add mth f4 binder pdf october 11 2007-6-18 am 376k

8/3/2019 Add Mth f4 Binder PDF October 11 2007-6-18 Am 376k

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ADDITIONAL MATHEMATICS FORM 4

2007 mozac 1

ADDITIONAL MATHEMATICS

FORM 4

MODULE 1

FUNCTIONS

SIMULTANEOUS EQUATIONS

MODUL KECEMERLANGAN AKADEMIK


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2007mozac 2

1 FUNCTIONS

PAPER 1

1 A relation from set P = {6, 7, 8, 9} to set Q = {0, 1, 2, 3, 4} is defined by subtract by 5 from.

State

(a) the object of 1 and 4,(b) the range of the relation.

Answer : (a)

(b)

2 The arrow diagram below shows the relation between Set A and Set B.

Set A Set B

State

(a) the range of the relation,

(b) the type of the relation.

Answer : (a)

(b)

3 The function f is defined by f: x 2 mx and f1 (8) = 2, find the value of m.

Answer: m = .

3

2

1

1

16

12

9

4

1


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4 Given the function : 3 4 f x x , find the value of m if1(2 1) . f m m

Answer: m = .

5 Given the functions : 2 4 f x x and10

: , 2,2

fg x xx

find

(a) the function g,

(b) the values of x when the function g mapped onto itself.

Answer : (a)

(b)

6 The function f is defined by : ,3

x a f x x h

x

. Given that

1(2) 8f

,

Find

(a) the value ofh,(b) the value ofa.

Answer : (a) h =

(b) a =


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7 Given the functions : 2 f x x and2: x mx n . If the composite function fg is given by

2: 3 12 8gf x x x , find

(a) the values ofm and n,

(b) 2 ( 1)g .

Answer : (a) m =

n =

(b) ..

8 Given the functions : f x px q where p > 0 and 2 9: f x x , find

(a) the values of p and q,(b)

1f

(5).

Answer : (a) p =

q =

(b) ..

9 If4

: , 33

f x xx

, : 3gf x x and3

: ,3 5 5

fh x xx

, find

(a) the function g,(b) the function h.

Answer : (a)

(b)


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10 Given the function f : 2 .x x Find(a) the range of f corresponds to the domain 1 3x ,

(b) the value ofx that maps onto itself.

Answer : (a)

(b) x = .

11 Given the function xxf 3: p and 15

: 2 f x qx

, where p and q are constants. Find the

values ofp and q.

Answer : (a) p =

(b) q =

12 Given 4 3 f x x , find

(a) the image of 3,

(b) the object which has the image of 5.

Answer : (a)

(b)


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13 The diagram below shows the mapping for the function1f and g.

`

Given that f(x) = ax + b and g(x) = , calculate the value of a and b.

Answer : a =

b =

14 Given that :h x | 5x 2 |, find

(a) the object of 6,

(b) the image which has the object 2.

Answer : (a)

(b)

15 Given that xxf 23: and 1)(2

xxg , find

(a) f g(x),(b) g f(1).

Answer : (a)

(b)

1

f g

2

6

4


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PAPER 2

16 The above diagram shows part of the function rqxpxxf 2)( .

Find

(a) the values of p, q and r,

(b) the values ofx which map onto itself under the function f.

17 Given that functions f and g are defined as2: xxf and :g x ax b where a and b are

constants.

(a) Given that f(1) = g(1) and f(3) = g(5), find the values of a and b.

(b) With the values a and b obtained from (a), find gg(x) and g1.

.

18 Given v(x) = 3x 6 and w(x) = 6x 1, find

(a) vw1

(x),

(b) values of x so that vw(2x) = x.

19 Given that the function2

1:

xxf , and the composite function 162:

21

xxxgf , find

(a) the function of g (x),(b) g f (3),

(c) f2 (x).

20 Given that : 3 2 f x x and : 15

xg x , find

(a) f1(x),

(b) f1g (x),

(c) h(x) such that hg(x) = 2x + 6.

x f(x)

2

1

0

10

1

4


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4 SIMULTANEOUSEQUATIONS

PAPER 2

1 Solve the equation 4x + y + 8 = x2 + x y = 2.

2 Solve the simultaneous equations

qp

1

3

2 = 2 and 3p + q = 3.

