addmath f4module

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MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4

2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 1

ADDITIONAL MATHEMATICSFORM 4

MODULE 1

FUNCTIONSSIMULTANEOUS EQUATIONS

PANEL

EN. KAMARUL ZAMAN BIN LONG SMK SULTAN SULAIMAN, K. TRG .EN. MOHD. ZULKIFLI BIN IBRAHIM SMK KOMPLEKS MENGABANG TELIPOT, K. TRGEN. OBAIDILLAH BIN ABDULLAH SM TEKNIK TERENGGANU, K. TRGPUAN NORUL HUDA BT. SULAIMAN SM SAINS KUALA TERENGGANU, K. TRG.PUAN CHE ZAINON BT. CHE AWANG SBP INTEGRASI BATU RAKIT, K. TRG.

MODUL KECEMERLANGAN AKADEMIKTERENGGANU TERBILANG 2007

PROG RAM PRAP EPER IKS A A N SPM

http://edu.joshuatly.comhttp://www.joshuatly.co


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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 f 1 (8) = 2, find the value of m.

Answer : m = .

3

2

1

1

161294

1



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

Answer : m = .

5 Given the functions : 2 4 f x x and 10

: , 2,2

fg x x x

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) 8 f ,

Find(a) the value of h ,(b) the value of a .

Answer : (a ) h =

(b) a =



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7 Given the functions : 2 f x x and 2: x mx n . If the composite function fg is given by2: 3 12 8 gf x x x , find

(a ) the values of m 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) 1 f (5).

Answer : (a ) p =

q =

(b) ..

9 If 4

: , 33

f x x x

, : 3 gf x x and3

: ,3 5 5

fh x x x

, 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 3 x ,(b ) the value of x that maps onto itself.

Answer : (a )

(b) x = .

11 Given the function x x f 3: p and 1 5: 2 f x qx , where p and q are constants. Find the

values of p 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 function 1 f 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 | 5 x 2 |, find

(a) the object of 6,(b) the image which has the object 2.

Answer : (a )

(b) 15 Given that x x f 23: and 1)( 2 x x g , find

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

Answer : (a )

(b)

1

f g 6

4



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PAPER 2

16 The above diagram shows part of the function r qx px x f 2)( .Find(a) the values of p, q and r,

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

17 Given that functions f and g are defined as 2: x x f and : x ax b where a and b areconstants.(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 g

1..

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

(a) vw 1( x),(b) values of x so that vw( 2 x) = x.

19 Given that the function2

1:

x x f , and the composite function 162: 21 x x x g f , find

(a) the function of g ( x),(b) g f (3),(c) f 2 ( x).

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

x g x , find

(a) f 1( x),

(b) f 1 g ( x),

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

x f ( x)

2

1

0

10

1

4



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4 SIMULTANEOUS EQUATIONS

PAPER 2

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

2 Solve the simultaneous equations

q p1

32

= 2 and 3 p + q = 3.

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

3 x y = 3 and y2 1 = 2 x.

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

x2

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


3 03 2

x y and

3 2 10

2 x y

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

y = 1.

Find the values of k and p.

9 Given the following equations :

A = x + y

B = 2 x 14

C = xy 9Find the values of x and y such that 3 A = B = C



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10 Solve the simultaneous equations and give your answers correct to four significant figures, x + 2 y = 22 y2 xy 7 = 0

11 The straight line 3 y = 1 2 x intersects the curve y2 3 x2 = 4 xy 6 at two points. Find the

coordinates of the points.

12 If x = 2 and y = 1 are the solutions to the simultaneous equations ax + b2 y = 2 and 2 2 12

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 andwidth of the rectangle.

