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Find the four complex roots of the equation z^4 = 8(3^0.5+i) in the form z = re^(i*theta)

We know that z=re^(i theta) from the definition of the exponential form of a complex number. Hence it follows that: z^4=(re^(i theta))^4=r^4 e^(4 i theta) We can find z^4 by converting 8(3^0.5+i) (cartesian ...
GG
6739 Views

Find, without using a calculator, integral of 1/sqrt(15+2x-x^2) dx, between 3 and 5, giving your answer as a multiple of pi

To get the denominator into something usable, you have to complete the square so you have it in one of the forms you can use a trig or hyperbolic substitution for. The minus sign in front of the x2 means we'...
LD
5277 Views

Prove by induction that n! > n^2 for all n greater than or equal to 4.

This is a fairly typical example of a question from the Further Maths syllabus. We wish to demonstrate that for all integers n greater than or equal to 4, n! > n 2 . We briefly recap that n! is the produc...
JB
20391 Views

If y=x^3+9x, find gradient of the tangent at (2,1).

To find the gradient of the tangent, we can differentiate to give dy/dx=3x^2+9. We can now put in x=2 to find the gradient at (2,1): 3(2)^2+9=21. Therefore the gradient is 21 at (2,1).
AM
3032 Views

Find the set of values of x for which (x+4) > 2/(x+3)

This is an example of an inequalities question from FP2. For this, we will need to use the tools learned in this chapter. To start with, it may be tempting to multiply both sides of the inequality by (x+3) t...
TD
11760 Views