[Edu.joshuatly.com] Gerak Gempur Perak STPM 2012 Maths T [5A930C99]

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1

Answer all questions

1. Find the set of values of x such that 2x

3x [4]

2. Using the laws of the algebra of sets, show that ')'(')( QQPQP [4]

3. Show that )3( x is a factor of .63234 x x x x State another linear factor

Hence,write down one quadratic factor of .63234 x x x x , and find a second

quadratic factor of this polynomial. Find the set of values of x so that 0)( x f . [10]

4. a) The real roots of the quadratic equation 02 cbxax are and .

Prove thata

b and

a

c . [5]

b) If and are roots of the equation 0322 x x , find a quadratic equation with

the roots 12 and 12 , expressing your answer in the form 0

2 cbyay

where a, b and c are integers. [6]

5. The functions f and g are defined by

0,,:

0,,ln:

x x x xh

x x x x f

i) Show that the composite function fh( x) exists and find the function. State the range

of fh( x). [5]

ii) Determine whether the composite function hf ( x) exists. [3]

iii) Sketch the graph of fh( x) and hence determine whether fh( x) is one-to-one. [4]

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2

6.. . If ,

211

121

112

A find 2 A [2]

find the values of m and n such that 02 nI mA A , where I is 3 x 3 identity

matrix and O is 3 x 3 null matrix. [3]

Hence find 3 A and [4]

7. a) Find the complex numbers z in the form bia if i z 24102 . [6]

b) Hence, find the arguments of . z [3]

8., Express4

12 x

in the form of partial fractions. Hence, show that

.2

3ln

4

1

4

16

4

2

dx x

[7]

9. The region R, in the first quadrant, is bounded by the line x y 3 and the curve

x y

1

3 between their points of intersection. Find the coordinates of these points of

intersection, and draw a sketch showing the region R. [4]

Find the area of R. [4]

Find the volume of the solid formed when R is rotated through one revolution about

the x-axis . [4]

10. The function f is defined by f : xk e x x , and k is a positive constant.

a) State the range of f

b) Find f ( ln k ) and simplify your answer

c) Find the inverse function of f and state its domain

d) On the same axes, sketch the curves of f and 1 f . [7]




3

11. Express

2

1

2

)x21(

)x31(

in ascending powers of x until and including the term in .2 x

State the range of values of x such that the expansion is valid. [4]

12. i) Express)1r 2)(1r 2(

2

in partial fractions [3]

ii) Show that 1n2

n2

)1r 2)(1r 2(

2n

1r

[3]

iii) Find the sum of the series3129

1

75

1

53

1

31

1

and find

1r )1r 2)(1r 2(

1 [5]




4




1

1 Express 4 sin x + cos x in the form r sin (x + α), where r > 0 and 0o < α < 90o.

Hence solve the equation 4 sin x + cos x = 2 for 0 o < x < 360o. (6 marks)

2 Given that y = , prove that y= - tan2 . Find the exact value of

tan2 15o in the form p+q , where p, q and r are integers. (6 marks)

3 The forces F1= (5i + 3j)N, F2= (4i -6j)N, F3= (-2i + 7j)N act at a point.

(a)

Calculate the magnitude of the resultant force.

(b)

Using the scalar product, calculate the angle between the resultant force

and the force F4= (5i + 3j)N

(2 marks)

(4 marks)

4 A certain substance evaporates at a rate which is proportional to the amount of

substance left. Given that the initial amount of the substance is A and the amount

which has evaporated at time t is x,

(a)

write a differential equation to show the rate of evaporation.

(b) Solve the differential equation and sketch the graph of x against t.

(c) Given that it took ln seconds for half the amount to be evaporated, find

how long it takes for of the initial amount to be evaporated.

(1 marks)

(6 marks)

(4 marks)

5 In triangle ABC, the point X divides BC internally in the ratio m:n, where m+n=1.

Express AX2 in terms of AB, BC, CA, m and n. (5 marks)

6 In the triangle ABC, the point P lies on the side AC such that BPC = ABC.Show that the triangles BPC and ABC are similar.

