(c) ) If the point O is the origin and point T is a point such that OPTQ is a parallelogram, prove [15 marks] that when m changes, the equation of the locus of T is 4 4.
(b)
8. The straight line y = mx – 2 intersects the curve 4 at two different points, P( , , ) and Q(, , . Show that (a) m ' 0 and m >
(b)Express
Answer allquestions in this section.
[1 marks] [2 marks] [3 marks]
7. Given that f ( x) = x 3 + px 2 + x + q, where p and q are constants, is divisible by ( x − 1). When divided by ( x − 2), the remainder is 17. Find the values of % and &. [4 marks] (a)Show that f ( x ) = 0 has only one real root. Find the set of values of such that [6 marks] f ( x) > 0 .
(a) State the range of f. (b) Find . (c) Sketch the graph of f and .
[4 marks]
2. Find the expansion of 9 as far as term in and state the range of values of x for which the expansion is valid. Hence, obtain the value of √9.05correct to four decimal places. [6 marks] 3. Find the inverse of matrix A by using elementary row operations 1 0 2 A = 3 1 6 8 9 2
[4 marks]
Hence, solve the simultaneous equations x +2z = 1 3x + y +6z = 2 8x +9y +2z = -1
[2 marks] [4 marks] [2 marks]
4. Find the roots, denoted by and , of the equation 2 4 0 in polar form. [10 marks] Using de Moivre’s theorem, show that . 5. The equation of an ellipse is 4 8 12 0. (a) Obtain the standard form of the equation of the ellipse. (b) Find the centre, foci and vertices of the ellipse. (c) Sketch the ellipse.
6. The vector v = ai + bj + ck is perpendicular to the vectors i – 2k and 2i + j – k. (a) Find a and b in terms of c. [4 marks] [3 marks] (b) Given that |#| = √56 , and that c is positive, find c.
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and then attended Harvard University, graduating in 1972 with an. MBA. While at Harvard, he met his beloved wife Susan Veda Langerhans, whom he married ...