3 Solve the equation x2 y + y2 = 2x + 2y = 10.

4 Solve the simultaneous equations and give your answers correct to three decimal places,

2m + 3n + 1 = 0,

m2 + 6 mn + 6 = 0.


1

3x y = 3 and y

2 1 = 2x.

6 Given ( 1, 2k) is the solution of the simultaneous equation

x

2

+ py 29 = 4 = px xy, where k andp are constants. Find the values of kand p.


3 03 2

x y and

3 2 1

2x y

8 Given (2k, 4p) is the solution of the simultaneous equations x 3y = 4 and9 7

y = 1.

Find the values of k andp.

9 Given the following equations :

A = x + y

B = 2x 14

C= xy 9

Find the values ofx andy such that 3A = B = C


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10 Solve the simultaneous equations and give your answers correct to four significant figures,

x + 2y = 2

2y2 xy 7 = 0

11 The straight line 3y = 1 2x intersects the curvey2

3x2 = 4xy 6 at two points. Find the

coordinates of the points.

12 Ifx = 2 and y = 1 are the solutions to the simultaneous equations ax + b2y = 2 and

2 21

2

b x ay ,

find the values of a and b.

13 The perimeter of a rectangle is 34 cm and the length of its diagonal is 13 cm. Find the length and

width of the rectangle.

14 The difference between two numbers is 8. The sum of the squares and the product of the numbers

is 19. Find the two numbers.

15 A piece of wire of length 24 cm is cut into two pieces, with one piece bent to form a square ABCD

and the other bent to form a right-angled triangle PQR. The diagram below shows the dimensions of

the two geometrical shapes formed.

The total area of two shapes is 15 cm2,

(a) show that 6x + y = 21 and 2x2 + y(x + 1) = 30.

(b) Find the value ofx and y.

x cmA

(x + 1) cm

y cm

B

x cm (x + 2) cm

P

D RC S


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2007mozac 1


FORM 4

MODULE 2

QUADRATIC EQUATIONS

QUADRATIC FUNCTIONS

MODUL KECEMERLAGAN AKADEMIK


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2 QUADRATIC EQUATIONS

PAPER 1

1 One of the roots of the quadratic equation 2x2 + kx 3 = 0 is 3, find the value of k.

Answer: k = ..

2 Given that the roots of the quadratic equation x2 hx + 8 = 0 arep and 2p, find the values of h.

Answer: h =

3 Given that the quadratic equation x2

+ (m 3)x = 2 m 6 has two equal roots, find the valuesof m.

Answer: m =

4Given that one of the roots of the quadratic equation 2x

2

+ 18x = 2 k is twice the other root, findthe value of k.

Answer: k = 5 Find the value of p for which 2y + x = p is a tangent to the curve y

2 + 4x = 20.


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Answer: p =

6 Solve the equation 2(3x 1)2 = 18.

Answer: ..

7 Solve the equation (x + 1)(x 4) = 7. Give your answer correct to 3 significant figures.

Answer: ..

8 Find the range of values ofm such that the equation 2x2

x = m 2 has real roots.

Answer: ..

9 Find the range of values of x for which (2x + 1)(x + 3) > (x + 3)(x 3).


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Answer: ..

10 Find the range of values ofk such that the quadratic equationx2 +x + 8 = k(2x k) has two real

roots.

Answer: ..

PAPER 2

11 The quadratic equation xqpxpx 10222 has roots

1

pand q.

(a) Find the values of p and q.

(b) Hence, form a quadratic equation which has the roots p and 3q.

12 (a) Given that and are the roots of the quadratic equation 2x2 + 7x 6 = 0, form a quadratic

equation with roots (+ 1) and (+ 1).

(b) Find the value ofp such that (p 4)x2 + 2(2 p)x + p + 1 = 0 has equal roots. Hence, find the

root of the equation based on the value ofp obtained.

13 (a) Given that 2 and m 1 are the roots of the equation x2 + 3x = k, find the values ofm and k.

(b) Find the range of values of p if the straight liney = px 5 does not intersect the curve

y = x2 1.

14 (a) Given that 3 and m are the roots of the quadratic equation 2(x + 1)(x + 2) = k(x 1).

Find the values of m and k .

(b) Prove that the roots of the equation x2

+ (2a 1)x + a2

= 0 is real when a 1

.

15 (a) Find the range of values of p wherepx2 + 2(p + 2)x + p + 7 = 0 has real roots.

