Time allowed: 1.5 hours

Simplify

2 8 + 24 .

x 2 − 5x − 6 , x 3 − 27 .

(

)

(4 marks)

Factorize (a)

(b)

(4 marks) (4 marks)

3.

Write FB116 in the expanded form and convert it to a decimal number.

4.

&& Express 0.8964 in the form

a where a and b are positive integers. b

(4 marks)

5.

Peter deposits $P in a bank at a rate of 2% p.a. compounded monthly. 2 years is $208964, find, correct to the nearest integer, the value of P.

If the expected amount after (4 marks)

6.

(a) (b)

Expand ( x + 5)( x − 3) . Solve the inequality ( x + 5)( x − 3) ≤ x 2 − 2 x + 5 and represent the solution graphically. (6 marks)

7.

Rationalize

6 3+ 2

and express the answers with the simplest surd form.

(5 marks)

8. 9.

Solve 27 2 x −1 = 9 x . In a class, there are 20 boys and n girls. are 71, 62 and 67 respectively. (a) (b) Find the value of n. After mark adjustment, each of the mark of x students is increased by 1. If the new mean mark is 67.25, find the value of x.

(5 marks) In a test, the mean mark of boys, girls and the whole class

(7 marks)

F.3 First Term Exam, 2009-2010

Mathematics 1

p.2 of 2

10. Simplify the following expressions and express the answers with positive indices. (a) (−2a 3 b −2 ) 3 4a −1b (b)

(xy

−1

− x −1 y

)

−1

(x − y )

(8 marks)

11. Solve x 3 + 3.4 × 101000 = 2.5 × 101001 and express the answer in scientific notation. A

(5 marks)

12. In Figure 1, AC intersects DE at B and ∠DAB = ∠CEB. (a) (b) Prove that ∆ABD ~ ∆EBC. Hence, prove that AB × BC = EB × BD . If AB = 8, BC = 12, DE = 22 and BE < BD, find BE. D B Figure 1 C

E

(9 marks)

13. In Figure 2, ED is a perpendicular bisector of ∆ABC. It is given that BD = 5, AB = 6 and ∠BAC = 90°. (a) (b) Find AC and ED. Find the area and the perimeter of the quadrilateral ABDE. B 6

A E Figure 2

5

D

C

(10 marks) 14. The mean of a, b, c, d and x is 2009. (a) (b) Find a + b + c + d + x. If x is unchanged while each of a, b, c and d is increased by 20%, the mean will be increased by 10%. Find the value of x. (6 marks) 15. In Figure 3, CD and BE are medians of ∆ABC. F is a point on ED produced such that FB // AC.

F B

(a) (b)

Prove that AEBF is a parallelogram. If ∠FDB = 90° and ∠DBE = 20°, find ∠ACB. A

D

E Figure 3 A

C

(7 marks)

A

16. In Figure 4a, G is the in-centre of ∆ABC and GB = GC. (a) (b) Show that ∆ABH ≅ ∆ACH. Hence, show that AH ⊥ BC.

H is a

point on BC such that A, G and H lie on the same straight line.

In Figure 4b, K is a point on AC such that GK ⊥ AC. (i) Show that GH = GK and HC = KC. B

G H Figure 4a C B

G H Figure 4b

K

C

(ii) If GH = 3 and HC = 4, find the perimeter of ∆ABC.

(12 marks) END OF PAPER