**Vectors**

*Problem 1.* Verify that for vectors in three dimensions,

.

**Lists and Matrices**

*Problem 2.* Using matrix operations,
find the values of x,y,z which solve the
following set of equations.

1.2 *x* + 4.5 *y* + 2.3 *z* = -1.3

-0.6 *x* + 3.1 *y* - 0.3 *z* = 0.6

1.9 *x* + 0.5 *y* + 1.3 *z* = 1.7

**Function definitions **

*Problem 3.* Define the function
*f*(*x*) = 0.8 *x* + *x*^{2} - 3.2 *x*^{4}. Find
the four roots, *x*_{r}, of *f*(*x*) and check that one of the
non-zero roots satisfies *f*(*x*_{r})=0.

**Plotting**

*Problem 4.* Define
*x*(*t*) = 10 *t* , *y*(*t*) = 10 *t* - 5 *t*^{2}. Plot
*x*(*t*), *y*(*t*) and also do a parametric plot of
(*x*(*t*),*y*(*t*)).
What sort of motion is this ?

**Ordinary differential equations**

*Problem 5.* A mass(m= 1.1 kg)/spring(k = 2.2 N/m)
system hangs vertically
at equilibrium in Earth's gravity. It is displaced
from equilirium by a small amount and then oscillates.
It experiences
a damping of , where *b*=0.1. Consider a small amplitude
initial displacement of 1 *cm* and initial velocity of zero.
Solve the linear differential equation for this problem.
(
*x*''(*t*) + 0.1 *x*'(*t*) + 2 *x* (*t*) = 0). Plot
*x*(*t*).

**Partial differential equations**

*Problem 6.* Show that
solves the
diffusion equation

provided that D is related to a. What is that relationship?