(b) Given that the roots of the equation x2

+ px + q = 0 are and 3, show that 3p2

= 16q.


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2007mozac 5

3 QUADRATIC FUNCTIONS

PAPER 1

1 Solve the inequality 2(x 3)2 > 8.

Answer: ..

2 Find the range of values ofp which satisfies the inequality 2p2 + 7p 4.

Answer: ..

3 Find the range of values ofm if the equation (2 3m)x2 + (4 m)x + 2 = 0 has no real roots.

Answer: ..

4 The quadratic function 4x2 + (12 4k)x + 15 5k= 0 has two different roots, find the range of

values of k.

Answer: ..


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5 Without using differentiation method find the minimum value of the function f(x) = 3x2

+ x + 2 .

Answer: f(x)min =

6 Given that g(x) = 3x2 2x 8, find the range of values ofx so that g(x) is always positive.

Answer: ..

7 The expression x2

x + p, where p is a constant, has a minimum value9

. Find the value of p.

Answer: p =

8 The quadratic functions2 3

( ) 3 ( 1)2

k f x x

has a minimum value of 6. Find the value of k.

Answer: k =


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9 (a) Express y = 1 + 20x 2x2 in the form y = a(x + p)2 + q.

(b) Hence, state

(i) the minimum value ofy,(ii) the corresponding value ofx.

Answer: (a) ...

(b) (i) ....

(ii)

10

Jawapan : p =

q =

r=

0

33

(4, 1)

x

y The diagram on the left shows the graph of the curve2( ) p x q r with the turning point at (4, 1).

Find the values of p, q and r.


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PAPER 211 Given the function f (x) = 7 mx x2 = 16 (x + n)2 for all real values ofx where m and n are

positive, find(a) the values ofm and n,

(b) the maximum point off(x),(c) the range of values ofx so that f(x) is negative. Hence, sketch the graph of f(x) and state the

axis of symmetry.

12 Given that the quadratic function f(x) = 2x2 12x 23,

(a) expressf(x) in the form m(x + n)2 + p, where m, n andp are constants.

(b) Determine whether the function f(x) has the minimum or maximum value and state its value.

13 Given that x2 3x + 5 = p(x h)

2 + kfor all real values of x, vherep, h and k are constants.

(a) State the values ofp, h and k,

(b) Find the minimum or maximum value of x2 3x + 5 and the corresponding value of x.(c) Sketch a graph off(x) = x

2 3x + 5.

(d) Find the range of values ofm such that the equation x2 3x + 5 = 2m has two different roots.


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FORM 4

MODULE 3

INDICES AND LOGARITHMS

COORDINATE GEOMETRY

MODUL KECEMERLANGAN AKADEMIK


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5 INDICES AND LOGARITHMS

PAPER 1

1 Simplify3

32

27

93

x

xx

.

Answer :

2 Express )5(155512212 xxx to its simplest form.

Answer :

3Show that 7

x

+ 7

x + 1

21(7

x 1

) is divisible by 5 for all positive integers ofn.

Answer :

4 Find the value of a if log a 8 = 3.

Answer : a = ..

5 Evaluate 55log5 .


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Answer :

6 Given ma 10log and nb 10log . Expressb

a

100log

3

10 in terms ofm and n.

Answer :

7 Given log 7 2 = p and q5log 7 . Express 7log 2 8 in terms of p and q.

Answer :

8 Simplify27log

243log13log

8

1364 .

Answer :

9 Solve the equationxx

9512 .


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Answer :

10 Solve the equation log 3 (2x + 1) = 2 + log 3 (3x 2).

Answer :

PAPER 2

11 The temperature of an object decreases from 80C to TC after tminutes.Given T= 80(08)t. Find(a) the temperature of the object after 3 minutes,

(b) the time taken for the object to cool down from 80C to 25C.

12 (a) (i) Prove that 9log b = 3 31

log log )2

( b .

(ii) Find the values ofa and b given that 3log 4 ab and2

1

log

log

4

4 b

a.

(b) Evaluate

1

1

5

3(5 )

n

n

.

13 The total amount of money deposited in a fixed deposit account in a finance company after a period

ofn years is given by RM20 000(104)n .Calculate the minimum number of years needed for theamount of money to exceed RM45 000.

14 (a) Solve the equation 5log

644x .