If AB=4 cm, AC=8 cm and BP=3 cm, find the area of the triangle BPC.

(3 marks)

(4 marks)

7 The probability that it rains in a certain area is 0.2. The probability that an

accident occurs at a particular corner of a road is 0.05 if it rains and 0.02 if it does

not rain. Find the probability that it rains if an accident occurs at the corner. (5 marks)

8 (a) X is a random variable such that X~B(n,p). If E(X) = 2 and Var (X) = ,

find the values of n and p, and P(X=3).

(b)

On average, the number of books read by an adult is 2 books per year.Using the Poisson distribution, find, correct to 4 decimal places, the

probability that

(i)

an adult reads exactly 3 books per year.

(ii)

an adult reads more than 5 books in 2 years.

(5 marks)

(5 marks)




2

9 The following data presented as a stem plot are the weekly expenditure of a group

of college students.

2

3

4

5

6

7

8

2 5

0 3 4 6 8

1 3 3 7 9

1

2

2

5

0 4

3

5

Key 3| 3 means RM33

(a) Find the percentage of students who spend less than RM50.

(b) Find the mean and standard deviation of the students’ expenditure.

(c) Find the median and semi-interquartile range of the students’ expenditure.

(d) Construct a boxplot for the above data and state the shape of the data.

(2 marks)

(4 marks)

(5 marks)

(4 marks)

10 Table below shows a probability distribution of a discrete random variable, X.X 0 3 6 9 12

P(X=x) 0.2 p q q 0.1

a) Given that E(X) = 6, find the value of p and q.

b) Calculate the variance of X.

(4 marks)

(3 marks)

11 X is the random variable of a normal distribution where X~N(µ, ). If P(X>120)

= 0.0415 and P(X<90) = 0.2114, find the values of µ and

(8 marks)

12 The cumulative distribution function of a continuous random variable is given by

a) Determine the value of k.

b) Determine the lower quartile of X

c) Find the probability density function of X and sketch the graph of y = f(x).

d) Find Var (2X – 1)

(2 marks)

(2 marks)

(4 marks)

(7 marks)




1

Answer all questions

1. 2x

3x

02x

3x M1

0322

x

x x

0)1)(3(

x

x x A1

Using a table for the sign of x

x x )1)(3(

The set values of x is ),3()0,1( x A1 [4]

2. Using the laws of the algebra of sets, show that ')'(')( QQPQP [4]

)'(')( QPQP =

')

'(

')( QPQP B1

= )'()''( QPQP M1

= )'('PPQ M1

= 'Q

=

'Q

A1

[4]

‐

‐

+

‐

+

‐

+

+

(x + 1)

(x – 3)

x

)3x)(1x(

x

30-1

M1




2

3. 3 x , 6)3(3)3()3()3()3( 234 f M1

= 6333339

= 0

A1

Another linear factor is 3 x B1

Quadratic factor is )3)(3( x x = 32 x B1

)2)(3()( 22 x x x x f M1

The second quadratic factor is 22 x x A1

0)( x f

0)2)(3( 22 x x x

24

1)

2

1()2( 22 x x x M1

=

4

7)

2

1( 2 x > 0 , For all values of . x A1

0)3( 2 x M1

0)3)(3( x x

.,33, x A1 [10]

4. Equation with the roots and can be written as

0))(( x x M1

0)(2 x x … (1) A1

Quadratic equation is 02 cbxax

02 a

c x

a

b x (2) B1

Comparing the equation (1) & (2),




3

a

b )( M1 ( comparing the coefficient x and constant )

a

c A1 (for each correct answers) [5]

OR

Roots are

a

acbb x

2

42

Let a

acbb

2

42 and

a

acbb

2

42 B1

a

b

a

acbb

a

acbb

)

2

4()

