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Unformatted text preview: nd M2 in terms of a.
(4) 23 (b) 5 − 4
If M2 is equal to − 4 5 , find the value of a. (2) Using this value of a, find M–l and hence solve the system of equations:
–x + 2y = –3
2x – y = 3
(Total 12 marks) 21. The function f is such that f″ (x) = 2x – 2.
When the graph of f is drawn, it has a minimum point at (3, –7).
(a) Show that f ′(x) = x2 – 2x – 3 and hence find f(x).
(6) (b) Find f (0), f (–1) and f′(–1).
(3) (c) Hence sketch the graph of f labelling it with the information obtained in part (b).
(4) (Note: It is not necessary to find the coordinates of the points where the graph
cuts the x-axis.)
(Total 13 marks) x 22. The diagram shows part of the graph of y = e 2 . y
x y = e2 P
ln2 x 24 (a) Find the coordinates of the point P, where the graph meets the y-axis.
(2) The shaded region between the graph and the x-axis, bounded by x = 0
and x = ln 2, is rotated through 360° about the x-axis.
(b) Write down an integral which represents the volume of the solid obtained.
(4) (c) Show that this volume is π.
(Total 11 marks) 23. The parabola shown has equation y2 = 9x. y 2 = 9x
P M Q (a) x Verify that the point P (4, 6) is on the parabola.
(2) The line (PQ) is the normal to the parabola at the point P, and cuts the x-axis at Q.
(b) (i) Find the equation of (PQ) in the form ax + by + c = 0.
(5) (ii) Find the coordinates of Q.
(2) 25 S is the point 9 , 0 . 4 (c) Verify that SP = SQ.
(4) (d) The line (PM) is parallel to the x-axis. From part (c), explain why
(QP) bisects the angle SPM.
(Total 16 marks) 24. A box contains 35 red discs and 5 black discs. A disc is selected at random and its colour noted.
The disc is then replaced in the box.
(a) In eight such selections, what is the probability that a black disc is selected
(i) exactly once?
(3) (ii) at least once?
(3) (b) The process of selecting and replacing is carried out 400 times.
What is the expected number of black discs that would be drawn?
(Total 8 marks) 26 25. When the function f (x) = 6x4 + 11x3 – 22x2 + ax + 6 is divided by (x + 1) the remainder is –20.
Find the value of a.
(Total 4 marks) 26. A bag contains 2 red balls, 3 blue balls and 4 green balls. A ball is chosen at random from the
bag and is not replaced. A second ball is chosen. Find the probability of choosing one green
ball and one blue ball in any order.
(Total 4 marks) 27 27. The second term of an arithmetic sequence is 7. The sum of the first four terms of the
arithmetic sequence is 12. Find the first term, a, and the common difference, d, of the sequence.
(Total 4 marks) 28. Find the coordinates of the point where the line given by the parametric equations x = 2λ + 4,
y = –λ – 2, z = 3λ + 2, intersects the plane with equation 2x + 3y – z = 2.
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- Fall '13
- The Land