# Engineering Mathematics GATE-2003

**Q 1:** A box contains 6 red balls and 4 green balls; one ball is randomly picked and then a second ball is picked without replacement of the first ball. The probability that both are green is

**Q 2:** The directional derivative of f(x,y,z)=x^2+y^2+z^2 at the point (1, 1, 1) in the direction \underline i-\underline k is

**Q 3:** The Taylor series expansion of the function: F(x)=x/(1+x) around x = 0 is

**Q 4:** The range of values for a constant ‘K’ to yield a stable system in the following set of time-dependent differential equations is

**Q 5:** The value of y as t → ∞ for the following differential equation for an initial value of y(1) = 0 is

**Q 6:** The most general complex analytical function f(z)=u(x,y)+i\;v(x,y) for u=x^2-y^2 is

**Q 7:** The differential equation \frac{d^2y}{dt^2}+10\frac{dy}{dt}+25x=0 will have a solution of the form NOTE: C_{1} and C_{2} are constants