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(b) Find the value of x given that log 5 log 135x x

= 3.

(c) Given5

loglog 42 ba . Express a in terms of b.

15 (a) Solve the equation 3 16log log (2 1) log 4x .

(b) Given that 3log 5 a and 3log 7 , find the value ofp if3

log 3ba

p

.


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6 COORDINATE GEOMETRY

PAPER 1

1 Given the distance between two pointsA(1, 3) andB(7, m) is 10 units. Find the value ofm.

Answer : m =

2 Given points P(2, 12), Q(2, a) and R(4, 3) are collinear. Find the value ofa.

Answer : a =

3 Find the equation of a straight line that passes through B(3, 1) and parallel to 5x 3y = 8.

Answer :


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4 Find the equation of the perpendicular bisector of points A(1, 6) andB(3,0).

Answer :

5 Given A(p, 3), B(3, 7), C(5, q) and D(3, 4) are vertices of a parallelogram. Find(a) the values ofp and q,

(b) the area ofABCD.

Answer: (a) p =

q =

(b) .

6 The points A(h, 2h), B(m, n) and C(3m, 2n) a re collinear. B divides ACinternally in the ratio of

3 : 2. Express m in terms ofn.

Answer :


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7 The equations of the straight lines AB and CD are as follows:

AB : y = hx + k

CD : 36

hx

ky

Given that the lines AB and CD are perpendicular to each other, express h in terms of k.

Answer :

8 Given point A is the point of intersection between the straight lines 31

xy andx + y = 9.

Find the coordinates of A.

Answer :

9 Find the equation of the locus of a moving point P such that its distance from point R(3, 6) is5 units.

Answer :


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10 Given points K(2, 0) and point L(2, 3). Point P moves such that PK: PL = 3 : 2.Find the equation ofthe locus ofP.

Answer :

PAPER 2

11 Given C(5, 2) and D(2, 1) are two fixed points. Point P moves such that the ratio of CP to PD is2 : 1.

(a) Show that the equation of the locus of point P is 034222 yxyx .

(b) Show that point E(1, 0) lies on the locus of point P.(c) Find the equation of the straight line CE.

(d) Given the straight line CEintersects the locus of point P again at point F, find the coordinates

of point F.

12Given points P(

2,

3), Q(0, 3) and R(6, 1).(a) Prove that angle PQR is a right angle.

(b) Find the area of triangle PQR.

(c) Find the equation of the straight line that is parallel to PR and passing through point Q.

13 The diagram above shows a quadrilateral KLMNwith vertices M(3, 4) andN(2, 4).Given theequation of KL is 5y = 9x 20. Find

(a) the equation ofML,

(b) coordinates of L,(c) the coordinates ofK,

(d) the area of the quadrilateral KLMN.

x

M(3, 4)

N(2,4) K

L

0

y


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14 In the above diagram, PQRS is a trapezium.QR is parallel to PS and QRS = PSR = 90.(a) Find

(i) the equation of the straight line RS,(ii) the coordinates ofS.

(b) The line PQ produced meets the line SR produced at T.Find(i) the coordinates ofT,

(ii) the ratio ofPQ : QT.

15 The above diagram shows a rectangle ABCD with vertices B(3, 3), A and Care points On the x-axis

and y-axis respectively. Given that the equation of the straight line AB is 2y = x + 3, find(a) the coordinates ofA,

(b) the equation ofBC,

(c) the coordinates ofC,(d) the area of triangleABC,

(e) the area of rectangle ABCD.

C

B(3,3)

A

D

0 x

y

Q(2, 7)

P(0, 1)

R(10, 11)

S

0 x

y


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2007mozac 1


FORM 4

MODULE 4

STATISTICS

CIRCULAR MEASURE


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2007mozac 2

7 STATISTICS

PAPER 1

1 The mean of a list of numbers x 1, x + 3, 2x + 4, 2x 3, x + 1 andx 2 is 7. Find

(a) the value ofx,(b) the variance of the numbers.

Answer: (a) x = .

(b)

2 The mean of a list of numbers 3k , 5k + 4, 3k + 4 , 7k 2 and 6k + 6 is 12. Find

(a) the value ofk,(b) the median of the numbers.

Answer: (a) k = .

(b)

3 Given a list of numbers 8, 9, 7, 10 and 6. Find the standard deviation of the numbers.

Answer: .


