2014 2 SELANGOR BandarUtama PJ MATHS QA
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CONFIDENTIAL 954/2 2014-2-SGOR-BandarUtamaPJ_MATHS QA TanSiangBeng Section A [ 45 marks ] Answer all questions in this section. 1. The function f : x 2 - x is defined in the domain x . Express f(x) in terms of x when (a) x 0. [ 2 ] (b) x < 0. [ 2 ] Determine whether f is continuous or not at x = 0 and sketch the graph of f. [ 3 ] 2. The curve C is defined parametrically by x = 4 t 2 , and y = t t ln where t > 0. (a) Find dx dy in terms of t and the value of t when dx dy = 0. [ 6 ] (b) Explain briefly why t t ln 0 when t . [ 1 ] (c) Deduce the nature of the stationary value in part (a). [ 2 ] 3. Using the substitution u = sin x, show that xdx sec = du u 2 1 1 . [ 3 ] Show that ) 3 ln( 2 1 sec 6 0 dx x . [ 4 ] 4. Show that 3 2 ln 14 ln 4 2 xdx x [ 7 ] 5. Consider the differential equation x 1 2 2 2 x x y dx dy . (a) Find the integrating factor for this differential equation. [ 3 ] (b) Given that y = 1 when x = 1, solve the differential equation in the form y = f(x). [ 4 ] 6. (a) Use the substitution u = x y to solve the differential equation 0 2 2 2 x y dx dy xy . [ 4 ] (b) Solve the differential equation x x y dx dy 2 sin tan , given that y = 1 when x = 0. [ 4 ]
Transcript of 2014 2 SELANGOR BandarUtama PJ MATHS QA
CONFIDENTIAL 954/2
2014-2-SGOR-BandarUtamaPJ_MATHS QA TanSiangBeng
Section A [ 45 marks ]
Answer all questions in this section.
1. The function f : x 2 - x is defined in the domain x .
Express f(x) in terms of x when (a) x 0. [ 2 ] (b) x < 0. [ 2 ] Determine whether f is continuous or not at x = 0 and sketch the graph of f. [ 3 ]
2. The curve C is defined parametrically by x = 4t 2 , and y = t
tlnwhere t > 0.
(a) Find dx
dy in terms of t and the value of t when
dx
dy= 0. [ 6 ]
(b) Explain briefly why t
tln 0 when t . [ 1 ]
(c) Deduce the nature of the stationary value in part (a). [ 2 ]
3. Using the substitution u = sin x, show that xdxsec = du
u 21
1. [ 3 ]
Show that )3ln(2
1sec6
0 dxx
. [ 4 ]
4. Show that 32ln14ln4
2 xdxx [ 7 ]
5. Consider the differential equation
x1
22
2
x
xy
dx
dy .
(a) Find the integrating factor for this differential equation. [ 3 ] (b) Given that y = 1 when x = 1, solve the differential equation in the form y = f(x). [ 4 ]
6. (a) Use the substitution u = x
yto solve the differential equation 02 22 xy
dx
dyxy .
[ 4 ]
(b) Solve the differential equation xxydx
dy2sintan , given that y = 1 when x = 0.
[ 4 ]
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Section B [ 15 marks ]
Answer any one question in this section.
7. Given that y = ln cos x, show that the first two non-zero terms of the Maclaurin series for y is
y = .122
42 xx Use this series to find the approximation in terms of for ln 2. [ 15 ]
8. By sketching a pair of graphs, show that the equation x = sin 2x has exactly one root in the
interval 4
1< x <
2
1. [ 6 ]
Using the iterative formula xn+1 = sin (2xn) and initial value x1 = 1, calculate successive values x2, x3, x4, x5, x6 and x7 correct to four decimal places. [ 5 ]
Show that the root in the interval 4
1< x <
2
1is 0.95, correct to two decimal places.[ 4 ]
END OF QUESTION PAPER
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