1.a) Given that the complex number Z and its conjugate satisfy the equation , find the possible values of Z.

b) Prove that if is real, then the locus of the point representing the complex number Z  is a straight line.

2. The coefficients of the first three terms of the expansion of  are in an Arithmetic Progression (AP). Find the value of n.

3. Solve the equation 3 tan²⁡θ + 2sec²⁡θ = 2 (5-3 tan⁡θ )for 0⁰ ≤ θ ≤ 180⁰.

4. Differentiate  with respect of x.

5. Solve for x in the equation 4^2x – 4 ^x+1 + 4 = 0.

6. Show that

7. The equation of a curve is given by y²- 6y+ 20x + 49 = 0.

1. Show the curve is a parabola.
2. Find the coordinates of the vertex.

8. A container is in the form of an inverted right circular cone. Its height is 100cm and base radius is 40cm. The container is full of water and has a small hole at its vertex. Water is flowing through the hole at a rate of 10cm³      S^-1. Find the rate at which the water level in the container is falling when the height of water in the container is halved.

9. The vertices of a triangle are P(4, 3), Q (6, 4) and R(5, 8). Find angle RPQ using vectors.

10. A circle whose centre is in the first quadrant touches the x and y axes and the line 8x – 15y = 120.

1. equation of the circle.
2. point at which the circle touches the x- axis.

11. A curve whose equation is x² y+ y²- 3x=3 passes through points A(1, 2) and B(-1, 0). The tangent at A and the normal to the curve at B intersect at point C. Determine the:

1. equation of the tangent.
2. coordinates of C

12. a) Express cos⁡(θ+30⁰ ) - cos⁡( θ+48⁰)  in the form Rsin Psin Q, where R is a constant. Hence solve the equation.
cos⁡(θ+30⁰ ) - cos⁡( θ+48⁰)=0.2

      b) Prove that in any triangle ABC;

13. a) Solve for x and y in the following simultaneous equations.
(x – 4y)² = 1
3x + 8y = 11

b) Find the set of values of x for which 4x² +2x < - 3x +6

14. a) The points A and B have position vectors a and b. A point C with a position vector c lies on AB such that 

Show that C =(1-λ ) a+ λ b.

b) The vector equations of two lines are

r₁=2i+j+λ(i+j+2k) and
r₂=2i+2j+t k+μ(i+2j+k)
where i, j and k are unit vectors and λ, μ and t are constatnts.

Given that the two lines intersect, find

1. the value of t
2. the coordinates of the point of intersection.

15. a) Sketch the curve y = x³ - 8.
b) The area enclosed by the curve in (a), the y- axis and the x-axis is rotated about the line y = 0 through 360⁰. Determine the volume of the solid generated.

16. a) Solve the differential equation
ln x, given that y = 1 when x =1.
Hence find the value of y when x = 4.