ADDITIONAL MATHEMATICSFORM 4

MODULE 3INDICES AND LOGARITHMSCOORDINATE GEOMETRY

MODUL KECEMERLANGAN AKADEMIK

5 INDICES AND LOGARITHMS

PAPER 1

1 Simplify 3

32

2793

x

xx

.

Answer : …………………………………

2 Express )5(1555 12212 xxx to its simplest form.

Answer : …………………………………

3 Show that 7 x + 7 x + 1 – 21(7 x – 1) is divisible by 5 for all positive integers of n.

Answer : …………………………………

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

Answer : a = ..…………………………

5 Evaluate 55log5 .

Answer : …………………………………

6 Given ma 10log and nb 10log . Expressb

a100

log3

10 in terms of m 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 equation xx 95 12 .

Answer : …………………………………

10 Solve the equation log 3 (2x + 1) = 2 + log3 (3x – 2).

Answer : …………………………………

PAPER 2

11 The temperature of an object decreases from 80C to TC after t minutes.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 ab = 3 3

1log log )

2( a b .

(ii) Find the values of a and b given that 3log 4 ab and21

loglog

4

4 ba

.

(b) Evaluate1

1

5 5

3(5 )

n 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(104)n .Calculate the minimum number of years needed for theamount of money to exceed RM45 000.

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

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

= 3.

(c) Given25

loglog 42 ba . Express a in terms of b.

15 (a) Solve the equation 9 3 16log log (2 1) log 4x .

(b) Given that 3log 5 a and 3log 7 b , find the value of p if2

3log3

bap

.

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 5x – 3y = 8.

Answer : …………………………………

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 of3 : 2. Express m in terms of n.

Answer : …………………………………

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 321

xy 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 : …………………………………

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 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 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 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 5y = 9x – 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

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 2y = 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