Advanced Mathematics

1. Determine whether the line x = 3+8t, y = 4+ 5t, z=-3-1 is parallel to the plane x-3y + 5z = 12.

Question 1 a) The time rate of change of a falling object's height can be given by the differential equation where y = height of the object, a = acceleration and t = time. An object is dropped from a height of 500m, with an acceleration due to gravity of -9.8m/sec², (i)Express the height of the object as a function of time. (ii) When will the object reach ground level? b) The acceleration (a =) of a car can be shown to obey the following linear differential equation = 0.05(250-v), where v is the velocity of the car. (i) Express the velocity of the car as a function of time. (ii) How long does it take the car to accelerate from rest to 200km/h? c) Find the general solution of d'r dr ds² ds - 2r=0 d) Find the first four non-zero terms of the power series solution for the differential equation y" + x²y = 0.

a) Use double integrals to find the volume of the solid lying under the surface z = 1+x+y and above the region in the xy-plane bounded by x = y² and .x = 4. b) Find the mass of the region bounded by the curve r = 2 sin given that the density across the region is constant. c) An ice cream cone can be described as being a solid bounded below by the cone ==√√3(x² + y²) and above by the hemisphere .x² + y² +2²=1, z 20. The volume of such a cone (ice cream and cone combined) can be calculated using triple integrals. (i) Set up the triple integral in rectangular co-ordinates to calculate the volume. (ii) Set up this triple integral in cylindrical co-ordinates to calculate the volume. (iii) Set up this triple integral in spherical co-ordinates to calculate the volume. (It may be helpful to know that the cone makes an angle of with the z axis.) (iv)Calculate the volume of this ice cream cone.

Assume a continuous function f(x) defined on x axis with a uniform grid of spacing h. Using appropriate Taylor series expansions2, find the leading order

Consider the function f(x)=2/x. The goal is to compute the first derivative of the function at = 2 numerically using the following approximations with different grid resolutions (Step sizes) and comparing the values to the exact solution.

In a vibration experiment, a block of mass m is attached to a spring of stiffness k, and a dashpot with damping coefficient c. To start the experiment, the block is moved from the equilibrium position and

Define an equivalence relation on S2 = {(x, y, z) | x2+y2++z2 = 1} by defining the equivalence classes to be {

Problem 4 Minimize the function f : R² → R given by f(x₁, x2) = x² + 2x₁x2 + x² on the set X = {(x₁, x₂) € R² : x₁ + x₂-1=0}.

Problem 5 Classify (local and global) the stationary points of f: R² → R, ƒ(x) = x¹Qx+b¹x where a/2 Q = (a/2 °1²) = and b = (1, 2) for a € R. Consider all possibilities depending on a.

Any countably infinite product of seprable spaces is separable. This Question shows that the box topology does not behave