2

4(

22

M1A1

a

c

a

acbb

a

acbb

)2

4)(2

4(

22

M1A1

[5]

b) For 0322 x x ; 2 , 3 B1

11 22 = 222

= 22)(2

M1

= 4 A1

)1)(1(22 = 1

2222

= 12)()(22 M1

= 12 A1

A quadratic equation with the roots 12 and 12 is

01242 y y A1 [6]




4

5. i) Range of h is ,0 B1

Domain of f is ,0 B1

Range of h = domain of f )( x fh exists. B1 (with reason)

0,,ln)( x x x x f x fh B1

Range of fh is , B1

ii) Range of f is , B1

Domain of

h is

,00,

B1

Range of f domain of g hf does not exists. B1 (with reason)

iii)

The function is not one‐to‐one B1

because any line that is drawn parallel to the x‐axis intersects the graph at 2

points

( or there are 2 values of x for the same value of y ) B1 reason) [12]

x

y

11

G1 (correct

shape)

G1 ( all correct )




5

6.. . 2 A

211

121

112

211

121

112

=

655

565

556

M1 A1

02

nI mA A

000

000

000

00

00

00

2

2

2

655

565

556

n

n

n

mmm

mmm

mmm

M1

05 m

5m

B1

026 nm

4n A1

A A A

I A A

I A A

45

45

045

23

2

2

=

211

121

112

4

655

565

556

5 M1

=

222121

212221

212122

A1




6

1

43

41

41

4

1

4

3

4

14

1

4

1

4

3

211

121

112

500

050

005

4

1

1)5(41

)5(4

1

)5(4

1

45

045

1

2

2

2

A

M A I A

I A I A

I A A

I A A

I A A

[9]

7. a i z 24102

ibia 2410)(2 M1

iabiba 2410222

)2........(..........12

)1......(..........1022

ab

ba M1

From (2) a

b12

, Subt. in (1)

102

2122 a

a M1

01442104 aa

0)82)(182( aa A1

182 a , .82 a

a is real , 182 a




7

23a A1(either one)

23a , 2223

12

b

2223 z i or 2223 i A1 [6]

b) 2223 z i

Arg. 23

22tan 1 z M1

= 0.588

rad.

A1

2223 z i

Arg.

23

22tan

1 z

= ‐2.55 rad. A1

8 Let 224

12

x

B

x

A

x

:. 1 = A(x+2) + B(x‐2) B1

x = 2, 1 = 4A ; A =

4

1

X = ‐2 1 = ‐4B ; B =

4

1

)2(4

1

)2(4

1

4

12

x x x

A1

M1




8

6

4

6

4

2)2(4

1

)2(4

1

4

1 dx x xdx

x B1

=

6

4

)2ln(4

1)2ln(

4

1

x x M1

=

6

42

2ln

4

1

x

x

Shown

2

3ln

4

1

4

6ln

4

1

2

6

8

4ln

4

1

6

2ln

4

1

8

4ln

4

1

9. x x

31

3 M1

02 x x

A1

x = 0 , 2 A1

x y 3

3

(2,1) 3

y

x

G1 curve correct

M1

A1 [7]




9

Area of R = dx x

x

2

01

33 M1

= 2

0

2

1ln32

3

x

x x A1

= 3ln326 M1

= 3ln34 A1

Volume = dx x

dx x

222

01

33

M1

=

2

0

12

0

32

1

19

339

x x

x x A1

= 3

8 M1 (subst.) A1 [12]

10. a) ,k B1

b) k k ek f k 2ln ln B1

c) yk e x M1

k y x ln

k x x f

ln:1

A1

domain is ,k B1




10

d)

