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SELANGOR 2011

PAPER 2

1. Solve the simultaneous equations 3x y =

2 and x2 + 2y2  xy = 4. Give the answerscorrect to 3 decimal places.

[x = 1.159, y = 1.478, x = 0.216, y = 1.353]

2. The diagram shows the graph of thequadratic function y = f(x).

(a)  Express the equation of f(x) in generalform.

(b)  Find

(i)  the minimum point of the function

by using the method of completingthe square,

(ii)  the range of values of p such that

f(x) = p does not have real roots.

[(a) x2 – 5x + 4 (b)(i)5 9

,2 4

 

(ii) p < 9

4

]

3. The curve y = px3 + qx2 + r, where p, q are

r are constants, has a gradient function

6x(x – 1) and passes through the point(2, 9).

Find

(a)  the value of p, q and r,(b)  the turning points of the curve.

[(a) p = 2, q = 3, r = 5 (b) (0, 5), (1, 4)]

4. (a) Sketch the graph of y = sin3

2x for

0≤ x ≤ 2.

(b) Hence, using the same axes, sketch a

suitable straight line to find thenumber of solutions for the equation

4sin3

2x + 3x = 0 for 0≤ x ≤ 2.

State the number of solutions.

5. The diagram shows a triangle OAB. The

straight line OM intersects the straight line

BL at N.

It is given OA = 3OL, AB = 2AM,

12OA x

and 8OB y

.

(a)  Express in term of   x

and/or  y

 

(i)  BL

 

(ii) OM 

 

(b)  It is given that ON hOM  

and

ON OL k LB

, where h and k are

constants. Find the value of h and of k.

[(a) (i) 4 8 x y

(ii) 6 4 x y

 

(b) h =1

2, k =

1

4]

6. The diagram shows the arrangement of the

first three of an infinite series of circles.The first circle has a circumference of 50

cm. The circumference of each subsequent

circle is half of the circumference of the

pervious circle.

3 solutions

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(a)  Show the areas of the circle form a

geometric progression and state thecommon ratio.

(b)  Find the sum to infinity of the areas, in

cm2

of all the circles, in terms of .

[(a)

1

4 (b)

2500

3  ]

SECTION B

7. The diagram shows the straight line PQ andthe straight line y = 2 intersecting the curve

y2 = x – 2 at point Q.

It is given that the area of the shaded region

is 122

3unit

2.

(a)  Find the value of k.

(b)  The region enclosed by the curve, the

straight line y = 2, the x-axis and the y-axis is revolved through 360

oabout the

y-axis. Find the volume of revolution,

in terms of .

[(a) 4 (b)376

15  unit

3]

8. The table shows the values of two viable, x

and y, obtained from an experiment. Thevariables x and y are related by the equation

y = hk x, where h and k are constants.

x 1 2 4 6 8 10

y 6.92 9.8 19.4 37.4 74.0 144.4

(a)  Plot log10 y against x, using a scale of 1

cm to 1 unit on the x-axis and 10 cm to1 unit on the log10y-axis. Hence, draw

the line of best fit.(b)  Use the graph in 8(a) to find the value

of 

(i) h(ii) k 

(iii)  x, when y = 50.

[(b) (i) 5.012 (ii) 1.4 (iii) 6.8]

9. The diagram shows a trapezium ABCD.

The line AB is perpendicular to the line AD.It is given that the equation of AB is 3y – x

– 6 = 0.

(a)  Find

(i)  the value of k,(ii) the equation of the straight line

CD.

(b)  The straight line CD is extended to a

point E such that CD : DE = 3 : 2.Find

(i) the coordinates of E,

(ii) the area of triangle ODE.(c)  A point P(x, y) moves such that PB =

PD. Find the equation of the locus of P.

[(a) (i) 13 (ii) y =1 13

3 3 x ]

10. The diagram shows a circle ABC with

centre O. BM and AMC are straight lines.

