2009 prelims A-Math 4EX/5NA P2 Solutons 1ai
60 2 x 30 60 ,120 ,420 ,480 x 15 ,45 ,195 ,225 .
1aii
1 3(1 sin 2 y ) 4 sin y 3 sin 2 y 4 sin y 4 0 (3 sin y 2)(sin y 2) 0 2 sin y or sin y 2 (NA) 3 41.8 x 41.8 ,138.2
1bi
sec (90o + A)
1 cos(90 A)
1 cos 90 cos A sin 90 sin A 1 sin A = - cosecA 1bii sin 5 B sin 3B sin B 2 sin 3B cos 2 B sin 3B cos 5B cos 3B cos B 2 cos 3B cos 2 B cos 3B sin 3B (2 cos 2 B 1) cos 3B (2 cos 2 B 1) tan 3B 2a AB 7 sin , BC 4 cos P 2(7 sin 4 cos ) 14 sin 8 cos 2b P 260 sin( 29.7 ) 10 10 sin( 29.7 ) 260 8.63
2c
Max = 260 cm = 16.1 cm when 29.7 90 60.3 .
3 3a
dy ln 2 x 1 dx ln 2 x 1 0
1 2e 0.184 d2y 1 2e > 0 dx 2 x min pt. dy 0.2 dt dx 1 0.2 dt ln 2 1 = 0.118 units2. dy 2 sec 2 2 x dx 2 cos 2 4 4 x
3b
4a
grad of normal =
4b
1 1 c 4 8 c 1 32 x y 1 4 32 dy 2 x 3 2 x dx (2 x 3) 2 3 (2 x 3) 2 0 no turning pt.
1 4
1 2
1
5a 5b
5c
YPZ PXY (angles in alt seg) PZY XZP (common angle) PZ YZ (similar ∆s) PX PY PZ PY PX YZ YZ XZ PZ 2 (tan-sec thm)
PX YZ PY 2 YZ XZ PX YZ 2 PY
2
2
6a
XZ PX YZ PY Let f ( x) x 3 3x 2 2 x k 2 f (3) f (2) 2(k 6) k 16
6b
k 4 x 0, s 3.
x 1, p 6. x 3 : q 5. x2 : 6 5 r r 1 7a
6 2 3
2 2 3 2 3 2 2 3 3 2 2 23 2 3 3 2 2
2 3
=
6
3 2 3 2 3 3 2 3 2 3 2 2 3 2 33 23 2 2 3 2 3 3
2 4 3 2 3 3
7b
3(3 x ) 2 9 3 x 6 0 (3 x ) 2 3(3 x ) 2 0 (3 x 1)(3 x 2) 0
3 x = 1 or 3 x =2 lg 2 x = 0 or x = lg 3 = 0.631
8a
8b
9a
2
2
Perpendicular height of isos ∆ = 13x 5 x = 12x. 1 10 x 12 x h 3840 2 64 h 2 x 1 A = 2h(13x ) h(10 x ) 10 x 12 x 2 2304 = 60 x 2 x dA 2304 2 120 x 0 dx x 3 x 19.2 x 2.68 d 2 A 4608 3 120 > 0 dx 2 x min A = 1290 cm2 1 Tr 1 10 C r (2 x 2 )10 r 3 3x
r
r
1 C r (2) x 205r 3 20 5r 5 r 5 5 1 coeff. = 10 C 5 (2) 5 3 896 = 27 6 (1 ax) 1 6ax 6C 2 a 2 x 2 ... 10 r
10
9b
(1 bx )(1 ax) 6 (1 bx)(1 6ax 6C 2 a 2 x 2 ...) x : 6a b 0 x 2 : 15a 2 6ab a2
1 9
1 a , b 2 3
7 3
10a
Centre = mid-pt = (0, -1) 32 Radius = 2 2 2 ( x 0) 2 ( y 1) 2 8 x2 y2 2y 7 0
10b
11a
( x 4) 2 ( y 1) 2 5 y x 2 8 x 11 5 dy x4 dx x 2 8 x 11 dy 1 1 x 3, or dx 4 2 1 3 (3) c 2 3 c 2 x 3 y 2 2 y a(1 x) n lg y lg a n lg(1 x ) plot lg y against lg(1 + x) lg(1 + x) lg y
0.301 0.5416
Gradient = normal ( x 4) 2 ( y 1) 2 5 centre = (4, 1) to circ = (3, 3) gradient = -2 dy 1 dx 2 1 3 (3) c 2 3 c 2 x 3 y 2 2
0.477 0.683
0.602 0.782
0.699 0.860
0.778 0.924
lg y
lg y lg a n lg(1 x )
lg(1 + x)
From graph, gradient = n = 0.8, Y-intercept = lg a = 0.3, a = 2.
11b
12
2 1 xy 2 2 5x 2 2 y 3 x 5x 2 ( x k ) 2( x k ) 1 4 x 0 x 2 ( 2k 6) x k 2 2 k 1 0
13 13ai
(2k 6) 2 4(k 2 2k 1) < 0 16k 32 < 0 2
( ) 2 2 = 2 2 2 36 4 4 8 13aii ( ) 3 3 3 2 3 2 3 3 3 ( ) 3 3 ( ) 6 3 3(2)(6) 180 13b New sum = 2 2 =8 1 new product = 4 4 1 new eqn: x 2 8 x 0 4