Maths Sa . Msa 1 Revision Package 2009

ST. ANDREW’S JUNIOR COLLEGE H2 MATHEMATICS/JC1/2009 MSA 1 Revision Package Partial Fractions Express the following algebraic fractions in partial fractions: 1.

2.

3.

7.

3x + 1 2x 2 − x −1

4.

3 x 2 + 23 x + 45 x ( x + 3)

2 + 5 x +15 x 2 (2 − x)(1 + 2 x 2 )

5.

2 x 3 − x − 16 x3 − 8

2 x 2 − x +1 x 2 (1 − x)

6.

3 x 3 +1 ( x 2 +1) 2

Express

x3 + 3x + 1 ( x + 2) 2

in partial fractions. Hence express

2 x3 + 5 x ( x + 2) 2

in partial

fractions. Answers: 1.

3x + 1 1 1 4  =  +  2 2x − x −1 3  2x +1 x −1 

2.

2 + 5 x + 15 x 2 8 x−3 = + 2 (2 − x)(1 + 2 x ) 2 − x 1 + 2 x2

3.

2 x2 − x + 1 1 2 = 2+ 2 x (1 − x) x 1− x

4.

3x 2 + 23x + 45 15 1 = 3+ − x( x + 3) x x+3

5.

6.

7.

∴

2 x3 − x − 16 1 1 2− x  = 2−  + 2  3 x −8 6  x − 2 x + 2x + 4 

3x3 + 1 3x 1 − 3x = 2 + 2 2 2 ( x + 1) x + 1 ( x + 1)2 x3 + 3x + 1 15 13 = x−4+ − 2 ( x + 2) x + 2 ( x + 2)2

; 2

Trigonometry

1.

It is given that

, and

Find the following, in terms of a.

is obtuse. ,

( x − 4) +

29 26 − x + 2 ( x + 2) 2

b. c. d. Prove the following identities (2 and 3) 2.

3.

4.

Sketch on the same diagram, the graphs of

and

for

.

Hence, state the number of solutions in the interval of the equation of .

5.

Given that value of

, show that

and state the

.

Prove the following identities (6, 7 and 8) 6.

7.

8.

9.

Prove that

.

Hence, solve for

10.

, the equation

The diagram shows a broken ladder PQR resting on a vertical wall PS. RS denotes the ground. The measurement of the ladder is as follows. .

a.

Obtain an expression for

b.

Express

in terms of

in the form of

and hence evaluate

and

, .

.

c.

State the maximum length of and the corresponding value of .

d.

Given that

, find the

value of

for which

.

Answers: 1.

2.

a.

− 1 −x2

b.

−

c.

−2 x 1 − x 2

d.

1 − 2x 2

7.

(Proof question)

8.

(Proof question)

x 1− x2

9.

x = 0,

2π π 4π 2π 8π , , , , or π 9 3 9 3 9

(Proof question) 10.

3.

(Proof question) a.

4.

3 solutions

b.

2 sin θ + cos θ 5 cos (θ −63 .4°) R=

h=6

6.

(Proof question)

d.

Number Systems, Surds & Indices

Simplify

2.

Simplify

3.

Simplify

α = 63 .4°

c. Maximum length is 5 when θ = α = 63 .4°

5.

1.

5,

  

 2 + 2 − 2  

(

) (

 10 + 51 − 

 2 − 2− 2  

)

10 − 51  

125 + 175 − 28 +

1 20

2

θ = 79 .4°

4.

Simplify

2 5+ 2 5− 2

, expressing your answer in the form

are integers to be determined.

5.

Give the solution set for the following problems,

a.

2 log (2 x +1) (2 x + 4) − log(2 x +1) 4 = 2

b.

5(8e 2 x − 3)3 = 625

Answers:

1.

4

2.

6

3.

51 5 +3 7 10

4.

4 + 10 ; a = 4, b = 10

5. a. b.

2

{ x∈¡ { x∈¡

: x = 1}

: x = 0}

a + b , where a and b