Complex Numbers & Quadratic Equations · JEE Main previous year questions

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}=$

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

Consider the system of linear equations; $$\begin{array}{rl}{x}_1+2x_2+x_3& =3\\ 2x_1+3x_2+x_3& =3\\ 3x_1+5x_2+2x_3& =1\end{array}$$ The system has :

1. A.exactly 3 solutions
2. B.a unique solution
3. C.no solution
4. D.infinite number of solutions

If the roots of the equation $b{x}^2+cx+a=0$ are imaginary, then for all real values of $x$, the expression $3b^2{x}^2+6bcx+2c^2$ is :

1. A.less than $4ab$
2. B.greater than $-4ab$
3. C.less than $-4ab$
4. D.greater than $4ab$

If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $|z|$ is equal to :

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

If $z^2+z+1=0$, where z is a complex number, then the value of $(z+\frac{1}{z}{)}^2+(z^2+\frac{1}{z^2}{)}^2+(z^3+\frac{1}{z^3}{)}^2+\cdots +(z^6+\frac{1}{z^6}{)}^2$ is :

1. A.18
2. B.54
3. C.6
4. D.12

If the roots of the quadratic equation $x^2+px+q=0$ are $\tan 30^{\circ }$ and $\tan 15^{\circ }$, respectively, then the value of $2+q-p$ is

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

All the values of $m$ for which both roots of the equation $x^2-2mx+m^2-1=0$ are greater than $-2$ but less than 4, lie in the interval

1. A.$-2<m<0$
2. B.$m>3$
3. C.$-1<m<3$
4. D.$1<m<4$

Angle between the tangents to the curve $y=x^2-5x+6$ at the points $(2,0)$ and $(3,0)$ is

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

If the cube roots of unity are 1, $\omega$, ${\omega }^2$ then the roots of the equation $(x-1{)}^3+8=0$, are :

1. A.$-1,-1+2\omega ,-1-2{\omega }^2$
2. B.$-1,-1,-1$
3. C.$-1,1-2\omega ,1-2{\omega }^2$
4. D.$-1,1+2\omega ,1+2{\omega }^2$

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then arg $z_1$ - arg $z_2$ is equal to :

1. A.$\frac{\pi }{2}$
2. B.$-\pi$
3. C.0
4. D.$\frac{-\pi }{2}$

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then $k$ lies in the interval

1. A.$(5,6]$
2. B.$(6,\infty )$
3. C.$(-\infty ,4)$
4. D.$[4,5]$

The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assumes the least value is

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

If the roots of the equation $x^2-bx+c=0$ be two consecutive integers, then $b^2-4c$ equals

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

If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$, then $4x^2-4xy\cos \alpha +y^2$ is equal to :

1. A.$2\sin 2\alpha$
2. B.$4$
3. C.$4{\sin }^2\alpha$
4. D.$-4{\sin }^2\alpha$

The system of equations $\alpha x+y+z=\alpha -1$ $x+\alpha y+z=\alpha -1$ $x+y+\alpha z=\alpha -1$ has no solutions, if $\alpha$ is :

1. A.$-2$
2. B.either $-2$ or $1$
3. C.not $-2$
4. D.$1$

Let z and w be complex numbers such that $\overline{z}+i\overline{w}=0$ and arg zw = $\pi$. Then arg z equals :

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

If $z=x-iy$ and $z^{\frac{1}{3}}=p+iq$, then $\frac{\left(\frac{x}{p}+\frac{y}{q}\right)}{(p^2+q^2)}$ is equal to :

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

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

1. A.$x^2-18x-16=0$
2. B.$x^2-18x+16=0$
3. C.$x^2+18x-16=0$
4. D.$x^2+18x+16=0$

If $(1-p)$ is a root of quadratic equation $x^2+px+(1-p)=0$ then its roots are

1. A.$-1,2$
2. B.$-1,1$
3. C.$0,-1$
4. D.$0,1$

If one root of the equation $x^2+px+12=0$ is 4, while the equation $x^2+px+q=0$ has equal roots, then the value of ${}^{\prime }{q}^{\prime }$ is

1. A.4
2. B.12
3. C.3
4. D.$\frac{49}{4}$

If the straight lines $x=1+s,y=-3-\lambda s,z=1+\lambda s$ and $x=\frac{t}{2},y=1+t,z=2-t$, with parameters $s$ and $t$ respectively, are co-planar, then $\lambda$ equals :

1. A.$0$
2. B.$-1$
3. C.$-\frac{1}{2}$
4. D.$-2$

If the sum of the slopes of the lines given by $x^2-2cxy-7y^2=0$ is four times their product, then $c$ has the value :

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

If one of the lines given by $6x^2-xy+4c{y}^2=0$ is $3x+4y=0$, then $c$ equals :

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

z and w are two nonzero complex numbers such that $|z|=|w|$ and Arg z + Arg w = $\pi$ then z equals

1. A.$\overline{\omega }$
2. B.$-\overline{\omega }$
3. C.$\omega$
4. D.$-\omega$

If $|z-4|<|z-2|$, its solution is given by :

1. A.$\mathrm{Re}(z)>0$
2. B.$\mathrm{Re}(z)<0$
3. C.$\mathrm{Re}(z)>3$
4. D.$\mathrm{Re}(z)>2$