Q 1: The following set of three vectors
\begin{pmatrix}1\\2\\1\end{pmatrix},\;\begin{pmatrix}x\\6\\x\end{pmatrix}\;and\;\begin{pmatrix}3\\4\\2\end{pmatrix}is linearly dependent when x is equal to
Q 2. For the matrix \begin{pmatrix}4&3\\3&4\end{pmatrix} , if \begin{pmatrix}1\\1\end{pmatrix} is an eigen-vector, the corresponding eigenvalue is ____________________.
Q 3. Consider a linear ordinary differential equation: \frac{dy}{dx}+p(x)y=r(x) . Functions p(x) and r(x) are defined and have a continuous first derivative. The integrating factor of this equation is non-zero. Multiplying this equation by integrating factor converts this into a:
Q 4. A complex-valued function, f(z), given below is analytic in domain D:
f(z)=u(x,y)+iv(x,y)\;\;\&\;\;z=x+iy
Which of the following is not correct?
Q 5. A scalar function in the xy-plane is given by \phi(x,y)=x^2+y^2 . If \widehat i and \widehat j are unit vectors in the x and y directions, the direction of maximum increase in the value of ф at (1,1) is along:
Q 6. A vector u=-2y\;\widehat i+2x\;\widehat j , where \widehat i and \widehat j are unit vectors in x and y directions, respectively. Evaluate the liner integral
I=\oint_Cu.drWhere C is a closed loop formed by connecting points (1,1), (3,1), (3,2), and (1,2) in that order. The Value of I is equal to _____________________.
Q 7. The solution of the non-linear equation x^3-x=0 is to be obtained using the Newton-Raphson method. If the initial guess is x = 0.5, the method converges to which one of the following values:
Q 8. For a complex variable z, the value of the contour integral \frac1{2\piⅈ}\int_c\frac{e^{-2z}}{z(z-3)}dz along the clockwise contour C: ∣z∣= 2 (up to two decimal places) is _____________.