# Engineering Mathematics GATE-2007

**Q 1: **Given i=\sqrt{-1} , the ratio \left(\frac{i+3}{i+1}\right) is given by

**Q 2:** The value of ‘a’ for which the following set of equations

y+2z=0

2x+y+z=0

ax+2y=0

have a non-trivial solution, is

**Q 3: **The initial condition for which the following equation

has infinitely many solutions, is

**Q4: **Given that the Laplace transform of the function below over a single period 0 < t < 2 is \frac1{s^2}\left(1-e^{-s}\right)^2 , the Laplace transform of the periodic function over 0 < t < ∞ is

**Q 5:** If z=x+iy is a complex number, where i=\sqrt{-1} then the derivative of z\overline z at 2 + *i* is

**Q 6:** \overset=A and \overset=B are two 3×3 matrix such that \overset=A=\begin{bmatrix}-2&4&6\\1&2&1\\0&4&4\end{bmatrix} , \overset=B=\overset=0 and \overset=A\overset=B=\overset=0 . Then the rank of matrix \overset=B is

**Q 7:** The solution of the following differential equation x\frac{dy}{dx}+y(y^2-1)=2x^3 is

**Q 8:** The directional derivative of f=\frac12\sqrt{x^2+y^2} at (1, 1) in the direction of \overrightarrow b=\overrightarrow i-\overrightarrow j is

**Q 9:** Evaluate the following integral (n ≠ 0) \int_c\left(-xy^ndx+x^nydy\right) within the area of a triangle with vertices (0, 0), (1, 0) and (1, 1) (counter-clockwise).

**Q 10: **The family of curves that is orthogonal to xy = c is

**Q 11:** The Laplace transform of f(t)=1/\sqrt t is

**Q 12:** The thickness of a conductive coating in micrometers has a probability density function of 600x^{-2} for 100 μm < x < 120 μm. The mean and the variance of the coating thickness is