2010-AL
PMATH
PAPER 2
HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG ADVANCED LEVEL EXAMINATION 2010
PURE MATHEMATICS A·LEVEL PAPER 2
•
1.30 pm - 4.30 pm (3 hours)
This paper must be answered in English
1. This paper consists of Section A and Section B.
2. Answer ALL questions in Section A, using the AL(E) answer book.
3. Answer any FOUR questions in Section B, using the AL(C) answer book.
4. Unless otherwise specified, all working must be clearly shown.
Not to be taken away before the
end of the examination session Hong Kong Examinations and Assessment Authority
All Rights Reserved 2010
201O-AL-P MATH 2-1
FORMULAS FOR REFERENCE
sin(A±B) =sin A cos B±cosAsin B
cos(A± B) = cos A cos B+sin A sin B
tan(A ± B) = tan A ± tan B
1Han A tan B
. A . B 2' A+B A Bsm +sm = sm--cos-
2 2
. A . B 2 A+B. A-B
sm -sm = cos--sm-
2 2
A+B A B
cos A + cos B =2 cos --cos-
2 2
· A+B . A-B
cos A -cos B - 2sm--sm-
2 2
2 sin Acos 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)
2010-AL-P MATH 2-2 2
,.
SECTION A (40 marks)
Answer ALL questions in this section.
Write your answers in the AL(E) answer book.
1. (a) Evaluate lim x ~in x
x~o xsmxV Let a and b b""l con,tant, and f' (-ff, ff) -> R be defmed by
{
f(x) =
sinx+~ when
sin x
3 + bx + x 2 when
n- 0 ,
(ii) f'(x) >0 ,
(iii) f"(x) > 0 .
(3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of the graph of y = f(x) .
(2 marks)
Cd) Find the asymptote(s) of the graph of y =: f(x) .
(2 marks)
(e) Sketch the graph of y f(x).
(3 marks)
(t) Let
line
nCk) be the number of points of intersection of the graph of y f(x)
y k. Using the graph of y =: f(x) , find n(k) for any k E R .
and the horizontal
(3 marks)
8. Let y =: I
1+
. For any positive integer n, define fn(x) (1 + x2 r+' in) .
(a) Prove that (1 + x2) i n+2) + 2(n +2) x y(n+l) + (n + 2)(n + l) y
2n 4 c 11 ,. .9, (a) Prove t at ::; - lor a posItIve Integers n.h
n! n
2n
Hence prove that lim "" 0 ,
n-~oo n!
(3 marks)
(b) For any positive integer n, define In= fx-\lnx)ndx,
(i) Evaluate II .
(ii) Express In+l in tenns of In .
(''') PhI - 1(_I__ ~In 1 1
III rove t at n - n. 12k 1 '
2n+ e k~O (n k)!2 +
(iv) Prove that e-2 X-I (lnxr1 ::; x-\lnx)n ::; x-'(lnxt for all x E [1, e] ,
Hence prove that ::; In::; _1_
o + 1) n + 1
(8 marks)
00 2k
(c) Using (a) and (b), evaluate I-,
k=O k!
(4 marks)
10. (a) Denote the interval [0, 1] by I.
(i) Let f: I --+ Rand g: I --+ R be continuous functions.
(1) Define H(x)=(f:f(t)g(t)dty (fox(f(t)idtj(f:(g(t»2dtJ forall xEI.
Prove that H is decreasing on I.
(2) Prove that (foxf(t) g(t)dt J::; (f:(f(t)i dtJ( f:(g(t)i dtJ for all x E I .
(ii) Let h be a real-valued function such that h' is continuous on I and h(O) = 0 .
Prove that
(1) (h(x)i ::; x f:(h/(t)idt for all x E I ,
(2) rl (h(x»2 dx ::;..!. ( (h/(x)i dxJo 2 Jo
(11 marks)
(b) Using (a)(ii), or otherwise, prove that Jorl (In(secx+ tanx»)2 dx ::; 2 1 tan 1 .
(4 marks)
2010-AL-P MATH 2-6 6
r
I,
11. A straight line passing through the point A(0, 16) cuts the ellipse E: + 1 at two distinct
144 400
points P(12sine,20cose) and Q(l2sin{6,20cos{6). The tangents to E at P and Q intersect at
the point R.
(a) Prove that sin 0:# 0 and sin {6 :# 0 .
(3 marks)
(b) (i) Find the equation of the tangent to E at P. Also write down the equation ofthe
tangent to E at Q.
(ii) Prove that sinCe - {6):# 0 .
(5 marks)
(c) Prove that
(i) 4(sin e - sin {b) == 5 sinCe - {b) ,
. ( 3 (4 - 5 cos e) )(ii) the coordmates of R are . , 25 .
\ sme
(4 marks)
(d) Is !:J.PAR a right-angled triangle? Explain your answer.
(3 marks)
END OF PAPER
20JO-AL-P MATH 2-7 7
I
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