Exam-Style Questions.

1.	GCSE Higher

Marilou and Sam had a skiing race. Here is Marilou’s speed-time graph from the start of the race.

(a) Marilou crossed the finishing line after a time of 40 seconds. How long was the race?

(b) Sam finished after a time of 50 seconds. What was his average speed, in kilometres per hour, for the race?

Worked Solution

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2.	GCSE Higher

Here is a speed-time graph for a drone.

(a) Work out an estimate for the distance the train travelled between the 10th and 15th second.

Is your answer to part (a) an underestimate or an overestimate of the actual distance the train travelled during that time? Give a reason for your answer.

Noah used the graph to find out an estimate for the deceleration of the drone at time 35 seconds. He realised that it took ten seconds for the drone's speed to fall to zero.
Here is Noah’s working.

$$ \text{deceleration = speed ÷ time} $$ $$= 16 \div 10$$ $$= 1.6 m/s²$$

(b) Noah’s method does not give a good estimate of the deceleration at time 35 seconds. Explain why.

Worked Solution

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3.	GCSE Higher

(a) Sketch the graph of \( y = f(x) \) for values of \( x \) between \(-5\) and \(5\) given that:

$$ f(x) = \frac{1}{x-1} - x $$

(b) Write down the equations of any asymptotes parallel to the \( y \)-axis.

(c) Find the coordinates of the y-axis intercept.

(d) Find the coordinates of the x-axis intercepts.

(e) Find a value of \( k \) for which \( g(x) = kx \) does not intercept \( f(x)\).

(f) Draw the graph of \(h(x)=x-3\) on the same set of axes.

(g) Solve the equation \( f(x) = h(x) \).

(h) Find the solutions to the inequality \( f(x) > h(x) \).

Worked Solution

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4.	GCSE Higher

The graph shows the distance travelled, in metres, of a commuter train as it pulls out of a station.

Estimate the speed of the train, in m/s, after 10 seconds. You must show your working.

Worked Solution

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5.	GCSE Higher

The diagram shows the graph of \( y = f(x) \) for \( -2.1 \leq x \leq 2 \). The y axis is vertical and the x axis is horizontal.

(a) Find \( f(1.2) \).

(b) Solve the equation \( f(x) = 0 \) for \( -2.1 \leq x \leq 2 \).

(c) \( f(x) = k \) has five solutions for \( -2.1 \leq x \leq 2 \) where \( k \) is an integer.

Find the smallest possible value of \( k \).

(d) On the grid, draw a line \( y = mx + x\) so that \( f(x) = mx + c\) has exactly one solution for \( -2.1 \leq x \leq 2 \).

Worked Solution

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