2013 stpm mathematics m past year questions p1 p2 p3

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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 year2013 using LATEX. Students should use this document as reference and try all the questions if possible.

Students are encourage to contact me via email1

or facebook2

. Students also encourage to send meyour collection of papers or questions by email because i am collecting various type of papers. Allpapers are welcomed.

Contents

1 PAPER 1 QUESTIONS 2

SPECIMEN PAPER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2STPM 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 PAPER 1 ANSWER 7

STPM 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

[email protected]://www.facebook.com/akeong

1
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PAPER 1 QUESTIONS Lee Kian Keong

1 PAPER 1 QUESTIONS

SPECIMEN PAPER

Section A [45 marks]

Answerall questions in this section.

1. The function f is defined by f(x) = ln(1 2x), x < 0.(a) Find f1, and state its domain. [3 marks]

(b) Sketch, on the same axes, the graphs of f and f1. [4 marks]

(c) Determine whether there is any value ofx for which f(x) = f1(x). [3 marks]

2. The sequence u1, u2, u3, . . . is defined by un+1 = 3un, u1 = 2.

(a) Write down the first five terms of the sequence. [2 marks]

(b) Suggest an explicit formula for ur. [2 marks]

3. Using an augmented matrix and elementary row operations, find the solution of the system of equa-tions

3x 2y 5z = 5,x + 3y 2z = 6,

5x 4y + z = 11.[9 marks]

4. Find the gradients of the curve y3 + y = x3 + x2 at the points where the curve meets the coordinateaxes. [6 marks]

5. Show that e

1

x ln x dx =1

4

e2 + 1

.

[4 marks]

Hence, find the value of

e

1

x(ln x)2 dx. [3 marks]

6. The variables x and y, where x, y > 0, are related by the differential equation

dydx

+ y2 = 2yx

.

Using the substitution y =u

x2, show that the differential equation may be reduced to

du

dx= u

2

x2.

[3 marks]

Solve this differential equation, and hence, find y, in terms of x, with the condition that y = 1 whenx = 1. [6 marks]

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PAPER 1 QUESTIONS Lee Kian Keong SPECIMEN PAPER

Section B [15 marks]

Answer anyone question in this section.

7. Expand (1+ x)2

3 and 1 + ax1 + bx

, where |b| < 1, in ascending powers ofx up to the term in x3. Determinethe 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,

(a) determine the values ofa and b, [3 marks]

(b) use x =1

8to obtain the approximation

3

81 21249

. [3 marks]

(c) find, correct to five decimal places, the difference between the terms in x3 for the two expansions

with x =1

8. [2 marks]

8. Sketch, on the same axes, the curve y2 = x and the straight line y = 2 x, showing the coordinatesof the points of intersection. [4 marks]

(a) State whether the curve y2 = x has a turning point. Justify your answer. [2 marks]

(b) Calculate the area of the region bounded by the curve y2 = 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 y2 = xand the straight line y = 2 x completely about the x-axis. [5 marks]

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PAPER 1 QUESTIONS Lee Kian Keong STPM 2013

STPM 2013

Section A [45 marks]

Answerall questions in this section.

1. The function f is defined as f(x) =1

2

ex ex, 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 0.1318(= 0.13181818 . . .) as the sum of a constant and a geometricseries. Hence, express the recurring decimal as a fraction in its lowest terms. [4 marks]

3. Given that matrix M = 5 4

2

4 5 22 2 2

.Show that there exist non-zero constants a and b such that M2 = aM + bI, where I is the 3 3identity matrix. [6 marks]

Hence, find the inverse of the matrix M. [3 marks]

4. Given that f(x) =3x2 + x

3x2 8x 3 . Find limx13

f(x) and limx

f(x). [4 marks]

5. Show that

e1

ln x

x5 dx =1

16

1 5

e4

.

[6 marks]

6. The variables x and y, where x, y > 0 are related by the differential equation

xydy

dx+ y2 = 3x4.

Show that the substitution u =y

x2transforms the above differential equation into

x

du

dx = 31

u2

u

,

and find u2 in terms of x. [9 marks]

Hence, find the particular solution of the original differential equation which satisfies the conditiony = 2 when x = 1. [3 marks]

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Section B [15 marks]

Answer anyone question in this section.

7. (a) Express 1(r2 1) in partial fractions, and deduce that

1

r(r2 1) 1

2

1

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, Sn1, of theseries

1

2 3 +1

3 8 +1

4 15 + . . . +1

r(r2 1) + . . . ,

and deduce Sn. [6 marks]

(b) Explain why the series converges to 14

, and determine the smallest value of n such that

1

4 Sn < 0.0025.

[5 marks]

8. The graph ofy =3x 1

(x + 1)3is shown below.

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. [9 marks]

Hence, state

i. the set of values of x when dydx

0, [1 marks]

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ii. the intervals where the graph is concave upward. [1 marks]

(c) Using the above graph of y =3x 1

(x + 1)3, determine the set of values of k for which the equation

3x

1

k(x + 1)3 = 0

i. has three distinct real roots, [2 marks]

ii. has only one positive root. [1 marks]

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PAPER 1 ANSWER Lee Kian Keong

2 PAPER 1 ANSWER

STPM 2013

1. Solution

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PAPER 1 ANSWER Lee Kian Keong STPM 2013

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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2013 stpm mathematics m past year questions p1 p2 p3

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