# Engineering Mathematics GATE-2004

**Q 1:** The inverse Laplace transform of the function f(s)=\frac1{s\left(s+1\right)} is

**Q 2:** The function f(x)=3(x-2) has a

**Q 3:** The complex number 2(1+i) can be represented in polar form is

**Q 4:** The differential equation \frac{d^2y}{dx^2}+\sinh x\;\frac{dy}{dx}+ye^x=\sinh x is

**Q 5:** For the time domain function f (t) = t, the Laplace transform of \int_0^tf(t)dt is given by

**Q 6:** The sum of the eigen values of the matrix \begin{pmatrix}3&4\\x&1\end{pmatrix} for real and negative values of x is

**Q 7:** The system of equations

4x+6y=8

7x+8y=9

3x+2y=1

has

**Q 8:** A box contains three blue balls and four red balls. Another identical box contains two blue balls & five red balls. One ball is picked at random from one of the two boxes and it is red. The probability that it comes from the first box is

**Q 9:** The series \sum_{n=1}^\infty\frac{\left(z+2\right)^n}{n!} Converges for

**Q 10:** The differential equation for the variation of the amount of salt x in a tank with time t is given by \frac{dy}{dt}+\frac x{20}=10 . Where x is in kg and t is in minutes. Assuming that there is no salt in the tank initially, the time (in minutes) at which the amount of salt increases to 100 kg is

**Q 11:** Value of the integral \int_{-2}^2\frac{dx}{x^2} is

**Q 12:** The differential equation \left(\frac{dy}{dx}\right)^2+y\frac{d^2y}{dx^2}=0 can be reduced to (where Ξ± is a constant)

**Q 13:** The value of \lim\limits_{x\rightarrow9}\frac{\sqrt x-3}{x^2-81} is