MT Past Year Question P1 P2 P3 PDF

Preview

Summary

STPM Mathematics (T) Past Year Questions Lee Kian Keong & LATEX [email protected] http://www.facebook.com/akeong Last Edited by April 27, 2014 Abstract This is a document which shows all the STPM questions from year 2013 to year 2013 using LATEX. Students should use this document as reference and...

Description

STPM Mathematics (T) Past Year Questions Lee Kian Keong & LATEX [email protected] http://www.facebook.com/akeong Last Edited by April 27, 2014

Abstract This is a document which shows all the STPM 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 email1 or facebook2 . Students also encourage to send me your collection of papers or questions by email because i am collecting various type of papers. All papers are welcomed.

Contents 1 Mathematics (T) Specimen SPECIMEN PAPER Paper 1 . SPECIMEN PAPER Paper 2 . SPECIMEN PAPER Paper 3 . 2 STPM Mathematics (T) STPM 2013 Paper 1 . . . . STPM 2013 Paper 1 (U1) . STPM 2013 Paper 2 . . . . STPM 2013 Paper 2 (U2) . STPM 2013 Paper 3 . . . .

Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2013 . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

2 2 4 6 9 9 11 13 15 17

3 STPM Mathematics (T) 2014 20 STPM 2014 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1 2

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

1

Mathematics (T) Specimen Paper

1

Lee Kian Keong

Mathematics (T) Specimen Paper

SPECIMEN PAPER Paper 1 Section A [45 marks] Answer all questions in this section. 1. The functions f and g are defined by f : x 7→ e2x , x ∈ R; g : x 7→ (ln x)2 , x > 0. (a) Find f −1 and state its domain. [3 marks]   1 (b) Show that g = g(2), and state, with a reason, whether g has an inverse. [4 marks] 2 2. A sequence is defined by ur = e−(r−1) − e−r for all integers r ≥ 1. Find n, and deduce the value of

∞ X

n X

ur in terms of

r=1

ur .

[5 marks]

r=1

   −1 1 0 2 −2 0 0 −1 are such that PQ = QP. 3. The matrices P = 0 0 2 and Q =  0 0 −2 2 a b c 

(a) Determine the values of a, b and c.

[5 marks]

(b) Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix. [5 marks] 4. Express the complex number z = 1 − 1 1 Hence, find z 5 + 5 and z 5 − 5 . z z

√

3i in polar form.

[4 marks] [4 marks]

5. The equation of a hyperbola is 4x2 − 9y 2 − 24x − 18y − 9 = 0. (a) Obtain the standard form for the equation of the hyperbola.

[3 marks]

(b) Find the vertices and the equations of the asymptotes of the hyperbola.

[6 marks]

6. Find the equation of the plane which contains the straight line x − 3 = is perpendicular to the plane 3x + 2y − z = 3.

2

z+1 y−4 = and 3 2

[6 marks]

Mathematics (T) Specimen Paper

Lee Kian Keong

SPECIMEN PAPER Paper 1

Section B [15 marks] Answer any one question in this section. 1 7. Express cos x + sin x in the form r cos(x − α), where r > 0 and 0 < α < π. Hence, find 2 the minimum and maximum values of cos x + sin x and the corresponding values of x in the interval 0 ≤ x ≤ 2π. [7 marks] (a) Sketch the graph of y = cos x + sin x for 0 ≤ x ≤ 2π.

[3 marks]

(b) By drawing appropriate lines on your graph, determine the number of roots in the interval 0 ≤ x ≤ 2π of each of the following equations. 1 i. cos x + sin x = − [1 marks] 2 ii. cos x + sin x = 2 [1 marks] (c) Find the set of values of x in the interval 0 ≤ x ≤ 2π for which | cos x + sin x| > 1.

[3 marks]

8. The position vectors a, b and c of three points A, B and C respectively are given by a = i + j + k, b = i + 2j + 3k, c = i − 3j + 2k. (a) Find a unit vector parallel to a + b + c.

[3 marks]

(b) Calculate the acute angle between a and a + b + c.

[3 marks]

(c) Find the vector of the form i + λj + µk perpendicular to both a and b.

[2 marks]

(d) Determine the position vector of the point D which is such that ABCD is a parallelogram having BD as a diagonal. [3 marks] (e) Calculate the area of the parallelogram ABCD.

3

[4 marks]

Mathematics (T) Specimen Paper

Lee Kian Keong

SPECIMEN PAPER Paper 2

SPECIMEN PAPER Paper 2 Section A [45 marks] Answer all questions in this section. 1. The function f is defined by f (x) =

(√

x + 1, |x| − 1,

x ≥ −1; otherwise.

(a) Find lim f (x).

[3 marks]

(b) Determine whether f is continuous at x = −1.

