1. This paper consists of TEN (10) questions.
2. Answer ALL questions.
3. Show your work clearly in each question.
4. Four figure and scientific calculators can be used.
Answer ALL Questions.
1. Using a non programmable scientific calculator. Find the value of the following expressions.
- correct to 5 decimal places
2. (a) Sets A, B and C are defined by
A = X E R: -3 < x ≤ 2
B = X E R: -4 ≤ x < 3
C = X E R; x ≥ 13
Where R is the set of all real numbers.
Express the following set in the same form.
[*]Anc (ii) A∩B′ (iii) AUB.
(b) Using the laws of algebra of propositions to simplify the following.
(i) A∩ (AUB)
(ii) A∩ (A′UB) U B∩ (A′UB′)
3. (a) Construct the truth table of
(p Λ q) Λ˜(pvq) and comment on the results.
(b) Simplify p ^ ( ~ q v ~ p).
4. (a) A straight line y – x + 1 = 0 intersects a circle of radius r = 2, and centre
P (2, -1
Find the coordinates of the points of intersection.
(b) Find the equation of the normal to the circle x2
– 26 x + 12y + 105 = 0 at a point (7,2).
5. (a) Given that (x – 2) and (x +2) are each factors of x3
+ a x2
+ bx - 4. Find the value of a˝ and b˝, hence for these valued find the third factor of the expression.
(b) If and are the roots of x2
- 4 x + 5 = 0. Find the equation with integral coeffient whose roots are 3
6. (a) Given two angles that are positive and acute, prove without using a calculator that
Cos + cos -1 (2) =
(b) If is a very small angle in radians. Find an approximation for
1 – cos
7. (a) Find the coefficient of x3
in the expansion of ( 1 + x – x2
(b) Express 2 x 2 + 8 x + 7
in partial fractions
(x + 2) (x + 3)
8. (a) Given the matrix
P = 1 3 0 Find
3 -1 4
8. (b) Solve the equation
x -3 1 -1
- 7 x + 5 -1 = 0
9. (a) Solve for x satisfying the inequality that
(b) Sketch the graph of
f (x) = x2
– x - 2
10. (a) If y = e2x
sin 3x . Find
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