It is given that ABC =1

4  radians, BM =

10.24 cm and area of sector OAC is 9cm2.Calculate

[Use = 3.142](a)  the radius, in cm, of the circle,

(b)  the perimeter, in cm, of sector OAC,(c)  the area, in cm

2, of the shaded region.

[(a) 6 cm (b) 21.426 cm (c) 144.208

cm2]

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11. (a) In a survey carried out in a school, 2

out of 5 teachers own laptops.(i) If 6 teachers are selected at

random from the school, calculate

the probability that at most 2

teachers do not own a laptop.

(ii) There are 120 teachers in theschool. Calculate the mean and

standard deviation of the numberof teachers that own laptops.

(b) The time taken by a worker to travel

from his house to the workplace has anormal distribution with a mean of 30

minutes and a variance of 81 minute2.

(i) Find the probability that a worker

chosen at random travels morethan 50 minutes.

(ii) If 70% of the time taken is less

than t minutes, find the value of t.[(a) (i) 0.1792 (ii)  = 48,   = 28

(b) (i) 0.01321 (ii) 34.716]

SECTION C12. The diagram shows a trapezium ABCD.

AB is parallel to DC.

Find

(a)  ADB,(b)  the length, in cm, of AC,

(c)  the length, in cm, of BD,(d)  the area, in cm2, of triangle BCD.

[(a) 22o 29 (b) 10.43 cm (c) 13.23 cm(d) 45.53 cm2]

13. The table shows the price indices andpercentages of four items, P, Q, R and S,

used in the production of a type of ‘popia’.Item Price

(RM)in the

year

Price

(RM)in the

year

Price

indexin the

year

2010based

on

Percent

-age(%)

2008 2010

2008

P 168 x 125 25

Q 150 125 120 m

R 105 100 105 n

S 90 99 y 40

(a)  Find the value of x and of y.

(b)  The composite index of the ‘popia’ inthe year 2010 based on the year 2008 is

115. Find the value of m and of n.

(c)  The composite index for the production

cost of the ‘popia’ is expected toincrease by 20% from the year 2010 to

2012.

Calculate the composite index for theproduction cost of ‘popia’ in the year

2012 based on the year 2008.

[(a) x = 210, y = 110 (b) m = 20, n = 15(c) 138]

14. A bakery plans to bake x number of 

chocolate cakes and y number of layercakes in a day. The production of the cakes

is based on the following constraints:

I: The number of chocolate cakes is atleast 200 but does not exceed 500.

II: The total number of both cakes does

not exceed 700.

III: The number of chocolate cakesexceeds 2 times the number of layer

cakes for at mot 200.

(a)  Write three inequalities, other than x≥ 0 and y ≥ 0 which satisfy all the aboveconstraints.

(b)  Using a scale of 2 cm to 100 cakes on

both axes, construct and shade the

region R which satisfies all the above

constraints.(c)  Using the graph constructed in 14(b),

find

(i) the maximum number of layercakes that can be baked if the

bakery wants to bake 250

chocolate cakes.(ii) the maximum profit per day if the

profit for each chocolate cake is

RM0.80 and the profit for each

layer cake is RM0.60.

[(a) 200≤ x ≤ 500, x + y≤ 700, 2y≥ x 200

(c) (i) 450 (ii) RM520 when x = 500, y =200]

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15. A particle moves in a straight line and

passes through a fixed point O. Thevelocity of the particle, v cm s-1, is given by

v = 2t2 – t – 6, where t is the time, in

seconds, after leaving O.

[Assume motion to the right as positive]

Find(a)  the initial velocity, in cm s-1, of the

particle,(b)  the minimum velocity, in cm s-1, of the

particle,

(c)  the range of values of t during whichthe particle moves to the right.

(d)  the total distance, in cm, travelled by

the particle in the first 4 seconds.

[(a) 6 cm s-1

(b) 49

8cm s

-1(c) t > 2

(d) 28 cm]