Complex Numbers & Quadratic Equations · JEE Main previous year questions

Let p, q and r be real numbers $(p\ne q,r\ne 0)$, such that the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

1. A.$\frac{p^2+q^2}{2}$
2. B.$p^2+q^2$
3. C.$2(p^2+q^2)$
4. D.$p^2+q^2+r^2$

If an angle A of a $\Delta \text{ABC}$ satisfies $5\cos \text{A}+3=0$, then the roots of the quadratic equation, $9x^2+27x+20=0$ are :

1. A.$\sec \text{A},\cot \text{A}$
2. B.$\sin \text{A},\sec \text{A}$
3. C.$\sec \text{A},\tan \text{A}$
4. D.$\tan \text{A},\cos \text{A}$

If $x^2+y^2+\sin y=4$, then the value of $\frac{d^{2}y}{d{x}^2}$ at the point $(-2,0)$ is :

1. A.$-34$
2. B.$-32$
3. C.4
4. D.$-2$

If $\tan \text{A}$ and $\tan \text{B}$ are the roots of the quadratic equation, $3x^2-10x-25=0$, then the value of $3{\sin }^2(\text{A}+\text{B})-10\sin (\text{A}+\text{B}).\cos (\text{A}+\text{B})-25{\cos }^2(\text{A}+\text{B})$ is :

1. A.$-10$
2. B.10
3. C.$-25$
4. D.25

If $\alpha ,\beta \in C$ are the distinct roots of the equation $x^2-x+1=0$, then ${\alpha }^{101}+{\beta }^{107}$ is equal to :

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

Let $\mathrm{g}(\mathrm{x})=\cos {\mathrm{x}}^2$, $\mathrm{f}(\mathrm{x})=\sqrt{x}$ and $\alpha ,\beta (\alpha <\beta )$ be the roots of the quadratic equation $18x^2-9\pi x+{\pi }^2=0$. Then the area (in sq. units) bounded by the curve $\mathrm{y}=(\mathrm{g}\circ \mathrm{f})(\mathrm{x})$ and the lines $x=\alpha$, $x=\beta$ and $y=0$ is :

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

If the curves $y^2=6x$, $9x^2+b{y}^2=16$ intersect each other at right angles, then the value of b is :

1. A.$\frac{9}{2}$
2. B.$6$
3. C.$\frac{7}{2}$
4. D.$4$

If $2x=y^{\frac{1}{5}}+y^{-\frac{1}{5}}$ and $(x^2-1)\frac{d^{2}y}{d{x}^2}+\lambda x\frac{dy}{dx}+ky=0$, then $\lambda +k$ is equal to :

1. A.$-23$
2. B.$-24$
3. C.$26$
4. D.$-26$

Let $p(x)$ be a quadratic polynomial such that $p(0)=1$. If $p(x)$ leaves remainder $4$ when divided by $x-1$ and it leaves remainder $6$ when divided by $x+1$; then :

1. A.$p(2)=11$
2. B.$p(2)=19$
3. C.$p(-2)=19$
4. D.$p(-2)=11$

Let $z\in \mathbb{C}$, the set of complex numbers. Then the equation, $2|z+3i|-|z-i|=0$ represents :

1. A.a circle with radius $\frac{8}{3}$.
2. B.a circle with diameter $\frac{10}{3}$.
3. C.an ellipse with length of major axis $\frac{16}{3}$.
4. D.an ellipse with length of minor axis $\frac{16}{9}$.

The number of real values of $\lambda$ for which the system of linear equations $2x+4y-\lambda z=0$ $4x+\lambda y+2z=0$ $\lambda x+2y+2z=0$ has infinitely many solutions, is :

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

If $y={\left[x+\sqrt{x^2-1}\right]}^{15}+{\left[x-\sqrt{x^2-1}\right]}^{15}$, then $(x^2-1)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}$ is equal to :

1. A.$125y$
2. B.$124y^2$
3. C.$225y^2$
4. D.$225y$

The area (in sq. units) of the smaller portion enclosed between the curves, $x^2+y^2=4$ and $y^2=3x$, is :

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

The system of linear equations $x+\lambda y-z=0$ $\lambda x-y-z=0$ $x+y-\lambda z=0$ has a non-trivial solution for :

1. A.infinitely many values of $\lambda$.
2. B.exactly one value of $\lambda$.
3. C.exactly two values of $\lambda$.
4. D.exactly three values of $\lambda$.

A complex number $z$ is said to be unimodular if $|z|=1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1-2z_2}{2-z_1{\bar{z}}_2}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a :

1. A.circle of radius $2$.
2. B.circle of radius $\sqrt{2}$.
3. C.straight line parallel to x-axis
4. D.straight line parallel to y-axis.

Let $\alpha$ and $\beta$ be the roots of equation $x^2-6x-2=0$. If $a_n={\alpha }^n-{\beta }^n$, for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is equal to :

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

The normal to the curve, $x^2+2xy-3y^2=0$, at $(1,1)$

1. A.meets the curve again in the third quadrant.
2. B.meets the curve again in the fourth quadrant.
3. C.does not meet the curve again.
4. D.meets the curve again in the second quadrant.

If $z$ is a complex number such that $|z|\ge 2$, then the minimum value of $\left|z+\frac{1}{2}\right|$ :

1. A.is strictly greater that $\frac{5}{2}$
2. B.is strictly greater that $\frac{3}{2}$ but less than $\frac{5}{2}$
3. C.is equal to $\frac{5}{2}$
4. D.lie in the interval $(1,2)$

Let $\alpha$ and $\beta$ be the roots of equation $p{x}^2+qx+r=0,p\ne 0$. If $p,q,r$ in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4$, then the value of $|\alpha -\beta |$ is :

1. A.$\frac{\sqrt{34}}{9}$
2. B.$\frac{2\sqrt{13}}{9}$
3. C.$\frac{\sqrt{61}}{9}$
4. D.$\frac{2\sqrt{17}}{9}$

If the sum of the square of the roots of the equation $x^2-(\sin \alpha -2)x-(1+\sin \alpha )=0$ is least, then $\alpha$ is equal to

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

The area of the triangle whose vertices are complex numbers $z,iz,z+iz$ in the Argand diagram is

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

The value of $K$ for which the equation $(K-2)x^2+8x+K+4=0$ has both roots real, distinct and negative is

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

Let $Z_1$ and $Z_2$ be any two complex number. Statement 1: $|Z_1-Z_2|\ge |Z_1|-|Z_2|$ Statement 2: $|Z_1+Z_2|\le |Z_1|+|Z_2|$

1. A.Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
2. B.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
3. C.Statement 1 is true, Statement 2 is false.
4. D.Statement 1 is false, Statement 2 is true.

If the system of equations $x+y+z=6$ $x+2y+3z=10$ $x+2y+\lambda z=0$ has a unique solution, then $\lambda$ is not equal to

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

The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals :

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