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4 The set of positive numbers 3, 4, 7, 8,12, x, y has a mean 6 and median 7. Find the possible values

ofx andy.

Answer: x= ..

y = ..

5 The test marks of a group of students are 15, 43, 47, 53, 65, and 59. Determine

(a) the range,

(b) the interquartile range of the marks.

Answer: (a)

(b)

6 The mean of five numbers is . The sum of the squares of the numbers is 120 and the standard

deviation of the numbers is 4m. Express q in terms of m.

Answer:


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7 The sum of the 10 numbers is 170 and the sum of the squares of the numbers is 2930. Find the

variance of the 10 numbers.

Answer:

8

Score 0 1 2 3 4

Frequency 7 10 p 15 8

The table shows the scores obtained by a group of contestants in a quiz. If the median is 2, find the

minimum value of p.

Answer:

9 The numbers 3, 9, y , 15, 17 and 21 are arranged in ascending order. If the mean is equal to themedian, determine the value of y.

Answer : y=

10

Number 41 45 46 50 51 55 56 60 61 65

Frequency 6 10 12 8 4

The table above shows the Additional Mathematics test marks of 40 candidates. Find the median of

the distribution.

Answer:.............................................

Number of goals 1 2 3 4 5


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11

The table above shows the number of goals score in each match in a football tournament. Calculate

the mean and the standard deviation of the data.

Answer: mean =

standard deviation = ...

12 Given the set of positive numbers n, 5, 11.

(a) Find the mean of the set of numbers in terms ofn.(b) If the variance is 14, find the values ofn.

Answer: (a)

(b) n = ..

13 The mean and standard deviation for the numbers x1, x2, , xn are 74 and 26 respectively.

Find the(a) mean for the numbers 3x1 + 5 , 3x2 + 5, , 3xn + 5,

(b) variance for the numbers 4x1 + 2 , 4x2 + 2, , 4xn + 2.

Answer: (a)

(b)

Frequency 7 6 4 2 1


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14 The mean of the data 2, h, 3h, 11, 12 and 17 which has been arranged in an ascending order, is p. If

each of the element of the data is reduced by 2, the new median is p. Find the values ofh andp.

Answer: h =

p =

15

The table above shows a set of numbers arranged in ascending order where p is a positive integer.

(a) Express the median of the set of the of numbers in terms ofp.

(b) Find the possible values ofp.

Answer: (a) ..

(b) p = ....

PAPER 2

16 A set of examination marks x1,x2, x3, x4, x5, x6 has a mean of 7 and a standard deviation of 1 4.

(a) Find(i) the sum of the marks, x.

(ii) the sum of the squares of the marks, x2.

(b) Each mark is multiplied by 3 and then 4 is added to it.

Find, for the new set of marks,

(i) the mean,

(ii) the variance.

Number 2 p 1 7 p + 4 10 12

Frequency 2 4 2 3 3 2


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2007mozac 7

17

Length (mm) 16 19 20 23 24 27 28 31 32 35 36 39

Frequency 2 8 18 15 6 1

The table above shows the lengths of 50 leaves collected from a tree.(a) Calculate

(i) the mean,

(ii) the variance length of the leaves.

(b) Without drawing an ogive, find the interquartile range length of the leaves.

18 Set R consists of 40 scores, y, for a certain game with the mean of 9 and standard deviation of 5.

(a) Calculate y and y2.

(b) A number of scores totaling 200 with a mean of 10 and the sum of the squares of these scores of

2700, is taken out from set R. Calculate the mean and variance of the remaining scores in set R.

19 A set of data consists of 10 number. The sum of the numbers is 150 and the sum of the squares of the

numbers is 2 472.

(a) Find the mean and variance of the 10 numbers.(b) Another number is added to the set of data and the mean is increased by 1.

Find(i) the value of this number,

(ii) standard deviation of the set of 11 numbers.

20 The table shows the frequency distribution of the scores of the scores of a group of pupils in a game.

Score Number of pupils

10 19 1

20 29 2

30 39 8

40 49 12

50 59 m

60 69 1

(a) It is given that the median score of the distribution is 42.

Calculate the value of m.

(b) Use the graph paper provided by the invigilator to answer this question .

Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the verticalaxis, draw a histogram to represent the frequency distribution of the scores.

Find the mode score.

