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
F(x)=\frac{1-\cos2x}{x^2}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 C1 and C2 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 = xe. The Newton-Raphson iterative method converges to the solution in one step, regardless of the initial guess, if