14 The difference between two numbers is 8. The sum of the squares and the product of the numbersis 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 ABCDand 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 cm 2,(a) show that 6 x + y = 21 and 2 x2 + y( x + 1) = 30.(b) Find the value of x and y.

x cm A

( x + 1) cm

y cm

B

x cm ( x + 2) cm

P

D RC S



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MODULE 2

QUADRATIC EQUATIONSQUADRATIC FUNCTIONS

PANEL

EN. KAMARUL ZAMAN BIN LONG SMK SULTAN SULAIMAN, K. TRG .EN. MOHD. ZULKIFLI BIN IBRAHIM SMK KOMPLEKS MENGABANG TELIPOT, K. TRGEN. OBAIDILLAH BIN ABDULLAH SM TEKNIK TERENGGANU, K. TRG

PUAN NORUL HUDA BT. SULAIMAN SM SAINS KUALA TERENGGANU, K. TRG.PUAN CHE ZAINON BT. CHE AWANG SBP INTEGRASI BATU RAKIT, K. TRG.





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2 QUADRATIC EQUATIONS

PAPER 1

1 One of the roots of the quadratic equation 2 x2 + 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 are p and 2 p, find the values of h.

Answer : h =

3 Given that the quadratic equation x2 + (m 3) x = 2m 6 has two equal roots, find the valuesof m.

Answer : m =

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

2

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

Answer : k = 5 Find the value of p for which 2 y + x = p is a tangent to the curve y2 + 4 x = 20.



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Answer : ..

10 Find the range of values of k such that the quadratic equation x2 + x + 8 = k (2 x k ) has two realroots.

Answer : ..

PAPER 2

11 The quadratic equation xq px px 10222 has roots 1

and 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 2 x2 + 7 x 6 = 0, form a quadraticequation with roots ( + 1) and ( + 1).

(b) Find the value of p such that ( p 4) x2 + 2(2 p) x + p + 1 = 0 has equal roots. Hence, find theroot of the equation based on the value of p obtained.

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

(b) Find the range of values of p if the straight line y = 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 where px2 + 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 3 p2 = 16 q.



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3 QUADRATIC FUNCTIONS

PAPER 1

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

Answer : ..

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

Answer : ..

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

Answer : ..

4 The quadratic function 4 x2 + (12 4 k ) x + 15 5 k = 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) = 3 x2 + x + 2 .

Answer : f ( x)min =

6 Given that g ( x) = 3 x2 2 x 8, find the range of values of x 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 functions 2 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 + 20 x 2 x2 in the form y = a ( x + p)2 + q.(b) Hence, state

(i) the minimum value of y,(ii) the corresponding value of x.

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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P PER

11 Given the function f ( x) = 7 mx x2 = 16 ( x + n)2 for all real values of x where m and n are positive, find(a) the values of m and n,(b) the maximum point of f ( x),(c) the range of values of x 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) = 2 x2 12 x 23,(a) express f ( x) in the form m( x + n)2 + p, where m, n and p are constants.(b) Determine whether the function f ( x) has the minimum or maximum value and state its value.

13 Given that x2 3 x + 5 = p( x h)2 + k for all real values of x, vhere p, h and k are constants.(a) State the values of p, h and k ,(b) Find the minimum or maximum value of x2 3 x + 5 and the corresponding value of x.(c) Sketch a graph of f ( x) = x2 3 x + 5.(d ) Find the range of values of m such that the equation x2 3 x + 5 = 2m has two different roots.



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MODULE 3

INDICES AND LOGARITHMS

COORDINATE GEOMETRY

PANEL






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Answer :

6 Given ma 10log and nb 10log . Express ba

100log

3

10 in terms of m and n.

Answer :

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

Answer :

8 Simplify27log

243log13log

8

1364 .

Answer :

9 Solve the equation x x 95 12 .



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Answer :

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

Answer :

PAPER 2

11 The temperature of an object decreases from 80 C to T C after t minutes.Given T = 80(0 8)t . Find(a) the temperature of the object after 3 minutes,(b) the time taken for the object to cool down from 80 C to 25 C.

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

log log )2

( b .

