Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions
1 Simplify , giving your answer in the form , where p and q
are positive rational numbers. (4 marks)
2
a Find the discriminant of f(x) in terms of k giving your answer as a simplified quadratic.
(3 marks)
b If the equation f(x) = 0 has two equal roots, find the possible values of k. (2 marks)
c Show that when k = 8, f(x) > 0 for all values of x. (3 marks)
3 A stone is thrown from the top of a cliff. The height h, in metres, of the stone above
the ground level after t seconds is modelled by the function
a Give a physical interpretation of the meaning of the constant term 115 in the model.
(1 mark)
2
b Write h(t) in the form A – B(t – C) , where A, B and C are constants to be found.
(3 marks)
c Using your answer to part b, or otherwise, find, with justification
i the time taken after the stone is thrown for it to reach ground level (3 marks)
ii the maximum height of the stone above the ground and the time after which this
maximum height is reached. (2 marks)
4 ,
a Solve the equation q(x) = 0. Write your answer in the form
where a and b are integers to be found. (2 marks)
b Sketch the graphs of y = p(x) and y = q(x) on the same set of axes. Label all
points where the curves intersect the coordinate axes. (4 marks)
c Use an algebraic method to find the coordinates of any point of intersection
of the graphs y = p(x) and y = q(x). (4 marks)
d Write down, using set notation, the set of values of x for which p(x) < q(x). (2 marks)
5 ,x ℝ
Sketch the graph y = g(x). Label any asymptotes and any points of intersection
with the coordinate axes. (5 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
1 Simplify , giving your answer in the form , where p and q
are positive rational numbers. (4 marks)
2
a Find the discriminant of f(x) in terms of k giving your answer as a simplified quadratic.
(3 marks)
b If the equation f(x) = 0 has two equal roots, find the possible values of k. (2 marks)
c Show that when k = 8, f(x) > 0 for all values of x. (3 marks)
3 A stone is thrown from the top of a cliff. The height h, in metres, of the stone above
the ground level after t seconds is modelled by the function
a Give a physical interpretation of the meaning of the constant term 115 in the model.
(1 mark)
2
b Write h(t) in the form A – B(t – C) , where A, B and C are constants to be found.
(3 marks)
c Using your answer to part b, or otherwise, find, with justification
i the time taken after the stone is thrown for it to reach ground level (3 marks)
ii the maximum height of the stone above the ground and the time after which this
maximum height is reached. (2 marks)
4 ,
a Solve the equation q(x) = 0. Write your answer in the form
where a and b are integers to be found. (2 marks)
b Sketch the graphs of y = p(x) and y = q(x) on the same set of axes. Label all
points where the curves intersect the coordinate axes. (4 marks)
c Use an algebraic method to find the coordinates of any point of intersection
of the graphs y = p(x) and y = q(x). (4 marks)
d Write down, using set notation, the set of values of x for which p(x) < q(x). (2 marks)
5 ,x ℝ
Sketch the graph y = g(x). Label any asymptotes and any points of intersection
with the coordinate axes. (5 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