(c) What is the mode score if the score of each pupil is increased by 5?


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2007mozac 8

8 CIRCULAR MEASURE

PAPER 1

1 Convert

(a) 54 20 to radians.

(b) 406 radians to degrees and minutes.

Answer : (a) ..........................................

(b) .........................................

2

Answer: ......................................

3 The area of a sector of a circle with radius 14 cm is 147 cm2. Find the perimeter of the sector.

Answer:.......................................

The diagram on the left shows a sector OAB with

centre O and radius 9 cm. Given that the perimeter of

the sector OAB is 30 cm. Find the angle ofAOB inradian.

O

A B

9 cm9 cm


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2007mozac 9

4

Answer :.......................................

5

Answer: .....................................

6

Answer: .....................................

2 rad 6 cm

O

A BThe diagram on the left shows a circle with

a sector OAB and centre O . Find the areaof the major sector OAB in cm2 and state

your answer in terms of.

O R Q

P

2 cm

10 cm

The diagram on the left shows a sector of acircle OPQ with centre O and OPR is a rightangle triangle. Find the area of the shaded

region.

O

A B

The diagram on the left shows an arc of a circle ABwith centre O and radius 4 cm. Given that the area of

the sector AOB is 6 cm2. Find the length of the arc AB.


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2007mozac 10

7

Answer: ......................................

8

Answer:.......................................

9

Answer: ......................................

O

P

Q

R

S2 cm

0.8 rad

The diagram shows two sectors OPQ and ORS ofconcentric circles with centre O. Given that

POQ = 0 8 radian and OP = 3PR, find the perimeter

of the shaded region.

The diagram shows a semicircle ofOPQRwith centre O. Given that OP = 10 cm and

QOR = 30. Calculate the area of the

shaded region.

P O R

Q

3010 cm

The diagram shows a circle with centre O.

Given that the major arc AB is 16 cm and the

minor arc AB is 4 cm. Find the radius of the

circle.

O

A

B


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10

Answer: ......................................

11

Answer : (a) r= ...................................

(b) = ...................................

12

Answer:

O

R

S

The diagram on the left shows a sector ROS with

centre O. Given the length of the arc RS is 724

cm and the perimeter of the sector ROS is 25 cm.

Find the value in radians.

O

A

B

rcm

The diagram on the left shows a sector withcentre O. Given that the perimeter and the

area of the sector is 14 cm and 10 cm2

respectively. Find

(a) the value ofr,

(b) the value of in radians.

O

A

B

60

8 cm

The diagram on the left shows a sector OAB of acircle with centre O. Find the perimeter of the

shaded segment.


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2007mozac 12

13

Answer : (a) ..........................................

(b) .........................................

PAPER 2

14 The above diagram shows two arcs AB andDE, of two circles with centre O. OBD and OCEarestraight lines. Given OB = BD,

find

(a) the length of arcAB,(b) the area of segment DE,

(c) the area of the shaded region.

15

The diagram on the right shows the position

of a simple pendulum which swings from P

and Q. Given that POQ = 25 and the

length of arc PQ is 12.5 cm, calculate

(a) the length ofOQ,(b) the area swept out by the pendulum.

O

P Q

O

P

Q

R

S

T

The diagram on the left shows a circle PRTSQ with

centre O and radius 3 cm.Given RS = 4 cm and

POQ = 130. Calculate

(a) ROS , in degrees and minutes,(b) the area of segment RST,

(c) the perimeter of the shaded region.

70

OA

B

C

D

E6 cm


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16 The diagram above shows a semicircle ACBE with centre Cand a sector of a circle OADB with

O. Given BAO = 35 and OA = OB = 7 cm. Calculate

(a) the diameter AB,

(b) the area of the triangle AOB,

(c) the area of the shaded region,(d) the perimeter of the shaded region.

17 The diagram above shows two circles PAQB with centres O andA respectively.

Given that the diameter of the circle PAQB = 12 cm and both of the circles have the same radius.

(a) Find POA in radians.

(b) Find the area of the minor sector BOP.

(c) Show that the area of the shaded region is (12 9 3 ) cm2

the perimeter of the shaded

region is (4+ 6 3 ) cm.

O AB

P

O

A BC

D

E

35


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18 The diagram below shows the plan of a garden. PCQ is a semicircle with centre O and has radius of

8 cm. RAQ is a sector of a circle with centre A and has a radius of 14 m.

