# Engineering Mathematics GATE-2011

**Q 1: **R is a closed planar region as shown by the shaded area in the figure below. Its boundary C consists of the circles C_{1} and C_{2}.

If F_{1}(x, y), F_{2}(x, y), \frac{\partial F_1}{\partial y} and \frac{\partial F_2}{\partial x} are all continuous everywhere in R, Green’s theorem states that

Which one of the following alternatives correctly depicts the direction of integration along C?

**Q 2:** Which one of the following functions y(x) has the slope of its tangent equal to \frac{ax}y ? Note: a and b are real constants.

**Q 3:** Let λ_{1} = -1 and λ_{2} = 3 be the eigenvalues and \overrightarrow{V_1}=\begin{pmatrix}1\\0\end{pmatrix} and \overrightarrow{V_2}=\begin{pmatrix}1\\1\end{pmatrix} be the corresponding eigenvectors of a real 2×2 matrix \overset=R . Given that \overset=R=\begin{pmatrix}\overrightarrow{V_1}&\overrightarrow{V_2}\end{pmatrix} , which one of the following matrices represents \overset={P^{-1}}\overset=R\overset=P ?

**Q 4:** Unit vectors in x and z directions are \overrightarrow i and \overrightarrow k respectively. Which one of the following is the directional derivative of the function F(x,z)=\ln\left(x^2+z^2\right) at the point P: (4, 0), in the direction of \left(\overrightarrow i-\overrightarrow k\right) ?

**Q 5:** Which one of the following choices is a solution of the differential equation given below?

Note: c is a real constant

**Q 6:** The value of the improper integral \int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)} is

**Q 7:** In the fixed point iteration method for solving equations of the form x = g(x), the (n+1)^{th} iteration value is x_{n+1} = g(x_{n}), where x_{n} represents the n^{th} iteration value. g(x) and corresponding initial guess value x_{0} in the domain of interest are shown in the following choices. Which one of these choices leads to a converged solution for x?