# Engineering Mathematics GATE-2008

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

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

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

**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)

**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 (I_{1}, I_{2}, I_{3} and I_{4}), the guess that is likely to lead to the root most rapidly is

**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)

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

**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)

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

**Q 10:** If \begin{bmatrix}1&2\\2&1\end{bmatrix} , then the eigen values of A^{3} are

**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

**Q 12:** The normal distribution is given by

The point of inflexion to the normal curve is

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

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

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

**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