(ii) Find the values of a and b given that 3log 4 ab and 21

loglog

4

4

ba

.

(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 periodof n years is given by RM20 000(1 04) n .Calculate the minimum number of years needed for theamount of money to exceed RM45 000.

14 (a) Solve the equation 5log 644 x .



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

= 3.

(c) Given25

loglog 42 ba . Express a in terms of b.

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

(b) Given that 3log 5 a and 3log 7 , find the value of p if 23

log 3ba

p

.



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6 COORDINATE GEOMETRY

PAPER 1

1 Given the distance between two points A(1, 3) and B(7, m) is 10 units. Find the value of m.

Answer : m =

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

Answer : a =

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

Answer :



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4 Find the equation of the perpendicular bisector of points A(1, 6) and B(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 of p and q,(b) the area of ABCD .

Answer : (a) p =

q =

(b) .

6 The points A(h , 2h), B(m, n) and C (3m, 2n) are collinear. B divides AC internally in the ratio of 3 : 2. Express m in terms of n.

Answer :



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7 The equations of the straight lines AB and CD are as follows: AB : y = hx + k

CD : 36

h xk

y

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 321 x y and x + 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 of the locus of P .

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 y x y x .(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 CE intersects the locus of point P again at point F , find the coordinates

of point F .

12 Given 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 KLMN with vertices M ( 3, 4) and N ( 2, 4).Given theequation of KL is 5 y = 9 x 20. Find(a) the equation of ML,(b) coordinates of L,(c) the coordinates of K ,(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 of S .

(b) The line PQ produced meets the line SR produced at T .Find(i) the coordinates of T ,(ii) the ratio of PQ : QT .

15 The above diagram shows a rectangle ABCD with vertices B(3, 3), A and C are points On the x-axisand y-axis respectively. Given that the equation of the straight line AB is 2 y = x + 3, find(a) the coordinates of A,(b) the equation of BC ,(c) the coordinates of C ,(d ) the area of triangle ABC ,(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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BIMBINGAN EMaS TAHUN 2007


MODULE 4

STATISTICS

CIRCULAR MEASURE

PANEL


JABATAN PELAJARAN TERENGGANU



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7 STATISTICS

PAPER 1

1 The mean of a list of numbers x 1, x + 3, 2 x + 4, 2 x 3, x + 1 and x 2 is 7. Find(a) the value of x,(b) the variance of the numbers.

Answer : (a) x = .

(b)

2 The mean of a list of numbers 3 k , 5k + 4, 3k + 4 , 7k 2 and 6k + 6 is 12. Find(a) the value of k ,(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 valuesof x and y.

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 standarddeviation of the numbers is 4 m. 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 thevariance of the 10 numbers.

Answer :

8Score 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 theminimum 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. Calculatethe 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 of n.(b) If the variance is 14, find the values of n .

Answer : (a)

(b) n = ..

13 The mean and standard deviation for the numbers x1 , x2 , , xn are 7 4 and 2 6 respectively.Find the(a) mean for the numbers 3 x1 + 5 , 3 x2 + 5, , 3 xn + 5,(b) variance for the numbers 4 x1 + 2 , 4 x2 + 2, , 4 xn + 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 of h and p.

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 of p.(b) Find the possible values of p.

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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8 CIRCULAR MEASURE

PAPER 1

1 Convert(a) 54 20 to radians.

(b) 4 06 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 cm 2. Find the perimeter of the sector.

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

The diagram on the left shows a sector OAB withcentre O and radius 9 cm. Given that the perimeter of the sector OAB is 30 cm. Find the angle of AOB inradian.

O

A B

9 cm9 cm



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4

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

5

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

6

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

2 rad 6 cm

O

A BThe diagram on the left shows a circle witha sector OAB and centre O . Find the areaof the major sector OAB in cm 2 and stateyour 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 shadedregion.

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 cm 2. Find the length of the arc AB.



