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Sma2160:Mathematics I Question Paper

Sma2160:Mathematics I

Course:Bachelor Of Horticulture

Institution: Meru University Of Science And Technology question papers

Exam Year:2013

QUESTION ONE – (30 MARKS)
(a) Solve the following trigonometric equations (i) 3 ??????2?? - sin??cos?? - 4 ??????2 ?? = 0 0 = ?? = 360 (3 Marks)
(ii) 4sin?? + 3cos?? = 4 0 = ?? = 360 (3 Marks) (b) A chord AB divides a circle of radius b into two segments. If AB subtends and angle of 450 at the centre of the circle: (i) Show that the area of the minor segment is ? ? 22 8 2 ? ? b (3 Marks) (ii) Find the area to 3 significant figures when ?? = 14????. (2 Marks)
(c) A body falls 16 fits during the first second, 48 fits during the second second, 80 fits during the third second and so on, in an arithmetic progress. (i) How far does it fall during the 10th second? (2 Marks) (ii) How far does it fall during the first ten seconds? (3 Marks)
(d) Define a complex plane. (2 Marks) (e) Given that ?? = (-1 + 3 ??), write z in polar form. (4 Marks) (f) Use de-moivres theorem to evaluate (1 + ??)20 (3 Marks) (g) Proof by mathematical induction that 1 + 2 + 3 + ?+ ?? = ?? 2 ?? + 1 (4 Marks)
2
QUESTION TWO – (20 MARKS)
(a) Draw on the same axis the graph of trigonometric function
2
cos3
x y ? and
0
30
2 sin5 ? ? ? ? ? ? ?? x y for range 0 = ?? = 360°. (6 Marks) (b) State the amplitude and the period of the function ?? = 3cos?? 2 . (2 Marks) (c) Use your graph to solve the equation 0 30 2 sin5 2 cos3 0 ? ? ? ? ? ? ? ?? xx (2 Marks) (d) Proof by mathematical induction that ? ?? ? 121 6 ...321 2 222 ? ?????? n n n n (5 Marks) (e) In triangle ??????,?? = 12???? ?? = 10???? ?????? ?? = 30°. Calculate the radius of the circumcircle of the triangle and remaining angles of the triangle. (5 Marks)
QUESTION THREE – (20 MARKS)
(a) Prove that cos3?? = 4 ??????2?? - 3cos?? (4 Marks) (b) Define the half angle t formulae for tan??,sin?? ??????cos?? and hence, solve the equation 7cos?? + sin?? - 5 = 0 0 = ?? = 360° (10 Marks) (c) Two circles shown below have centres ?? & ?? and radii 24cm and 7cm respectively. Given that ???? = 25????, determine the area of the shaded region and the perimeter of ??????????. (6 Marks)
3
QUESTION FOUR – (20 MARKS) (a) Define a polar co-ordinate (2 Marks) (b) Given the equation ?? = 10cos??, change it to a rectangular co-ordinate system. (4 Marks) (c) (i) State and proof the de-moivres theorem. (6 Marks) (ii) Use the formulae derived in above to evaluate ( 3 - 2??)'' (8 Marks)

Sma2160:Mathematics I Question Paper

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