1 CONFIDENTIAL*
2
Section A[45
marks] marks]
Answer all questions in this section. section.
x − k , x < −3 + − 3 ≤ x < 0 x p, e , x ≥ 0 1The function his defined by h( x) x) = , 2
x − q
a) deter determi mine ne the the a! a!ue ue of p of p if
!im h ( x ) # →"1
=2
[2marks] marks]
!im h ( x ) !im h ( x ) b) find the a!ues of k and q if # →−3 and # →0 e# e#ist. c) $enc $ence, e,s% s%etc etch h the the &ra'h &ra'h of of y= y= h(#).
[4marks] marks] [3 marks] marks]
t the instant hen the de'th of ater in a !ar&e ater stora&e container is xmeter, xmeter, the π x 3 V = 3 12 . *f the o!ume o!ume, V ,in ,in m , of the ater is &ien by o!ume of ater in the container is 3 "1 increasin& at the constant rate of 3m s , ca!cu!ate, in terms of +, a) the rate rate of of increas increasee in de't de'th h of ate aterr in ms ms"1 at the instant hen the de'th de'th is 3m [3marks [3marks]] b) the time ta%en in seconds for the de'th to increase from 5m to 10m. 10m. [3 marks] marks] 2
x 2 + 2 x − 14 3
#'ress ( x − 3)( x + 5) in 'artia! fractions. 5
$ence, sho that 4
d y d x
x 2 + 2 x − 14
∫ ( x − 3)( x + 5) 4
dx = dx = 1 -
1
/
.
!n 5 .
[marks [marks]]
y usin& substitution v = y2, sho that the non"!inear differentia! euation +
y
2 x
=
e
x
d
+
v
= 2e
x
may be reduced into the !inear differentia! euation d x x .o!e this 2 !inear differentia! euation, and hence, obtain y in terms of x of x,, &ien that y that y = = 1 hen x hen x = 1. [/marks [/marks]] 5
y
sin x) ien that y = !n( 1 + sin d 2 y
d y
−1 d x d x a) sho th that . [4marks [4marks]] b) ind the 6ac!aurin7s series for y for yin in ascendin& 'oers of x of x,, u' to and inc!udin& the 3 term x term x . [4marks [4marks]] 2
+(
)2 = e
− y
1 2 ho that x that x3 = 1 + x has a root beteen 0 and 1.8se the 9eton":a'hson method, ith the initia! a''ro#imation x a''ro#imation x0= 0., to find the root correct to to decima! '!aces. [;marks [;marks]]
6
2
954/2
9*?9T*@ unti! t he e#amination is oer. CONFIDENTIAL*
CONFIDENTIAL*
3
Section B[15
mar%s]
nser ONLY oneuestion in this section. >n
the same a#es,
(a) %etch the cures y = ; A e# and y = 5e "#
[3marks]
(b) ind the coordinates of the 'oints of intersection
[4 mar%s]
(c) a!cu!ate the area of the re&ion bounded by the cures
[4 mar%s]
(d) a!cu!ate the o!ume of the so!id formed hen the re&ion is rotated throu&h 2 π radians about the #"a#is [4mar%s]
!ien
that y = e "# sin 2# 2
(a) ho that
d y dx
2
dy dx
= " 5y "2
[4 mar%s]
(b) ind the6ac!aurin series for e"# sin 2# in ascendin& 'oers of # u' to and inc!udin& the term in # 4
[5 mar%s]
(c) $ence, find the 6ac!aurin series for e"#cos 2# in ascendin& 'oers of # u' to and inc!udin& the term in # 3 [3 mar%s] 2
(d) $ence, find the a!ue of
x
∫ sin 2 x 2
1
x e
dx
<<<<<<<<<< 9? > B8T*>9 CC: <<<<<<<<<<
[3marks]
3
