GATE IN 2026 Question Paper (07-Feb-2026) (Shift-1)

A matrix $P\in {\mathbb{R}}^{3\times 3}$ satisfies $P^3-4P^2+5P-2I=0$ where $I$ is a $3\times 3$ identity matrix.

The set of all possible eigenvalues of matrix $P$ is ___.

Which one among the following statements is true for the function $f(x)=x|x|$?

Consider a function $f(z)=z^2+z+1$ where $z\in \mathbb{C}$ is a complex variable. A simple closed contour $\gamma$ in $z$ -plane encloses the point $z=1+0j$.

The value of integral ${\oint }_{\gamma }\frac{f(z)}{z-1}dz=$ _______.

Consider a matrix
$$A=\left[\begin{array}{ccc}1& 0& 2\\ -1& 1& 0\\ 0& 1& 2\end{array}\right]$$
Let $b=\left[\begin{array}{c}1\\ 3\\ 0\end{array}\right]$ and $x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$

The number of solutions to the linear system of equations $Ax=b$ is _____.

An electric field has a potential of $V(x,y,z)=\sqrt{x^2+y^2+z^2}\text{ V}$. A charge of 1 coulomb placed at $(\hat{i}+\hat{j}+\hat{k})$ experiences a force of

$$\vec{F}=(a\hat{i}+b\hat{j}+c\hat{k})\text{ N}$$

The values of $(a,b,c)$ are ______.