Complex Numbers & Quadratic Equations · JEE Advanced previous year questions

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument of $z$, with $-\pi <\arg (z)\le \pi$. Let $\omega$ be the cube root of unity for which $0<\arg (\omega )<\pi$. Let $$\alpha =\arg \left(\sum _{n=1}^{2025}(-\omega {)}^n\right).$$ Then the value of $\frac{3\alpha }{\pi }$ is ____.

Let $f(x)=x^4+a{x}^3+b{x}^2+c$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3+3a{x}^2+2bx=0$, where $i=\sqrt{-1}$. If ${\alpha }_1,{\alpha }_2,{\alpha }_3$, and ${\alpha }_4$ are all the roots of the equation $f(x)=0$, then $|{\alpha }_1{|}^2+|{\alpha }_2{|}^2+|{\alpha }_3{|}^2+|{\alpha }_4{|}^2$ is equal to _____.

Let $\alpha ,\beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $$x+2y+z=7$$ $$x+\alpha z=11$$ $$2x-3y+\beta z=\gamma$$ Match each entry in List-I to the correct entries in List-II. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) If }\beta =\frac{1}{2}(7\alpha -3)\text{ and }\gamma =28\text{, then the system has}& \text{(1) a unique solution}\\ \text{(Q) If }\beta =\frac{1}{2}(7\alpha -3)\text{ and }\gamma \ne 28\text{, then the system has}& \text{(2) no solution}\\ \text{(R) If }\beta \ne \frac{1}{2}(7\alpha -3)\text{ where }\alpha =1\text{ and }\gamma \ne 28\text{, then the system has}& \text{(3) infinitely many solutions}\\ \text{(S) If }\beta \ne \frac{1}{2}(7\alpha -3)\text{ where }\alpha =1\text{ and }\gamma =28\text{, then the system has}& \text{(4) }x=11,y=-2\text{ and }z=0\text{ as a solution}\\ & \text{(5) }x=-15,y=4\text{ and }z=0\text{ as a solution}\end{array}$$ The correct option is:

1. A.$(P)\to (3)(Q)\to (2)(R)\to (1)(S)\to (4)$
2. B.$(P)\to (3)(Q)\to (2)(R)\to (5)(S)\to (4)$
3. C.$(P)\to (2)(Q)\to (1)(R)\to (4)(S)\to (5)$
4. D.$(P)\to (2)(Q)\to (1)(R)\to (1)(S)\to (3)$

Let $z$ be a complex number satisfying $|z{|}^3+2z^2+4\overline{z}-8=0$, where $\overline{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) }|z{|}^2\text{ is equal to}& \text{(1) }12\\ \text{(Q) }|z-\overline{z}{|}^2\text{ is equal to}& \text{(2) }4\\ \text{(R) }|z{|}^2+|z+\overline{z}{|}^2\text{ is equal to}& \text{(3) }8\\ \text{(S) }|z+1{|}^2\text{ is equal to}& \text{(4) }10\\ & \text{(5) }7\end{array}$$ The correct option is:

1. A.$(P)\to (1)(Q)\to (3)(R)\to (5)(S)\to (4)$
2. B.$(P)\to (2)(Q)\to (1)(R)\to (3)(S)\to (5)$
3. C.$(P)\to (2)(Q)\to (4)(R)\to (5)(S)\to (1)$
4. D.$(P)\to (2)(Q)\to (3)(R)\to (5)(S)\to (4)$