**SPM Additional Mathematics (Model Test Paper)**

**Section A**

[40 marks]

Answer

**all**questions.**Question 1**

Solve the simultaneous equations:

2(

*x*–*y*) =*x*+*y*– 1 = 2*x*^{2}– 11*y*^{2 }[5 marks]**Question 2**

$\text{Giventhat}f:x\to \frac{5x}{mx-3},x\ne \frac{3}{2}\text{and}g:x\to 3-4x.\text{Find}$

(a) the value of

*m,*[2 marks](b)

(c) function*gf*^{-1}(–2), [3 marks]*h*if

*hg*(

*x*) = 12

*x*+ 5 [3 marks]

**Question 3**

Diagram 1 shows four rectangles. The largest rectangle has a length of

*L*cm and a width of*W*cm. The measurement of the length and width of each subsequent rectangle are half of the measurements of its previous one. The areas of the rectangles form a geometric progression. The terms of the progression are in descending.(a) State the common ratio, hence find the area of the first rectangle given the sum of four rectangles is 510 cm

^{2}. [4 marks](b) Determine which rectangle has an area of 96 cm

^{2}. [2 marks]**Question 4**

(a) A worker is pumping air into a spherical shape balloon at the rate of 25 cm

^{3}s^{-1}.$\left[\text{Volume of a sphere},\text{}V=\frac{4}{3}\pi {r}^{3}\right]$

Leaving answer in terms of π, find,

(i) rate of change of radius of the balloon when its radius is 10 cm. [3 marks]

(ii) approximate change of volume when radius of the balloon decrease from 10 cm to 9.95 cm. [2 marks]

(i) rate of change of radius of the balloon when its radius is 10 cm. [3 marks]

(ii) approximate change of volume when radius of the balloon decrease from 10 cm to 9.95 cm. [2 marks]

(b) A curve $y=h{x}^{3}-\frac{3}{x}$ has turning point

*x*= 1, find value of*h*. [3 marks]**Question 5**

Diagram 2 shows a triangle

*EFG*.

$\text{Itisgiven}ER:RF=1:2,\text{}FG:TG=3:1,\text{}\overrightarrow{ER}=4\underset{\u02dc}{x}\text{and}\overrightarrow{EG}=6\underset{\u02dc}{y}.$

$\begin{array}{l}\text{(a)Expressintermsof}\underset{\u02dc}{x}\text{and}\underset{\u02dc}{y}:\\ \text{(i)}\overrightarrow{GR}\\ \text{(ii)}\overrightarrow{GT}\end{array}$

[3 marks]

(b) If line GR is extended to point K such that $\overrightarrow{GK}=h\overrightarrow{GR}\text{and}\overrightarrow{EK}=6\underset{\u02dc}{x}-3\underset{\u02dc}{y},\text{findthevalueof}h.$

$\begin{array}{l}\text{(a)Expressintermsof}\underset{\u02dc}{x}\text{and}\underset{\u02dc}{y}:\\ \text{(i)}\overrightarrow{GR}\\ \text{(ii)}\overrightarrow{GT}\end{array}$

[3 marks]

(b) If line GR is extended to point K such that $\overrightarrow{GK}=h\overrightarrow{GR}\text{and}\overrightarrow{EK}=6\underset{\u02dc}{x}-3\underset{\u02dc}{y},\text{findthevalueof}h.$

[3 marks]

**Question 6**

Solution by scale drawing is not accepted.

Diagram 3 shows three points,

*A*,*B*and*D*in a Cartesian plane. The straight line*AB*is perpendicular to straight line*BD*which intersect the*y*-axis at*D*. The equation of the line*BD*is*y*= 2*x*– 5.(a) Find the equation of the straight line

*AB*. [2 Marks](b) The straight line

*AB*is extended to a point*C*such that*AB*:*BC*= 2 : 3. Find the coordinates of*C*. [3 Marks](c) Point

*P*moves such that its distance from*A*is equal to its distance from*B*. Find the equation of the locus*P*. [2 Marks]