Sector COQ is a lawn. The shaded region is a flower bed and has to be fenced. It is given that

AC= 8 m and COQ = 1 956 radians. Using = 3 142, calculate(a) the area, in m

2, of the lawn,(b) the length, in m, of the fence required for fencing the flower bed,

(c) the area, in m2, of the flower bed.

R

Q

C

P A O


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MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4

2007mozac 1


FORM 4

MODULE 5

DIFFERENTIATIONS


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9 DIFFERENTIATIONS

PAPER 1

1 Given y = 4(1 2x)3, find

y

dx.

Answer :

2 Differentiate 3x2(2x 5)4 with respect tox.

Answer :

3 Given that1

3 5)( )

xh x

, evaluate h(1).

Answer :


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4 Differentiate the following expressions with respect to x.

(a) (1 + 5x2)3

(b)43

4

x

Answer : (a)

(b)

5 Given a curve with an equation y = (2x + 1)5, find the gradient of the curve at the point x = 1.

Answer :

6 Given y = (3x 1)5, solve the equation

2

212 0

d y dy

dx dx

Answer :


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7 Find the equation of the normal to the curve 53 2 xy at the point (1, 2).

Answer :

8 Given that the curve qxpxy 2

has the gradient of 5 at the point (1, 2), find the values of

p and q.

Answer : p =

q =

9 Given ( 2, t) is the turning point of the curve 142 xkxy . Find the values ofk and t.

Answer : k=

t =

10 Given22 yxz and y 1 , find the minimum value of z.

Answer :


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11 Given 12 tx and t . Find

(a)dx

dyin terms of t , where tis a variable,

(b) dx

dy

in terms of y.

Answer : (a)

(b)

12 Given that y = 14x(5 x), calculate

(a) the value of x wheny is a maximum,

(b) the maximum value of y.

Answer : (a)

(b)

13 Given that y = x2

+ 5x, use differentiation to find the small change in y when x increases from

3 to 301.

Answer :


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14 Two variables, x and y, are related by the equation y = 3x + . Given that y increases at a constant

rate of 4 units per second, find the rate of change of x when x = 2.

Answer :

15 The volume of water, Vcm3 , in a container is given by 3

18

3V h h , where h cm is the height of

the water in the container. Water is poured into the container at the rate of 10 cm3s1.

Find the rate of change of the height of water, in cm s1, at the instant when its height is 2 cm.

Answer :


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PAPER 2

16 (a) Given that graph of function2

3)(q

pxxf , has gradient function 23

192( ) 6 f x x

x

wherep and q are constants, find

(i) the values of p and q ,(ii) x-coordinate of the turning point of the graph of the function.

(b) Given3 29

( 1)2

p t t .

Finddt

dp, and hence find the values of t where 9.

dp

dt

17 The gradient of the curve 4y xx

at the point (2, 7) is1

2, find

(a) value of k,

(b) the equation of the normal at the point (2, 7),(c) small change in y when x decreases from 2 to 197.

18 The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2x m sides are

cut out from its four vertices.The zinc sheet is then folded to form an open square box.(a) Show that the volume, V m

3, is V = 128x 128x

2+ 32x

3.

(b) Calculate the value ofx when Vis maximum.

(c) Hence, find the maximum value ofV.

8 m

8 m

2x m

2x m2x m

2x m

2x m

2x m

2x m

2x m


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19 (a) Given that 12p q , where 0p and .q Find the maximum value of .2qp

(b) The above diagram shows a conical container of diameter 8 cm and height 6 cm. Water

is poured into the container at a constant rate of 3 cm3 s1. Calculate the rate of change of the

height of the water level at the instant when the height of the water level is 2 cm.

[Use = 3 142 ; Volume of a cone = hr2

3

1 ]

20 (a) The above diagram shows a closed rectangular box of widthx cm and height h cm. The lengthis two times its width and the volume of the box is 72 cm 3 .

(i) Show that the total surface area of the box, A cm2 is

xxA

2164 2 ,

(ii) Hence, find the minimum value ofA.

(b) The straight line 4y + x = k is the normal to the curve y = (2x 3) 2 5 at point E. Find

(i) the coordinates of point E and the value of k,

(ii) the equation of tangent at point E.

6 cm

8 cm

h cm

x cm

2x cm

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