[2 marks]

x→−1

2. Find the equation of the normal to the curve with parametric equations x = 1 − 2t and 2 [6 marks] y = −2 + at the point (3, −4). t 2

3. Using the substitution x = 4 sin u, evaluate R

Z

0

1r

x dx. 4−x

x2 . x−1 Hence, find the particular solution of the differential equation

4. Show that e

x−2 dx x(x−1)

=

[6 marks]

[4 marks]

dy x−2 1 + y=− 2 dx x(x − 1) x (x − 1) which satisfies the boundary condition y =

3 when x = 2. 4

[4 marks]

 3  3  5 d2 y dy dy d3 y 2 dy = x = and + 3x . [5 marks] dx2 dx dx3 dx dx Using Maclaurin’s theorem, express sin−1 x as a series of ascending powers of x up to the term in x5 . State the range of values of x for which the expansion is valid. [7 marks]

5. If y = sin−1 x, show that

6. UseZ the trapezium rule with subdivisions at x = 3 and x = 5 to obtain an approximation 7 x3 dx, giving your answer correct to three places of decimals. [4 marks] to 4 1 1+x By evaluating the integral exactly, show that the error of the approximation is about 4.1%. [4 marks]

4

Mathematics (T) Specimen Paper

Lee Kian Keong

SPECIMEN PAPER Paper 2

Section B [15 marks] Answer any one question in this section. 7. A right circular cone of height a + x, where −a ≤ x ≤ a, is inscribed in a sphere of constant radius a, such that the vertex and all points on the circumference of the base lie on the surface of the sphere. 1 (a) Show that the volume V of the cone is given by V = π(a − x)(a + x)2 . [3 marks] 3 (b) Determine the value of x for which V is maximum and find the maximum value of V . [6 marks]

(c) Sketch the graph of V against x.

[2 marks]

1 1 (d) Determine the rate at which V changes when x = a if x is increasing at a rate of a 2 10 per minute. [4 marks] 8. Two iterations suggested to estimate a root of the equation x3 −4x2 +6 = 0 are xn+1 = 4− 1 1 and xn+1 = (x3n + 6) 2 . 2

(a) Show that the equation x3 − 4x2 + 6 = 0 has a root between 3 and 4.

6 x2n

[3 marks]

(b) Using sketched graphs of y = x and y = f (x) on the same axes, show that, with initial approximation x0 = 3, one of the iterations converges to the root whereas the other does not. [6 marks] (c) Use the iteration which converges to the root to obtain a sequence of iterations with x0 = 3, ending the process when the difference of two consecutive iterations is less than 0.05. [4 marks] (d) Determine whether the iteration used still converges to the root if the initial approximation is x0 = 4. [2 marks]

5

Mathematics (T) Specimen Paper

Lee Kian Keong

SPECIMEN PAPER Paper 3

SPECIMEN PAPER Paper 3 Section A [45 marks] Answer all questions in this section. 1. The number of ships anchored at a port is recorded every week. The results for 26 particular weeks are as follows: 32 26 (a) (b) (c) (d)

28 27

43 38

21 42

35 18

19 37

25 50

45 46

35 23

32 40

18 20

26 29

30 46

Display the data in a stem-and-leat diagram. [2 marks] Find the median and interquartile range. [4 marks] Draw a box-and-whisker plot to represent the data. [3 marks] State the shape of the frequency distribution, giving a reason for your answer.[2 marks]

2. The events A and B are such that P(A) 6= 0 and p(B) 6= 0. (a) Show that P(A′ |B) = 1−P(A|B). (b) Show that P(A′ |B) =P(A′ ) if A and B are independent.

[2 marks] [3 marks]

3. The number of defective electrical components per 1000 components manufactured on a machine may be modelled by a Poisson distribution with a mean of 4. (a) Calculate the probability that there are at most 3 defective electrical components in the next 100 components manufactured on the machine. [3 marks] (b) State the assumptions that need to be made about the defective electrical components in order that the Poisson distribution is a suitable model. [2 marks] 4. The masses of bags of flour produced in a factory have mean 1.004 kg and standard deviation 0.006 kg. (a) Find the probability that a randomly selected bag has a mass of at least 1 kg. State any assumptions made. [4 marks] (b) Find the probability that the mean mass of 50 randomly selected bags is at least 1 kg. [4 marks]

5. The proportion of fans of a certain football club who are able to explain the offside rule correctly is p. A random sample of 9 fans of the football club is selected and 6 fans are able to explain the offside rule correctly. Test the null hypothesis H0 : p = 0.8 against the alternative hypothesis Hl : p < 0.8 at the 10% significance level. [6 marks] 6. It is thought that there is an association between the colour of a person’s eyes and the reaction of the person’s skin to ultraviolet light. In order to investigate this, each of a random sample of 120 persons is subjected to a standard dose of ultraviolet light. The degree of the reaction for each person is noted, ”-” indicating no reaction, ”+” indicating a slight reaction and ”++” indicating a strong reaction. The results are shown in the table below. 6

