Add Maths 2000 Paper 2

5.
y A(1, 2)
P
O Figure 2
B(2, 0)
x
In Figure 2, the coordinates of points A and B are (1, 2) and (2, 0) respectively. Point P divides AB internally in the ratio 1 : r. (a) (b) (c) Find the coordinates of P in terms of r . Show that the slope of OP is 2r . 2+r (6 marks)
(c)
C1
C2 O
C3 x Figure 5
Figure 5 shows the circles C1 and C2 in (b). L is a common tangent to C1 and C2. C3 is a circle touching C2 , L and the x-axis but it is not in F. (See Figure 5.) Find the equation of C3. (Hint : The centres of the three circles are collinear.) (5 marks)
π intersect at the point ( , − 1) . Find the area of the shaded region 4 π bounded by C1 , C 2 and the line x = . 3 (5 marks)
2000-CE-A MATH 2–6 –5–
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11.
(a)
y
O –h x2 + y2 = r2
x
9.
Given a family of circles
F : x 2 + y 2 + (4 k + 4) x + (3k + 1) y − (8k + 8) = 0 ,
where k is real. C1 is the circle x 2 + y 2 − 2 y = 0 . (a) Show that (i) (ii) (b) C1 is a circle in F, C1 touches the x-axis. (4 marks)
sin A + sin B = 2 sin
sin A − sin B = 2 cos
cos A + cos B = 2 cos
cos A − cos B = −2 sin
2 sin A cos B = sin ( A + B) + sin ( A − B) 2 cos A cos B = cos ( A + B) + cos ( A − B) 2 sin A sin B = cos ( A − B) − cos ( A + B)
2000-CE A MATH
PAPER 2
HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 2000
ADDITIONAL MATHEMATICS PAPER 2
11.15 am – 1.15 pm (2 hours) This paper must be answered in English
two distinct points on P, where t1 < 0 < t 2 . L1 and L2 are tangents to P at A and C respectively and they intersect at point B. Let M be the midpoint of AC. (a) Show that (i) (ii) (iii) the equation of L1 is x − t1 y + t12 = 0 , the coordinates of B are (t1t 2 , t1 + t 2 ) , BM is parallel to the x-axis. (7 marks)
1. 2. 3. 4.
Answer ALL questions in Section A and any THREE questions in Section B. All working must be clearly shown. Unless otherwise specified, numerical answers must be exact. The diagrams in the paper are not necessarily drawn to scale.
3.
y
Q(4 cos θ , 3 sin θ )
P(–4, 0) O R E
x
Figure 1 y2 x2 P(−4, 0) and + = 1. 16 9 π Q (4 cos θ , 3 sin θ ) are points on E, where 0 < θ < . R is a point such 2 that the mid-point of QR is the origin O. Figure 1 shows the ellipse E : (a) (b) Write down the coordinates of R in terms of θ . If the area of ∆ PQR is 6 square units, find the coordinates of Q . (6 marks)
∫ cos 3x cos x d x .
sin 5 x − sin x = 4 cos 3 x cos x . sin x
(3 marks)
(b)
Show that
Hence, or otherwise, find
∫
sin 5 x dx . sin x
(4 marks)
(c)
Using a suitable substitution, show that
7.
(a)
By expressing cos x − 3 sin x in the form r cos ( x + θ ) , or otherwise, find the general solution of the equation cos x − 3 sin x = 2 .
(b)
Find the number of points of intersection of the curves y = cos x and y = 2 + 3 sin x for 0 < x < 9π . (8 marks)
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2000-CE-A MATH 2–1
FORMULAS FOR REFERENCE
sin ( A ± B ) = sin A cos B ± cos A sin B cos ( A ± B) = cos A cos B # sin A sin B tan ( A ± B) = tan A ± tan B 1 # tan A tan B A+ B A− B cos 2 2 A+ B A− B sin 2 2 A+ B A− B cos 2 2 A+ B A− B sin 2 2
If ∠ AOP = 45° , find the value of r .
2000-CE-A MATH 2–4
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6.
y
C
A
O Figure 3
x y = –x + 1
dy = 2 x + 3 . The dx line y = − x + 1 is a tangent to the curve at point A. (See Figure 3.) Find The slope at any point ( x, y ) of a curve C is given by (a) (b) the coordinates of A, the equation of C. (7 marks)
∫
(d) y
π 4 π 6
sin 5 x dx = sin x
∫
π 3 π 4
cos 5 x dx . cos x
(π 3
x
(
π , − 1) 4
Figure 4 sin 5 x cos 5 x and C 2 : y = sin x cos x
In Figure 4, the curves C1 : y =
2000-CE-A MATH 2–2
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Section A (42 marks) Answer ALL questions in this section.
1.
Find
∫
2x +1 d x . (4 marks)
2.
Expand (1 + 2 x) 7 (2 − x) 2 in ascending powers of x up to the term x 2 . (5 marks)
2000-CE-A MATH 2–8
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