Engineering Mathematics GATE-2015

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

A) 0
B) 1
C) 2
D) 3

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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 ____________________.

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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:

A) Homogenous differential equation
B) Second order differential equation
C) Non-Linear differential equation
D) Exact differential equation

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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?

A) \frac{df}{dz}=\frac{\partial v}{\partial y}+i\frac{\partial u}{\partial y}
B) \frac{df}{dz}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}
C) \frac{df}{dz}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}
D) \frac{df}{dz}=\frac{\partial v}{\partial y}+i\frac{\partial v}{\partial y}

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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:

A) -2\;\widehat i+2\;\widehat j
B) 2\;\widehat i+2\;\widehat j
C) -2\;\widehat i-2\;\widehat j
D) 2\;\widehat i-2\;\widehat j

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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.dr

Where 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 _____________________.

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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:

A) -1
B) 0
C) 1
D) 2

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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 _____________.

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