Mathematics (T) Specimen Paper ❵❵❵

❵❵❵

Reaction

Lee Kian Keong

colour ❵❵Eye Blue ❵❵❵ ❵❵❵

+ ++

7 29 21

SPECIMEN PAPER Paper 3

Grey or green

Brown

8 10 9

18 16 2

Test whether the data provide evidence, at the 5% significance level, that the colour of a person’s eyes and the reaction of the person’s skin to ultraviolet light are independent. [10 marks]

7

Mathematics (T) Specimen Paper

Lee Kian Keong

SPECIMEN PAPER Paper 3

Section B [15 marks] Answer any one question in this section. 7. A random variable T , in hours, represents the life-span of a thermal detection system. The probability that the system fails to work at time t hour is given by 21

P (T < t) = 1 − e− 5500 t . (a) Find the probability that the system works continuously for at least 250 hours.[3 marks] (b) Calculate the median life-span of the system.

[3 marks]

(c) Find the probability density function of the life-span of the system and sketch its graph. [4 marks]

(d) Calculate the expected life-span of the system.

[5 marks]

8. A random sample of 48 mushrooms is taken from a farm. The diameter in centimetres, of 48 48 X X each mushroom is measured. The results are summarised by xi = 300.4 and xi 2 = i=1

i=1

2011.01.

(a) Calculate unbiased estimates of the population mean and variance of the diameters of the mushrooms. [3 marks] (b) Determine a 90% confidence interval for the mean diameter of the mushrooms.[4 marks] (c) Test, at the 10% significance level, the null hypothesis that the mean diameter of the mushrooms is 6.5cm. [6 marks] (d) Relate the confidence interval obtained in (b) with the result of the test in (c).[2 marks]

8

STPM Mathematics (T) 2013

2

Lee Kian Keong

STPM Mathematics (T) 2013

STPM 2013 Paper 1 Section A [45 marks] Answer all questions in this section. 1. Sketch the graph of y = sin 2x in the range 0 ≤ x ≤ π. Hence, solve the inequality 1 | sin 2x| < , where 0 ≤ x ≤ π. [6 marks] 2 2. A sequence a1 , a2 , a3 , . . . is defined by an = 3n2 . The difference between successive terms of the sequence forms a new sequence b1 , b2 , b3 , . . .. (a) Express bn in terms of n.

[2 marks]

(b) Show that b1 , b2 , b3 , . . . is an arithmetic sequence, and state its first term and common difference. [3 marks] (c) Find the sum of the first n terms of the sequence b1 , b2 , b3 , . . . in terms of an and bn . [2 marks]

3. A system of linear equations is given by x + y + z = k, x − y + z = 0, 4x + 2y + λz = 3. where λ and k are real numbers. Show that the augmented matrix for the system may be reduced to   1 1 1 k  0 −2 −k  . 0 0 0 λ − 4 3 − 3k

[5 marks]

Hence, determine the values of λ and k so that the system of linear equations has (a) a unique solution,

[1 marks]

(b) infinitely many solutions,

[1 marks]

(c) no solution.

[1 marks]

4. The complex number z is given by z = 1 + (a) Find |z| and arg z.

√

3i.

√ (b) Using de Moivre’s theorem, show that z = 16 − 16 3i. (c) Express

z4

z∗

5

in the form x + yi, where z ∗ is the conjugate of z and x, y ∈ R.

9

[3 marks] [3 marks] [3 marks]

STPM Mathematics (T) 2013

Lee Kian Keong

STPM 2013 Paper 1

c 5. Show that the parametric equations x = ct and y = , where c is a constant, define a point t on the rectangular hyperbola xy = c2 . [2 marks] The points P , Q, R and S, with parameters p, q, r and s respectively, lie on the rectangular hyperbola xy = c2 . (a) Show that pqrs = −1 if the chords P Q and RS are perpendicular.

[4 marks]

(b) Find the equation of the line passing through the points P and Q. Deduce the equation of the tangent to the rectangular hyperbola at the point P . [4 marks] 6. Show that the point A(2, 0, 0) lies on both planes 2x − y + 4z = 4 and x − 3y − 2z = 2. Hence, find the vector equation of the line of intersection of both planes. [5 marks] Section B [15 marks] Answer any one question in this section. 7. The polynomial p(x) = hx4 + kx3 + 2x − 1, where h and k are constants, leaves a remainder of 4 when divided by x − 1, and a remainder of −2 when divided by x + 1. (a) Determine the values of h and k.

