# Engineering Mathematics GATE-2009

**Q 1:** The direction of largest increase of the function xy^{3} – x^{2} at the point (1, 1) is

**Q 2: **The modulus of the complex number \frac{1+i}{\sqrt2} is

**Q 3:** The system of linear equations Ax = 0, where A is an n×n matrix, has a non-trivial solution only if

**Q 4:** The value of the limit \lim\limits_{x\rightarrow\mathrm\pi/2}\frac{\cos x}{\left(\mathrm x-\mathrm\pi/2\right)^3} is

**Q 5:** The general solution of the differential equation \frac{d^2y}{dx^2}-\frac{dy}{dx}-6y=0 , where C_{1} and C_{2} are constants of integration, is

**Q 6:** Using the residue theorem, the value of the integral (counter-clockwise)

around a circle with center at z = 0 and radius = 8 (where z is a complex number and i=\sqrt{-1} , is

**Q 7:** Consider the integral

over the surface of a sphere of radius = 3 with center at the origin and surface unit normal \widehat n pointing away from the origin. Using the Gauss divergence theorem, the value of this integral is

**Q 8:** Using the trapezoidal rule and 4 equal intervals (n = 4), the calculated value of the integral (rounded to the first place of the decimal) \int_0^\pi\sin\theta d\theta is

**Q 9:** The eigen values of matrix A=\begin{bmatrix}1&2\\4&3\end{bmatrix} are 5 and -1 then the eigen values of -2A + 3I (I is a 2×2 identity matrix) are

**Q 10: **A fair die is rolled. Let R denotes the event of obtaining a number less than or equal to 5 and S denotes the event of obtaining an odd number. Then which one of the following about the probability (P) is true?

**Q 11:** The inverse Laplace transform of \frac1{2s^2+3s+1} is