# Engineering Mathematics GATE-2012

**Q 1: **Consider the following set of linear algebraic equations

The system has

**Q 2: **If *a *and *b *are arbitrary constants, then the solution to the ordinary differential equation \frac{d^2y}{dx^2}-4y=0 is

**Q 3:** For the function f(t)=e^{-t/\tau} , the Taylor series approximation for *t *<< t is

**Q 4:** A box containing 10 identical compartments has 6 red balls and 2 blue balls. If each compartment can hold only one ball, then the number of different possible arrangements are

**Q 5: **Consider the following (2×2) matrix \begin{pmatrix}4&0\\0&4\end{pmatrix} . Which one of the following vectors is NOT a valid eigenvector of the above matrix?

**Q 6: **If *a *is a constant, then the value of the integral a\int_0^\infty xe^{-ax}\operatorname dx is

**Q 7: **The Newton – Raphson method is used to find the roots of the equation

If the initial guess for the root is 0.5, then the value of *x *after the first iteration is

**Q 8: **If i=\sqrt{-1} , the value of the integral

using the Cauchy residue theorem is