[3 marks] 2

(b) Express the polynomial p(x) in the form (x − 1)q(x) + r(x), where q(x) is quadratic and r(x) is linear. [4 marks] (c) Express q(x) in a completed square form a(x + b)2 + c. i. Deduce that q(x) is always positive for all real values of x. ii. Deduce the minimum value of q(x) and the corresponding value of x. (d) Determine the set of values of x for which p(x) > 3x + 1.

[2 marks] [1 marks] [2 marks] [3 marks]

−→ −−→ 8. A tetrahedron OABC has a base OAB and a vertex C, with OA = 2i+j+k, OB = 4i−j+3k −−→ and OC = 2i − j − 3k. −−→ −→ −−→ (a) Show that OC is perpendicular to both OA and OB.

[3 marks]

◦

(b) Calculate, to the nearest 0.1 , the angle between the edge AC and base OAB of the tetrahedron. [5 marks] (c) Calculate the area of the base OAB and the volume of the tetrahedron.

10

[7 marks]

STPM Mathematics (T) 2013

Lee Kian Keong

STPM 2013 Paper 1 (U1)

STPM 2013 Paper 1 (U1) Section A [45 marks] Answer all questions in this section. 1. Sketch, on the same axes, the graphs of y = |2x + 1| and y = 1 − x2 . Hence, solve the inequality |2x + 1| ≥ 1 − x2 . [8 marks] √ √ √ √ 6 6 6 6 2. Use the binomial expansions √ of ( 63 + 2) and ( 3 − 2) to evaluate ( 3 + 2) + ( 3 − 2) . Hence, show that 2701 < ( 3 + 2) < 2702. [7 marks] 3. A matrix P is given by  −5 0 2 P =  0 2 −1 . −1 4 −2 

By using elementary row operations, find the inverse of P. 4. Express √ √ the complex number 6 − i 2.

√

[5 marks]

√ 6 − i 2 in polar form. Hence, solve the equation z 3 =

[9 marks]

5. The equation of an ellipse is 3x2 + y 2 + 30x + 10y + 79 = 0. (a) Obtain the standard form for the equation of the ellipse.

[3 marks]

(b) Find the coordinates of the centre C, the focus F1 , and the focus F2 of the ellipse. [4 marks]

(c) Sketch the ellipse, and indicate the points C, F1 and F2 on the ellipse.

[2 marks]

6. Three vectors a = pi + qj, b = −5i + j and c = 4+7j are such that a and b are perpendicular and the scalar product of a and c is 78. (a) Determine the values of p and q.

[4 marks]

(b) Find the angle between a and c.

[3 marks]

11

STPM Mathematics (T) 2013

Lee Kian Keong

STPM 2013 Paper 1 (U1)

Section B [15 marks] Answer any one question in this section. 7. (a) Show that for a fixed number x 6= 1, 3x2 + 3x3 + . . . + 3n is a geometric series, and find its sum in terms of x and n. [4 marks] (b) The series Tn (x) is given by Tn (x) = x + 4x2 + 7x3 + . . . + (3n − 2)xn , for x 6= 1. By considering Tn (x) − xTn (x) and using the result from (a), show that Tn (x) =

x + 2x2 − (3n + 1)xn+1 + (2n − 2)xn+2 . (1 − x)2 [5 marks]

Hence, find the value of

20 X r=1

2r (3r − 2), and deduce the value of

19 X

2r+2 (3r + 1).[6 marks]

r=0

8. A parallelepiped for which OABC, DEF G, ABF E and OCGD are rectangles is shown in the diagram below.

−→ −−→ The unit vectors i and j are parallel to OA and OC respectively, and the unit vector k is −→ −−→ −−→ perpendicular to the plane OABC, where O is the origin. The vectors OA, OB and OD are 4i, 4i + 3j and i + 5k respectively. √ 13 35 (a) Show that cos ∠BEG = . [6 marks] 175 (b) Calculate the area of the triangle AEG. [6 marks] (c) Find the equation of the plane AEG.

12

[3 marks]

STPM Mathematics (T) 2013

Lee Kian Keong

STPM 2013 Paper 2

STPM 2013 Paper 2 Section A [45 marks] Answer all questions in this section. 1. The function f is defined by  4  √ ,   √ 4 − x f (x) = 2,   x  √ , 1+x−1

x < 0, x = 0, x > 0.

(a) Show that lim f (x) exists.

[5 marks]

(b) Determine whether f is continuous at x = 0.

[2 marks]

x→0

2. The parametric equations of a curve are x = θ − sin θ and y = 1 − cos θ. Find the equation 1 of the normal to the curve at a point with parameter π. [7 marks] 2 3. The equations of two curves are given by y = x2 − 1 and y =

6 . x2

(a) Sketch the two curves on the same coordinate axes.

[3 marks]

(b) ...