Top answers

Prove that the sum of squares of the first n natural numbers is n/6(n+1)(2n+1)

In order to do this we must follow the standard procedure for a proof by induction which is to first check a base case:Let n = 1, then the sum can be written as 1 2 = 1/6(1+1)(2+1) = 1 as required. Next, ass...
TD
6448 Views

Show that the set of real diagonal (n by n) matrices (with non-zero diagonal elements) represent a group under matrix multiplication

We must show that the set satisfies the group requirements: Identity, Closure, Associativity and Invertibility.Identity: Contains identity matrixAssociativity: Follows from the rules of matrix multiplication...
NP
3293 Views

The curve C has parametric equations x=cos(t)+1/2*sin(2t) and y =-(1+sin(t)) for 0<=t<=2π. Find a Cartesian equation for C. Find the volume of the solid of revolution of C about the y-axis.

Note the simplest relation to eliminate t is the fact cos 2 (t)+sin 2 (t)=1 for all t, so we need only find x and y in terms of cos(t) and sin(t).Note we have sin(t)=-(y+1) from the equation for y already.Fr...
LP
5602 Views

Given M = [[-2,6],[1,3]], find P and D such that M = PDP^(-1) where D is a diagonal matrix

We are given M = [[-2,6],[1,3]], with columns [-2,6] and [1,3]. To find P and D, eigenvalues and eigenvectors must be calculated, as D is defined to be the matrix whose diagonal is comprised of the eigenvalu...
HR
3595 Views

Given that y = cosh^-1 (x) , Show that y = ln(x+ sqrt(x^2-1))

Have a picture of full working with annotation to go through during interview.Here is rough outline:y = cosh -1 (x)x = cosh(y)x = (e y +e -y )/22x = e y +e -y e y +e -y -2x = 0Turn into hidden quadratic by m...
MW
6163 Views