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 C1 and C2.
If F1(x, y), F2(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
\iint_R\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\operatorname dx\operatorname dy=\int_C\left(F_1\operatorname dx+F_2\operatorname dy\right)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 λ1 = -1 and λ2 = 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?
\frac{dy}{dx}=\frac{y^2}x+\frac yx-\frac2xNote: 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 xn+1 = g(xn), where xn represents the nth iteration value. g(x) and corresponding initial guess value x0 in the domain of interest are shown in the following choices. Which one of these choices leads to a converged solution for x?