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7

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

8

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

9

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

O

P

Q

R

S 2 cm0.8 rad

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

POQ = 0 8 radian and OP = 3 PR , find the perimeter of the shaded region.

The diagram shows a semicircle of OPQRwith centre O. Given that OP = 10 cm andQOR = 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 theminor arc AB is 4 cm. Find the radius of thecircle.

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 withcentre O . Given the length of the arc RS is 7 24cm and the perimeter of the sector ROS is 25 cm.Find the value in radians.

O

A

B

r cm

The diagram on the left shows a sector withcentre O . Given that the perimeter and thearea of the sector is 14 cm and 10 cm 2

respectively. Find(a) the value of r ,(b) the value of in radians.

O

A

B

60

8 cm

The diagramon the left shows a sector OAB of acircle with centre O. Find the perimeter of theshaded segment.



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13

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

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

PAPER 2

14 The above diagram shows two arcs AB and DE, of two circles with centre O. OBD and OCE arestraight lines. Given OB = BD ,find(a) the length of arc AB,(b) the area of segment DE ,(c) the area of the shaded region.

15

The diagram on the right shows the positionof a simple pendulum which swings from P and Q. Given that POQ = 25 and thelength of arc PQ is 12.5 cm, calculate(a ) the length of OQ,(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 withcentre 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

O A

B

C

D

E 6 cm



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16 The diagram above shows a semicircle ACBE with centre C and a sector of a circle OADB withO. 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 and A 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 A B

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 m 2, of the flower bed.

R

Q

C

P A O



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BIMBINGAN EMaS TAHUN 2007


MODULE 5

DIFFERENTIATIONS

PANEL


JABATAN PELAJARAN TERENGGANU



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9 DIFFERENTIATIONS

PAPER 1

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

dx.

Answer :

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

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 + 5 x2)3

(b)2

434

x

Answer : (a)

(b)

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

Answer :

6 Given y = (3 x 1)5 , solve the equation2

2 12 0d y dy

dx dx

Answer :



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

Answer :

8 Given that the curve qx px y 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 xkx y . Find the values of k and t.

Answer : k =

t =

10 Given 22 y x z and 21 , find the minimum value of z.

Answer :



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11 Given 12 t x and t . Find

(a)dxdy

in terms of t , where t is a variable,

(b) dxdy

in terms of y.

Answer : (a)

(b)

12 Given that y = 14 x(5 x), calculate(a) the value of x when y is a maximum,

(b) the maximum value of y.

Answer : (a)

(b)

13 Given that y = x 2 + 5x , use differentiation to find the small change in y when x increases from

3 to 3 01.

Answer :



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14 Two variables, x and y, are related by the equation y = 3 x + 2

. 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, V cm3 , in a container is given by 31

83

V 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 cm 3s 1.Find the rate of change of the height of water, in cm s 1 , 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

px x f , has gradient function 2 3192

( ) 6 f x x x

where p 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) Given 3 29

( 1)2

p t t .

Finddt dp

, and hence find the values of t where .dpdt

17 The gradient of the curve 4 y x x

at the point (2, 7) is 1

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 1 97.

18 The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2 x m sides arecut out from its four vertices.The zinc sheet is then folded to form an open square box.(a) Show that the volume, V m3, is V = 128 x 128 x2 + 32 x3.(b) Calculate the value of x when V is maximum.(c) Hence, find the maximum value of V.

8 m

8 m

2 x m

2 x m2 x m

2 x m

2 x m

2 x m

2 x m

2 x m



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

(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 cm 3 s 1. Calculate the rate of change of theheight of the water level at the instant when the height of the water level is 2 cm.

[Use = 3 142 ; Volume of a cone = hr 231

]

20 (a) The above diagram shows a closed rectangular box of width x 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 x

x A 216

4 2 ,

(ii) Hence, find the minimum value of A.

(b) The straight line 4 y + x = k is the normal to the curve y = (2 x 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 cm2 x cm

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