maths

1. Prove using mathematical induction that 1+2+3+...+n=[ n(n+1)]/2

2. Find the derivative of 2^x^2+log2(2x^2-1)

3. Use DeMeoivre's theorem to simplify the following expressions

[(Cos pie/3 + isin pie/3)^5 (cos 2pie/3- isin 2pie/3)^4]/ (cos pie/6 - isin pie/6)^2

4.Find the radius of the circle with center at C (-2,5) if the line x+3y=9 is a tangent line.

5. Find the equation of the line of a circle if A(6,4) and B(-2,7) are the end points of the diameter.

6.(a). Let f(x)=x^4-x^2-2
(i) Find the relative maximum and minimum points of f(x).

(ii) Determine the intervals where the function is increasing.

(iii) Hence sketch the graph of the function.
(b) (i) sketch the graph of the function f(x)=x(2-x_ for values of x in the interval -2<_x<_4.

(ii) Find the area bounded by the curve f(x)=x(2-x) and the x- axis between the lines x= -1 and x=2.

7.Solve the equation (1-cos2A)/(3+cos2A)=1/7 for values of A in the interval 0<_ A <_180

8.(a) sketch the graph of f(x)= -2cos(x/2) for values of x in the interval 0<_ x <_ pie

(b) determine the general solution of -2cos(x/2)=0

9. The root of the equation 2x^2-4x+5=0 are a and b.

(i) Find the value of (1)/(a^2+1) + (1)/b^2+1

(ii) Find an equation whose roots are 2/a and 2/b.

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