Complex Numbers & Quadratic Equations · JEE Main previous year questions

Let $\alpha ,\beta ,\gamma$ be the three roots of the equation $x^3+bx+c=0$. If $\beta \gamma =1=-\alpha$, then $b^3+2c^3-3{\alpha }^3-6{\beta }^3-8{\gamma }^3$ is equal to :

1. A.$21$
2. B.$19$
3. C.$\frac{169}{8}$
4. D.$\frac{155}{8}$

If for $z=\alpha +i\beta ,|z+2|=z+4(1+i)$, then $\alpha +\beta$ and $\alpha \beta$ are the roots of the equation :

1. A.$x^2+2x-3=0$
2. B.$x^2+3x-4=0$
3. C.$x^2+x-12=0$
4. D.$x^2+7x+12=0$

Let $[t]$ denote the greatest integer $\le t$. If the constant term in the expansion of ${\left(3x^2-\frac{1}{2x^5}\right)}^7$ is $\alpha$, then $[\alpha ]$ is equal to ___________.

The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of ${\left(2x^2+\frac{1}{2x}\right)}^{11}$ is equal to :

1. A.$11^3-11$
2. B.$13^3-13$
3. C.$12^3-12$
4. D.$10^3-10$

Let $\mathrm{m}$ and $\mathrm{n}$ be the numbers of real roots of the quadratic equations $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest integer $\le x$. Then ${\mathrm{m}}^2+\mathrm{mn}+{\mathrm{n}}^2$ is equal to ___________.

The sum of all the roots of the equation $|x^2-8x+15|-2x+7=0$ is :

1. A.$11+\sqrt{3}$
2. B.$9+\sqrt{3}$
3. C.$9-\sqrt{3}$
4. D.$11-\sqrt{3}$

For the system of equations $$x+y+z=6$$ $$x+2y+\alpha z=10$$ $$x+3y+5z=\beta ,$$ which one of the following is NOT true?

1. A.System has a unique solution for $\alpha =3,\beta \ne 14$.
2. B.System has infinitely many solutions for $\alpha =3,\beta =14$.
3. C.System has no solution for $\alpha =3,\beta =24$.
4. D.System has a unique solution for $\alpha =-3,\beta =14$.

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $$\lambda x+y+z=1$$ $$x+\lambda y+z=1$$ $$x+y+\lambda z=1$$ is inconsistent, then ${\sum }_{\lambda \in S}(|\lambda {|}^2+|\lambda |)$ is equal to

1. A.$12$
2. B.$2$
3. C.$4$
4. D.$6$

Let $f(x)=2x+{\tan }^{-1}x$ and $g(x)={\log }_e(\sqrt{1+x^2}+x),x\in [0,3]$. Then

1. A.there exists $\bar{x}\in [0,3]$ such that $f^{\prime }(\bar{x})<g^{\prime }(\bar{x})$
2. B.there exist $0<x_1<x_2<3$ such that $f(x)<g(x),\forall x\in (x_1,x_2)$
3. C.$\min f^{\prime }(x)=1+\max g^{\prime }(x)$
4. D.$\max f(x)>\max g(x)$

If the orthocentre of the triangle, whose vertices are $(1,2),(2,3)$ and $(3,1)$ is $(\alpha ,\beta )$, then the quadratic equation whose roots are $\alpha +4\beta$ and $4\alpha +\beta$, is :

1. A.$x^2-20x+99=0$
2. B.$x^2-22x+120=0$
3. C.$x^2-19x+90=0$
4. D.$x^2-18x+80=0$

The combined equation of the two lines $ax+by+c=0$ and $a^{\prime }x+b^{\prime }y+c^{\prime }=0$ can be written as $$(ax+by+c)(a^{\prime }x+b^{\prime }y+c^{\prime })=0.$$ The equation of the angle bisectors of the lines represented by the equation $2x^2+xy-3y^2=0$ is :

1. A.$3x^2+xy-2y^2=0$
2. B.$x^2-y^2-10xy=0$
3. C.$x^2-y^2+10xy=0$
4. D.$3x^2+5xy+2y^2=0$

The number of real roots of the equation $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$, is :

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

The complex number $z=\frac{i-1}{\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}}$ is equal to :

