Solution of S.4 Mathematics Paper I
Section A
1.Solve the following equations.
(a) (2 marks)
x =1 , 3
(b)(5 marks)
By (a)
n = 0, 1
(c)(5 marks)
By (a)
x = 10, 1000
2.The figure shows the graph of for one period.
Give the coordinates of the points P, Q and R.
(6 marks)
P is the point (120, 4)
Q is the point (210, 0)
R is the point (0, -2)
- In , AB= 10 cm , and .
(a) find the length of BC (4 marks)
By sine formula
BC = 6.527
BC is 6.53 cm, correct to 3 sig fig
(b)find the area of the triangle.(4 marks)
Area of triangle ABC =
= 32.139
area is 32.1 cm (correct to3 sig fig)
4.ABCDHGFE is a rectangular box with AB = 3, BC = 4 and BF = 2.
Find the angles between
(a)line AG and the plane ABFE and
(b)the planes AHG and EFGH.
.(8 marks)
(a)angle between line AG and the plane ABFE = GAF
GAF =
(b)the angle between the planes AHG and EFGH = AHE
5.The weight (w) of a cylindrical metal varies jointly as its height (h) and the square of its
radius (r).When h = 10 and r = 1; w = 20.
(a)`Find the weight of the metal in terms of h and r.(4 marks)
where k is a non-zero constant
sub h = 10, r = 1 and w = 20
(b)Two cylindrical metals with heights and ; radii and respectively. If and . Find the ratio of their weights. (4 marks)
=
=
The ratio of their weights are 16 to 27.
6.(a)If , find the values of A, B and C.(6 marks)
By comparing coefficients
(A, B, C) = (9, 13, 4)
(b)Hence or otherwise, solve . (3marks)
by (a)
x(9x-5) = 0
7.The parabola touches the x-axis at point A.
(a) Find the y-intercept, value of k and the coordinates of point A (8 marks)
y intercept = -3
it touches the x-axis,
sub y = 0,
point A is (
(b)Use the above information to sketch the graph of . (2 marks)
8.ABCD is a quadrilateral with A : B : C : D = 6 : 5 : 3 : 4.
(a)Find B and D.
(b)Is ABCD is a cyclic quadrilateral? Explain your answer.(9 marks)
(a)Let A= 6k,B=5k,C=3k and D = 4k
(where k is a non-zero constant)
alternatively,
(b)B+D = 180
ABCD is a cyclic quadrilateral (opp supp)
9.In the figure, BN is a tangent to the circle passing points A, B, C and with centre at O.
(a)If , find .(8 marks)
(tgrad)
(at centre twiceat circumference)
(b)From , find a pair of angles which are equal.(2 marks)
( in alt segment)
or ACB = BCM = ( in semicircle)
Section B
10. y varies partly as and partly as the square of .
When , ; when , .
(a)Find the relationship between y and.(6 marks)
(b)If y = 3, find the values of where .(6 marks)
(c)(i) Express in the form of .
(ii)By using (i) or otherwise, find the maximum value of y.
(iii)Does y has a minimumvalue? If no, explain your answer; if yes, find it. (8 marks)
(a) where are non-zero constants
sub
From (2),
Sub into (1),
Hence
(b)(i)
(c)
(ii)By (i)
Hence max y =
(iii)y attains a minimum value of -3 when
11A kite V held by Peter with a string at A. V is vertically above B and the angle of elevation of V from A is. Paul stands at a point C on the circle ABC observes the kite with an angle of elevation. C is due south of point B and AC is the diameter of the circle. VA = 100 m.
(a)Find AB, VB, BC, VC and AC. (give the answers in surd form if necessary.)(6 marks)
(b)Find the compass bearing of C from A. (3 marks)
(c)Find, correct to the nearest degree, . (3 marks)
(d)Mary, after walking along the circle ABC for a complete revolution, says that from a point P on the circumference, she can see point V with the smallest angle of elevation. Where is point P and find,correct to the nearest degree,this smallest angle of elevation. (5 marks)
(e)Mary further claims that , is she correct? Explain your answer.
(3 marks)
(a)
since ,
(b)
The compass bearing of C from A is SW
(c)
correct to the nearest degree
(d)Point P should be the end point of the diameter drawn from B.
PB is a diameter,
Let the angle of elevation from P be
(e)Since ,
1