# Engineering Mathematics GATE-2005

**Q 1:** Match the following, where x is the spatial coordinate and t is the time

Group I | Group II |
---|---|

P. Wave function | 1. \frac{\partial C}{\partial t}=\alpha\;\frac{\partial C}{\partial x} |

Q. Heat equation | 2. \frac{\partial C}{\partial t}=\alpha^2\frac{\partial^2C}{\partial x^2} 3. \frac{\partial^2C}{\partial t^2}=\alpha^2\frac{\partial C}{\partial x} 4. \frac{\partial^2C}{\partial t^2}=\alpha^2\frac{\partial^2C}{\partial x^2} |

**Q 2:** Two bags contain ten coins each, and the coins in each bag are numbered from 1 to 10. One coin is drawn at random from each bag. The probability that one of the coins has a value of 1, 2, 3, or 4, while the other has a value of 7, 8, 9, or 10, is

**Q 3:** Given i=\sqrt{-1} the ratio \frac{1+2i}{i-2} is given by

**Q 4:** How many solutions does the following system of equations have?

4x+2y+z=7

x+3y+z=3

3x+4y+2z=2

**Q 5:** The matrix A is given by A=\begin{bmatrix}1&4\a&2\end{bmatrix} . The eigenvalues of the matrix A are real and non-negative for the condition.

**Q 6:** The divergence of a vector field A is always equal to zero, if the vector field A can be expressed as

**Q 7:** In the limit x→0, what is the limiting value of the function F (x) given below

**Q 8:** If Z=x=iy is a complex number, where i=\sqrt{-1} , then which of the following is an analytic function of z?

**Q 9:** What condition is to be satisfied so that the solution of the differential equation \frac{d^2y}{dx^2}+a\frac{dy}{dx}+by=0 is of the form y=(C_1+C_2x)e^{mx} , where C_{1} and C_{2} are constants of integration?

**Q 10:** In the domain -\infty<x<\infty , the function y(x)=x^3e^{-x} has

**Q 11:** If f(x) is the solution of the equation \frac{dy}{dx}+2xy+2x=0 and g(x) is the solution of the equation \frac{dy}{dx}+2xy-2x=0 , and the constant of integration in f (x) is equal to that in g (x) then which of the following is true?

**Q 12:** The function f (x) satisfies the equation f (x) = 0 at x = x_{e}. The Newton-Raphson iterative method converges to the solution in one step, regardless of the initial guess, if