of the roots of the equation $x^3-2x^2+2x-1=0$ is _________.

A natural number has prime factorization given by $n=2^x{3}^y{5}^z$, where y and z are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6}$, $y>z$. Then the number of odd divisors of n, including 1, is :

1. A.11
2. B.6
3. C.12
4. D.6x

Let z be those complex numbers which satisfy $|z+5|\le 4$ and $z(1+i)+\bar{z}(1-i)\ge -10,i=\sqrt{-1}$. If the maximum value of $|z+1{|}^2$ is $\alpha +\beta \sqrt{2}$, then the value of $(\alpha +\beta )$ is _________.

Let a be an integer such that all the real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $(a,a+1)$. Then, $|a|$ is equal to _________.

Let L be a common tangent line to the curves $4x^2+9y^2=36$ and $(2x{)}^2+(2y{)}^2=31$. Then the square of the slope of the line L is _________.

If the curve $x^2+2y^2=2$ intersects the line $x+y=1$ at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

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

The region represented by $\{z=x+iy\in \mathrm{C}:|z|-\mathrm{Re}(z)\le 1\}$ is also given by the inequality :

1. A.$y^2\le 2\left(x+\frac{1}{2}\right)$
2. B.$y^2\le x+\frac{1}{2}$
3. C.$y^2\ge 2(x+1)$
4. D.$y^2\ge x+1$

The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$ $x+2y+3z=5$ $x+3y+\lambda z=\mu$ has infinitely many solutions are, respectively:

1. A.6 and 8
2. B.5 and 8
3. C.5 and 7
4. D.4 and 9

If $\alpha$ and $\beta$ be two roots of the equation $x^2-64x+256=0$. Then the value of ${\left(\frac{{\alpha }^3}{{\beta }^5}\right)}^{1/8}+{\left(\frac{{\beta }^3}{{\alpha }^5}\right)}^{1/8}$ is :

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