Algebra - Mathematics Previous Year Questions

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$

If $S$ is the circumcentre, $O$ is the orthocentre and $G$ is the centroid of a $\triangle A B C$, then match the items of the List-I with those of the items of List-II given below.

Then, the correct match is

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4 i}\right)=0$ and $\lim \left(\frac{z-2}{z-4 i}\right)=1$ simultaneously is

For $n \in \mathbf{N}$, If $A_n=\cos \left(\frac{\pi}{2^n}\right)+i \sin \left(\frac{\pi}{2^n}\right)$, then $\left(A_1 A_2 A_3 A_4\right)^4=$

Sum of the modulii of the complex roots of the equation $\left(x^2+\frac{1}{x^2}\right)-5\left(x+\frac{1}{x}\right)+6=0$ is

The roots of the equation $(x-1)^5=32(x+1)^5$ are