11. 2

1

2 )x21()x31(

= ...)x2(

21

)2

3)(

2

1(

)x2)(

2

1(1[)x31( 22

] M1

= ...])x4(8

3x1)[x9x61( 22

= ...]2

31)[961( 22 x x x x M1

= ...x9x6x6x2

3x1 222

= ...2

3371 2 x x A1

)2

1,

2

1( x B1 [4]

k

k

k+1

k+1

f

1 f

D1 correct graph for f

D1 correct

graph

for

f 1




11

12.i))1r 2)(1r 2(

2

=

1r 2

B

1r 2

A

=)1r 2)(1r 2(

B)1r 2( A)1r 2(

B1

When2

1r ; 2 A2 1 A

When2

1r ; 2B2 1B

Therefore1r 2

1

1r 2

1

)1r 2)(1r 2(

2

A1

ii)

n

1r )1r 2)(1r 2(2 =

n

1r 1r 21

1r 21 M1

=

n

1r 1r 2

1

1r 2

1

= )0(f )n(f M1

=1n2

11

=1n2

n2

A1

iii)3129

1

75

1

53

1

31

1

=

15

1r )1r 2)(1r 2(

1 M1

=

15

1r )1r 2)(1r 2(2

21

=

1n2

n2

2

1 M1

=31

15 A1

M1




12

1r )1r 2)(1r 2(

1 =

1r

n )1r 2)(1r 2(

1lim M1

=

1n2

nlimn

=

n

12

1limn

=2

1 A1




13




1

1.

=

r2 = 42 + 12

4 sin

x + cos

x =

=2

X+14.04O = 29.02O, 150.98O

X = 15.0O, 1370O

B1

M1

M1

A1

M1

A1

2.

=

=

= ‐tan2 ( )

Let α= 30o

, ‐ tan2

( )=

=

tan 15o= 7‐4

M1

M1

A1

M1

M1

A1

3.a) FR= (5i+3j) + (4i‐6j) + (‐2i +7j) = 7i + 4j

|FR| = = 8.062 N

B1

B1

3.b) cos θ=

=

=

M1

M1

M1




3

In ΔABC, by using the cosine rule

In ΔABX, by using the cosine rule

=

=

=

=

=

=

M1

M1

M1

A1

6.

In ΔBPC and ΔABC

given

common angle

remaining angle

are similar AAA. Shown.

B1

B1

B1

M1




4

By using Heron’s formula,

Area of ΔBPC = = 6.54cm2

M1

M1

A1

7 Let R: Event that it rains

Let A: Event that an accident occurs

P(R/A) = = M1 M1




5

=

= = 0.3846

M1 M1

A1

8.a) X

8C3

0.2076

B1(both)

M1

A1(both)

M1

A1

8. b)i Let X= number of books read per year

X

= 0.1804

M1

A1

8.b)ii Let Y = number of books read in two years

Y

M1

A1

9.a) % students who spent less than RM50 =

= 60%

M1

A1

9.b) Mean =

Var = = 237.1

Std Dev = 15.4

M1A1

M1

A1

9.c) Median=

Semi interquartile range = = 9.25

B1

M1A1

9.d)




6

box , median, Q 1, Q3 correct

whiskers seen and drawn on graph paper all correct

skewed to the left (negative)

D1

D1 D1

B1

10.a) E(X)=6 = 3p +15q + 1.2

1.6 = p+5q ......(1)

p+2q=0.7 ......(2)

p=0.1

q=0.3

M1

M1

A1

A1

10.b) Var (X) = E(X2)‐E2(X)

= 50.4‐36 = 14.4

B1M1 A1

11. P(X>120) = 0.0415

From table, P(Z>1.733) = 0.0415

From table, P(Z>0.802) = 0.2114

(1)/(2)

= ‐194.49 + 2.161

3.161

From (1)....

=11.83

B1

M1

B1

M1

M1

M1

A1

A1

12.a) k(6‐3) =1

k=

M1

A1

12.b) Q 1= 3+( =3.75 M1A1




7

12.c)

f(x) =

On the graph,

values

horizontal line

M1A1

D1

D1

12.d) E(X2) =

=

=

= 21

E(X) =

=

=

= 4.5

Var(X) = 21‐

=0.75

Var(2X‐1) = 4 Var(X) = 4x0.75

M1

M1

A1

M1 A1




= 3

h // d h l /

[Edu.joshuatly.com] Gerak Gempur Perak STPM 2012 Maths T [5A930C99]

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