Engineering Mathematics GATE-2008

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Q 1: Which one of the following is not an integrating factor for the differential equation xdy-ydx=0 ?





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Q 2: Which one of the following is not a solution of the differential equation \frac{d^2y}{dx^2}+y=1 ?





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Q 3: The limit of \frac{\sin x}x as x∞ is





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Q 4: The unit normal vector to the surface of the sphere x^2+y^2+z^2=1 at the point \left(\frac1{\sqrt2},0,\frac1{\sqrt2}\right) is ( \widehat i,\widehat j,\widehat k are unit normal vectors in the cartesian coordinate system)





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Q 5: A non-linear function f (x) is defined in the interval – 1.2 < x < 4 as illustrated in the figure below. The equation f(x) = 0 is solved for x within this interval by using the Newton-Raphson iterative scheme. Among the initial guesses (I1, I2, I3 and I4), the guess that is likely to lead to the root most rapidly is





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Q 6: Which one of the following transformations u=f\left(y\right) reduces \frac{dy}{dx}+Ay^3+By=0 to a linear differential equation?  (A and B are positive constants)





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Q 7: The Laplace transform of the function f(t)=t\;\sin t is





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Q 8: The value of the surface integral \oiint_s\left(x\widehat i+y\widehat j\right).\widehat ndA evaluated over the surface of a cube having sides of length a is ( \widehat n is a unit normal vector)





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Q 9: The first four terms of the Taylor series expansion of cos x about the point x = 0 are





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Q 10: If \begin{bmatrix}1&2\\2&1\end{bmatrix} , then the eigen values of A3 are





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Q 11: An analytic function w(z) is defined as w=u+iv , where i=\sqrt{-1} and z=x+iy . If the real part is given by u=\frac y{x^2+y^2} , w(z) is





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Q 12: The normal distribution is given by

f(x)=\frac1{\sigma\sqrt{2\mathrm\pi}}exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right),\;\;-\infty<x<\infty

The point of inflexion to the normal curve is





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Q 13: Using Simpson’s 1/3 rule and four equally spaced intervals (n = 4), estimate the value of the integral

\int_0^{\mathrm\pi/4}\frac{\sin x}{\cos^3x}\operatorname dx




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Q 14: The following differential equation is to be solved numerically by Euler’s explicit method.

\frac{dy}{dx}=x^2y-1.2y\;\;with\;y(0)=1

A step size of 0.1 is used. The solution for y at x = 0.1 is





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Q 15: The Poisson distribution is given by P(r)=\frac{m^r}{r!}exp\left(-m\right) . The first moment about the origin for the distribution is





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