Algebra

a) A straight line passes through the points (3, 10) and (−3, 2).
(i) Calculate the gradient of the line. [1]
(ii) Find the equation of the line. [2]
(iii) Find the x-intercept of the line. [2]
(b) Does the line y = 4x + 9 intersect with the line that you found in
part (a)? Explain your answer. [2]
(c) Find the coordinates of the point of intersection of the lines with the
equations
2x + 2y = 4,
7x − 6y = 1. [3]
(d) A skateboarder ramp at a local park can be modelled by the equation
y = 0.2x
2 + 0.15x + 0.8,
where x (−5 ≤ x ≤ 5) is the horizontal distance from the centre line of
the ramp, and y is the vertical height of the ramp from the ground
(both measured in metres). A sketch of the ramp is given in Figure 2.
y
x
−5 5
ramp centre line
Figure 2
(i) Find the y-intercept of the parabola
y = 0.2x
2 + 0.15x + 0.8
(the height of the ramp at its centre line). [1]
(ii) (1) By substituting x = −5 into the equation of the parabola,
find the coordinates of the point where the line x = −5 meets
the parabola. [2]
(2) Using your answer to part (d)(ii)(1), explain whether a
ladder of height 3.5 m is long enough to reach the top of the
left-hand side of the ramp. [2]
(iii) (1) Show that the parabola does not have x-intercepts. [4]
(2) Explain what this means in the context of the situation being
modelled. [1]
page 6 of 9
Question 4 – 18 marks
You should use algebra in all parts of this question, showing your working
clearly.
(a) Solve the following equations, giving your answers as integers, surds or
fractions in their simplest form.
(i) 16x − 3 = 9 − 20x [2]
(ii) 1
2
(6x − 2) − 7 = 4 + 3
5
x [3]
(iii) 2
x + 6
+
1
x − 3
= 1 [3]
(b) Factorise the expression 2x
2 + 6x + 4, and hence solve the equation
2x
2 + 6x + 4 = 0. [3]
(c) A student was asked to rearrange the formula
a = b +
1
3
(bc − 6)
to make b the subject (assuming that 3 + c “= 0).
Below is the student’s incorrect attempt.
% ž ” Ö Ü”ý Õ µÚ È
¼
% ž ¼” Ö Ü”ý Õ µÚ
% Õ µ ž ¼” Ö “ý
“ܼ Ö ýÚ ž % Õ µ
” ž
% Õ µ
¼ Ö ý
Ó %]]Y~…kŠ ¼ Ö ý a Ê
(i) Write out a correct rearrangement of the formula. [3]
(ii) Identify the two mistakes made by the student. Explain, as if
directly to the student, why their working is incorrect. [4]
page 7 of 9
Question 5 – 20 marks
Throughout this question, take care to explain your reasoning carefully.
You should round your answers, where necessary, to two significant figures.
A student is making a desktop crane for an investigation. She makes two
models. Model 1 is of the caterpillar track wheel base, and Model 2 is of
the hydraulic arm.
(a) In Model 1 the wheels have radius 5 cm. The distance between the
centre of the two wheels is 46 cm. A cross-section view of the
caterpillar track is given in Figure 3.
Figure 3
(i) What is the length of caterpillar track, that is, the solid line in
Figure 3? [2]
(ii) What is the area of the circular face of one of the wheels? [2]
(b) A sketch of Model 2, the hydraulic arm, is given in Figure 4 and
consists of a base AB of length 15 cm and a vertical strut BG. The
angle ABG is a right angle. The vertical strut is secured in place by a
bar of length 17 cm, represented by the line AC. A bar represented by
the line F G can be raised or lowered by increasing or decreasing the
length of a vertical hydraulic tube represented by the line ED. In its
present position, ED is set at 20 cm so that F G and AC are parallel.
The length of F E is 30 cm.
Figure 4
(i) Use Pythagoras’ Theorem to calculate the length BC, ignoring
the thickness of the material. [2]
(ii) Calculate ∠ACB. [3]
(iii) What are the angles ∠F GB and ∠F ED? Explain your answer. [2]
(iv) Add the line F D to form the triangle F ED, and use the cosine
rule to calculate the length F D. [4]
(v) Using the sine rule and your answer to part (c)(ii), calculate
∠EF D. [3]
(c) In a planned full-size version of the crane, the bar AC has length
3.4 m. What is the length corresponding to F E in the full-size version? [2]
page 8 of 9
Question 6 – 20 marks
(a) A journalist observes that the total number of Facebook users,
between 2010 and 2015, can be modelled by the linear equation
u = 213t − 1740 (10 ≤ t ≤ 15),
where u is the total number of Facebook users in millions, and t is the
number of years since the start of 2000.
(i) Find the number of users at the start of 2012. [2]
(ii) Calculate the year in which the total number of users reaches
1000 million. [3]
(iii) Write down the gradient of the straight line represented by the
equation
u = 213t − 1740 (10 ≤ t ≤ 15).
What does this measure in the practical situation being modelled? [2]
(iv) Explain why the value −1740 in the model does not imply that
there were negative numbers of Facebook users at the start of
2000. [2]
(v) Either using Graphplotter or by hand, sketch the graph of
u = 213t − 1740 (10 ≤ t ≤ 15),
putting u on the vertical axis, and covering the time interval
10 ≤ t ≤ 15. [2]
(vi) How could you use your graph from part (a)(v) to estimate the
year in which the total number of users passes 1000 million, and
thus check your answer to part (a)(ii)? [1]
(b) A researcher claims that between the start of 2005 and the end of
2009, the number of Facebook users can be modelled by the
exponential equation u = 0.43 × (3.17)t
, where u is the number of
users in millions, and t is the number of years since the start of 2004
(1 ≤ t < 6). So after 1 year (t = 1) there are 1.4 million users (to one
decimal place).
(i) Copy and complete the table below, giving the number of
Facebook users in millions each year. [2]
Number of years Number of Facebook users
since 2004 in millions (to 1 d.p.)
1 1.4
2
3
(ii) Use the method shown in Unit 13, Subsection 5.2, to find the year
in which the exponential model predicts that the number of users
would first reach 200 million. [4]
(iii) Write down the value of the scale factor for the exponential
model. Use this to find the percentage increase in the number of
users each year.

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