STPM Mathematics (M) Past Year Questions Lee Kian Keong & LATEX
[email protected] http://www.facebook.com/akeong Last Edited by November 8, 2012 Abstract
This is a document which shows all the STPM Mathematics (M) questions from year 2013 to year 2013 using LATEX. Students should use this document as reference and try all the questions if possible. Students are encourage to contact me via emai l1 or facebook2 . Students also encourage to send me your your collectio collection n of papers or questi questions ons by email because because i am collec collecting ting vario various us type of papers. papers. All papers are welcomed. welcomed.
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PAPER APER 1 QUES QUESTIO TIONS NS
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SPECIMEN PAPER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . STPM 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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PAPER APER 1 ANSWE ANSWER R
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STPM 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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[email protected]
2
http://www.facebook.com/akeong
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PAPER 1 QUESTIONS
1
Lee Kian Keong
PAPER 1 QUESTIONS
SPECIMEN PAPER Section A [45 marks]
Answer all questions in this section.
1. The function f is defined by f (x) = ln(1 −1
(a) Find f
− 2x), x < 0.
, and state its domain.
[3 marks]
(b) Sketch, on the same axes, the graphs of f and f 1 . −
[4 marks]
(c) Determine whether there is any value of x for which f (x) = f 1 (x). −
2. The sequence u1 , u2, u3 , . . . is defined by u
+1
n
[3 marks]
= 3u , u1 = 2. n
(a) Write down the first five terms of the sequence.
[2 marks]
(b) Suggest an explicit formula for u .
[2 marks]
r
3. Using an augmented matrix and elementary row operations, find the solution of the system of equations 3x 2y 5z = 5, x + 3y 2z = 6, 5x 4y + z = 11.
− −
− −
− −
[9 marks]
4. Find the gradients of the curve y 3 + y = x3 + x2 at the points where the curve meets the coordinate axes. [6 marks] 5. Show that
e
x ln x dx =
1
1 2 e +1 . 4
[4 marks] e
Hence, find the value of
x(ln x)2 dx.
[3 marks]
1
6. The variables x and y , where x, y > 0, are related by the differential equation dy + y2 = dx
Using the substitution y =
− 2xy .
u , show that the differential equation may be reduced to x2 du = dx
− ux
2 2
. [3 marks]
Solve this differential equation, and hence, find y , in terms of x, with the condition that y = 1 when x = 1. [6 marks]
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PAPER 1 QUESTIONS
Lee Kian Keong
SPECIMEN PAPER
Section B [15 marks]
Answer any one question in this section.
1 + ax , where b < 1, in ascending powers of x up to the term in x3 . Determine 1 + bx the set of values of x for which both the expansions are valid. [7 marks] If the two expansions are identical up to the term in x2, 2 3
7. Expand (1+ x) and
||
(a) determine the values of a and b, [3 marks] 1 212 (b) use x = to obtain the approximation 81 . [3 marks] 8 49 (c) find, correct to five decimal places, the difference between the terms in x3 for the two expansions 1 with x = . [2 marks] 8
√ 3
≈
8. Sketch, on the same axes, the curve y2 = x and the straight line y = 2 of the points of intersection.
− x, showing the coordinates
(a) State whether the curve y 2 = x has a turning point. Justify your answer. 2
[4 marks] [2 marks]
(b) Calculate the area of the region bounded by the curve y = x and the straight line y = 2
− x.
[4 marks]
(c) Calculate the volume of the solid formed by revolving the region bounded by the curve y 2 = x and the straight line y = 2 x completely about the x-axis. [5 marks]
−
3
PAPER 1 QUESTIONS
Lee Kian Keong
STPM 2013
STPM 2013 Section A [45 marks]
Answer all questions in this section.
1. The function f is defined as f (x) =
1 e 2
x
−x
−e
, where
x
∈ R.
(a) Show that f has an inverse.
[3 marks]
(b) Find the inverse function of f , and state its domain.
[7 marks]
˙ 2. Write the infinite recurring decimal 0.131˙ 8(= 0.13181818 . . .) as the sum of a constant and a geometric series. Hence, express the recurring decimal as a fraction in its lowest terms. [4 marks]
3. Given that matrix M
5 = 4
4 5 2
−2 −
−2 −2. 2
Show that there exist non-zero constants a and b such that M2 = aM + bI, where I is the 3 3 identity matrix. [6 marks] Hence, find the inverse of the matrix M. [3 marks]
×
3x2 + x 4. Given that f (x) = 2 . Find lim f (x) and lim f (x). 3x 8x 3
− −
5. Show that
x→−
e
1
ln x x5
1 3
[4 marks]
x→∞
1 dx = 16
1
5
−e
4
. [6 marks]
6. The variables x and y , where x, y > 0 are related by the differential equation xy
Show that the substitution u =
dy + y 2 = 3 x4 . dx
y transforms the above differential equation into x2 du x =3 dx
u2
1 − u
,
and find u2 in terms of x. [9 marks] Hence, find the particular solution of the original differential equation which satisfies the condition y = 2 when x = 1. [3 marks]
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PAPER 1 QUESTIONS
Lee Kian Keong
STPM 2013
Section B [15 marks]
Answer any one question in this section.
7. (a) Express
1 (r 2
− 1) in partial fractions, and deduce that 1 r(r2
− 1) ≡
1 1 2 r (r 1)
− −
1 . r(r + 1) [4 marks]
Hence, use the method of differences to find the sum of the first ( n 1) terms, S series 1 1 1 1 + + + ... + +..., r(r2 1) 2 3 3 8 4 15
−
×
and deduce S .
×
×
1
n−
, of the
−
[6 marks]
n
(b) Explain why the series converges to
1 , and determine the smallest value of n such that 4 1 4
− S
n
< 0.0025. [5 marks]
8. The graph of y =
3x 1 is shown below. (x + 1)3
−
The graph has a local maximum at A and a point of inflexion at B . (a) Write the equations of the asymptotes of the graph.
[1 marks]
(b) Determine the coordinates of the points A and B . Hence, state
[9 marks]
i. the set of values of x when
dy dx
≥ 0,
[1 marks]
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PAPER 1 QUESTIONS
Lee Kian Keong
STPM 2013
ii. the intervals where the graph is concave upward. [1 marks] 3x 1 (c) Using the above graph of y = , determine the set of values of k for which the equation (x + 1)3 3x 1 k (x + 1)3 = 0
−
− −
i. has three distinct real roots, ii. has only one positive root.
[2 marks] [1 marks]
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PAPER 1 ANSWER
2
Lee Kian Keong
PAPER 1 ANSWER
STPM 2013 1. Solution
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2. Solution
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3. Solution
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4. Solution
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5. Solution
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6. Solution
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7. Solution
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8. Solution
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