Session 1: Coordinate Geometry (C1 CH5)
Q1.
Figure 1
The line l1 has equation 2x − 3y + 12 = 0
(a) find the gradient of l1.
(1)
The line l1 crosses the x-axis at the point A and the y-axis at the point B, as shown in Figure 1.
The line l2 is perpendicular to l1 and passes through B.
(b) Find an equation of l2.
(3)
The line l2 crosses the x-axis at the point C.
(c) Find the area of triangle ABC.
(4)
(Total 8 marks)
Q2.
Figure 2
The line l1, shown in Figure 2 has equation 2x + 3y = 26
The line l2 passes through the origin O and is perpendicular to l1
(a) Find an equation for the line l2
(4)
The line l2 intersects the line l1 at the point C.
Line l1 crosses the y-axis at the point B as shown in Figure 2.
(b) Find the area of triangle OBC.
Give your answer in the form a⁄b, where a and b are integers to be determined.
(6)
(Total 10 marks)
Q3.
Figure 2
Figure 2 shows a right angled triangle LMN.
The points L and M have coordinates (−1, 2) and (7, −4) respectively.
(a) Find an equation for the straight line passing through the points L and M.
Give your answer in the form ax + by + c = 0, where a, b and c are integers.
(4)
Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90°,
(b) find the value of p.
(3)
Given that there is a point K such that the points L, M, N, and K form a rectangle,
(c) find the y coordinate of K.
(2)
(Total 9 marks)
Session 2: Tangents & Normals (C1 CH8)
Q4.
The curve C has equation
y=
x3 − 9x +
(a) Find
+ 30,
x>0
.
(4)
(b) Show that the point P(4,−8) lies on C.
(2)
(c) Find an equation of the normal to C at the point P, giving your answer in the form ax + by + c = 0 ,
where a, b and c are integers.
(6)
(Total 12 marks)
Q5.
The curve C has equation
y = 2x − 8√ x + 5 , x ≥ 0
(a) Find
, giving each term in its simplest form.
(3)
The point P on C has x-coordinate equal to
(b) Find the equation of the tangent to C at the point P, giving your answer in the form y = ax + b, where a
and b are constants.
(4)
The tangent to C at the point Q is parallel to the line with equation 2x − 3y + 18 = 0
(c) Find the coordinates of Q.
(5)
(Total 12 marks)
Q6.
The curve C has equation y = f(x), x > 0, where
Given that the point P(4, 5) lies on C, find
(a) f(x),
(5)
(b) an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where
a, b and c are integers.
(4)
(Total 9 marks)
Session 3: Probability (S1)
Q7.
(a) State in words the relationship between two events R and S when P(R∩S) = 0
(1)
The events A and B are independent with P(A) =
and P(A∪B) =
Find
(b) P(B)
(4)
(c) P(A'∩B)
(2)
(d) P(B' A)
(2)
(Total 9 marks)
Q8.
The following shows the results of a survey on the types of exercise taken by a group of 100 people.
65 run
48 swim
60 cycle
40 run and swim
30 swim and cycle
35 run and cycle
25 do all three
(a) Draw a Venn Diagram to represent these data.
(4)
Find the probability that a randomly selected person from the survey
(b) takes none of these types of exercise,
(2)
(c) swims but does not run,
(2)
(d) takes at least two of these types of exercise.
(2)
Jason is one of the above group.
Given that Jason runs,
(e) find the probability that he swims but does not cycle.
(3)
(Total 13 marks)
Q9.
Figure 1
Figure 1 shows how 25 people travelled to work.
Their travel to work is represented by the events
B bicycle
T train
W walk
(a) Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
(2)
(b) Determine whether or not B and T are independent events.
(3)
One person is chosen at random.
Find the probability that this person
(c) walks to work,
(1)
(d) travels to work by bicycle and train.
(1)
(e) Given that this person travels to work by bicycle, find the probability that they will also take the train.
(2)
(Total 9 marks)
Q10.
A manufacturer carried out a survey of the defects in their soft toys. It is found that the probability of a toy
having poor stitching is 0.03 and that a toy with poor stitching has a probability of 0.7 of splitting open. A
toy without poor stitching has a probability of 0.02 of splitting open.
(a) Draw a tree diagram to represent this information.
(3)
(b) Find the probability that a randomly chosen soft toy has exactly one of the two defects, poor stitching
or splitting open.
(3)
The manufacturer also finds that soft toys can become faded with probability 0.05 and that this defect is
independent of poor stitching or splitting open. A soft toy is chosen at random.
(c) Find the probability that the soft toy has none of these 3 defects.
(2)
(d) Find the probability that the soft toy has exactly one of these 3 defects.
(4)
(Total 12 marks)
Mark Scheme
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Q10.