1. A.$\cos \frac{\pi }{12}-i\sin \frac{\pi }{12}$
2. B.$\sqrt{2}\left(\cos \frac{\pi }{12}+i\sin \frac{\pi }{12}\right)$
3. C.$\sqrt{2}i\left(\cos \frac{5\pi }{12}-i\sin \frac{5\pi }{12}\right)$
4. D.$\sqrt{2}\left(\cos \frac{5\pi }{12}+i\sin \frac{5\pi }{12}\right)$

Let the system of linear equations $x+y+kz=2$ $2x+3y-z=1$ $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$ $(2k+1)x+(k+5)y=10$ has :

1. A.unique solution satisfying $x-y=1$
2. B.infinitely many solutions
3. C.no solution
4. D.unique solution satisfying $x+y=1$

The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines are parallel to $x+90y+2=0$ is :

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

Let $z=1+i$ and $z_1=\frac{1+i\bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$. Then $\frac{12}{\pi }\arg (z_1)$ is equal to ____.

Let $\lambda \ne 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14x^2-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35x^2-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation

1. A.$7x^2-245x+250=0$
2. B.$49x^2-245x+250=0$
3. C.$49x^2+245x+250=0$
4. D.$7x^2+245x-250=0$

For two non-zero complex numbers $z_1$ and $z_2$, if $\mathrm{Re}(z_1{z}_2)=0$ and $\mathrm{Re}(z_1+z_2)=0$, then which of the following are possible? A. $\mathrm{Im}(z_1)>0$ and $\mathrm{Im}(z_2)>0$ B. $\mathrm{Im}(z_1)<0$ and $\mathrm{Im}(z_2)>0$ C. $\mathrm{Im}(z_1)>0$ and $\mathrm{Im}(z_2)<0$ D. $\mathrm{Im}(z_1)<0$ and $\mathrm{Im}(z_2)<0$ Choose the correct answer from the options given below :

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

Consider the following system of equations $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ for some $\alpha ,\beta \in \mathbb{R}$. Then which of the following is NOT correct.

1. A.It has a solution for all $\alpha \ne -1$ and $\beta =2$
2. B.It has no solution if $\alpha =-1$ and $\beta \ne 2$
3. C.It has no solution for $\alpha =-1$ and for all $\beta \in \mathbb{R}$
4. D.It has no solution for $\alpha =3$ and for all $\beta \ne 2$

Let ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_7$ be the roots of the equation $x^7+3x^5-13x^3-15x=0$ and $|{\alpha }_1|\ge |{\alpha }_2|\ge \dots \ge |{\alpha }_7|$. Then ${\alpha }_1{\alpha }_2-{\alpha }_3{\alpha }_4+{\alpha }_5{\alpha }_6$ is equal to ____.

Let ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ be respectively the sets of all $a\in \mathbb{R}-\{0\}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

1. A.$\mathrm{n}({\mathrm{S}}_1)=2$ and ${\mathrm{S}}_2$ is an infinite set
2. B.${\mathrm{S}}_1=\Phi$ and ${\mathrm{S}}_2=\mathbb{R}-\{0\}$
3. C.${\mathrm{S}}_1=\mathbb{R}-\{0\}$ and ${\mathrm{S}}_2=\Phi$
4. D.${\mathrm{S}}_1$ is an infinite set and $\mathrm{n}({\mathrm{S}}_2)=2$

Let $S=\left\{\alpha :{\log }_2(9^{2\alpha -4}+13)-{\log }_2\left(\frac{5}{2}\cdot 3^{2\alpha -4}+1\right)=2\right\}$. Then the maximum value of $\beta$ for which the equation $x^2-2{\left(\sum _{\alpha \in S}\alpha \right)}^{2}x+\sum _{\alpha \in S}(\alpha +1{)}^2\beta =0$ has real roots, is ____.

Let $a\in \mathbb{R}$ and let $\alpha ,\beta$ be the roots of the equation $x^2+60^{\frac{1}{4}}x+a=0$. If ${\alpha }^4+{\beta }^4=-30$, then the product of all possible values of $a$ is ____.

Let $p,q\in \mathbb{R}$ and $(1-\sqrt{3}i{)}^{200}=2^{199}(p+iq)$, $i=\sqrt{-1}$ then $p+q+q^2$ and $p-q+q^2$ are roots of the equation.

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

If $z\ne 0$ be a complex number such that $\left|z-\frac{1}{z}\right|=2$, then the maximum value of $|z|$ is :

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