Straight line graphs

Straight line graphs - Edexcel Past Exam Questions

1.
y
A(1, 7) B(20, 7)

D(8, 2)

C( p, q)

The points A(1, 7), B(20, 7) and C( p, q) form the vertices of a triangle ABC, as shown in
Figure 2. The point D(8, 2) is the mid-point of AC.

(a) Find the value of p and the value of q.

(2)

The line l, which passes through D and is perpendicular to AC, intersects AB at E.

(b) Find an equation for l, in the form ax + by + c = 0, where a, b and c are integers.
(5)
(c) Find the exact x-coordinate of E.
(2)
 Straight line graphs

2. Given that
f(x) = x2 – 6x + 18, x  0,

(a) express f(x) in the form (x – a)2 + b, where a and b are integers.
(3)

The curve C with equation y = f(x), x  0, meets the y-axis at P and has a minimum point at Q.

(b) Sketch the graph of C, showing the coordinates of P and Q.

(4)

The line y = 41 meets C at the point R.

(c) Find the x-coordinate of R, giving your answer in the form p + q2, where p and q are
integers.
(5)

3. The line l1 passes through the point (9, –4) and has gradient 1
3 .

(a) Find an equation for l1 in the form ax + by + c = 0, where a, b and c are integers.
(3)

The line l2 passes through the origin O and has gradient –2. The lines l1 and l2 intersect at the
point P.

(b) Calculate the coordinates of P.

(4)

Given that l1 crosses the y-axis at the point C,

(c) calculate the exact area of OCP. (3)

4. The line L has equation y = 5 – 2x.

(a) Show that the point P (3, –1) lies on L.

(1)
(b) Find an equation of the line perpendicular to L, which passes through P. Give your answer
in the form ax + by + c = 0, where a, b and c are integers. (4)
 Straight line graphs

5. The line l1 passes through the points P(–1, 2) and Q(11, 8).

(a) Find an equation for l1 in the form y = mx + c, where m and c are constants. (4)

The line l2 passes through the point R(10, 0) and is perpendicular to l1. The lines l1 and l2
intersect at the point S.

(b) Calculate the coordinates of S.

(5)
(c) Show that the length of RS is 35.
(2)
(d) Hence, or otherwise, find the exact area of triangle PQR. (4)

4
6. The curve C has equation y = x2(x – 6) + , x > 0.
x

The points P and Q lie on C and have x-coordinates 1 and 2 respectively.

(a) Show that the length of PQ is 170.

(4)

7. The line l1 has equation y  3x  2 and the line l2 has equation 3x  2 y  8  0 .

(a) Find the gradient of the line l2 . (2)

The point of intersection of l1 and l2 is P.

(b) Find the coordinates of P. (3)

The lines l1 and l2 cross the line y  1 at the points A and B respectively.

(c) Find the area of triangle ABP. (4)

 Straight line graphs

8. The point A(–6, 4) and the point B(8, –3) lie on the line L.

(a) Find an equation for L in the form ax + by + c = 0, where a, b and c are integers.
(4)
(b) Find the distance AB, giving your answer in the form k5, where k is an integer. (3)

9.

Figure 2

The points Q (1, 3) and R (7, 0) lie on the line l1 , as shown in Figure 2.

The length of QR is a√5.

(a) Find the value of a. (3)

The line l 2 is perpendicular to l1 , passes through Q and crosses the y-axis at the point P, as
shown in Figure 2. Find

(b) an equation for l 2 ,

(5)
(c) the coordinates of P,
(1)
(d) the area of ΔPQR. (4)
 Straight line graphs

10. The line l1 passes through the point A(2, 5) and has gradient – 12 .

(a) Find an equation of l1, giving your answer in the form y = mx + c.

(3)

The point B has coordinates (–2, 7).

(b) Show that B lies on l1.

(1)
(c) Find the length of AB, giving your answer in the form k5, where k is an integer.
(3)

The point C lies on l1 and has x-coordinate equal to p.

The length of AC is 5 units.

(d) Show that p satisfies

p2 – 4p – 16 = 0. (4)
 Straight line graphs

11.

Figure 1

The points A and B have coordinates (6, 7) and (8, 2) respectively.

The line l passes through the point A and is perpendicular to the line AB, as shown in Figure
1.

(a) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers.
(4)

Given that l intersects the y-axis at the point C, find

(b) the coordinates of C,

(2)
(c) the area of ΔOCB, where O is the origin.
(2)
 Straight line graphs

12. The line l1 has equation 3x + 5y – 2 = 0.

(a) Find the gradient of l1.

(2)

The line l2 is perpendicular to l1 and passes through the point (3, 1).

(b) Find the equation of l2 in the form y = mx + c, where m and c are constants. (3)

13. (a) Factorise completely x3 – 4x.

(3)
(b) Sketch the curve C with equation

y = x3 – 4x,

showing the coordinates of the points at which the curve meets the axis.
(3)

The point A with x-coordinate –1 and the point B with x-coordinate 3 lie on the curve C.

(c) Find an equation of the line which passes through A and B, giving your answer in the form
y = mx + c, where m and c are constants.
(5)
(d) Show that the length of AB is k10, where k is a constant to be found.
(2)

14. (a) Find an equation of the line joining A(7, 4) and B(2, 0), giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(3)
(b) Find the length of AB, leaving your answer in surd form.
(2)

The point C has coordinates (2, t), where t > 0, and AC = AB.

(c) Find the value of t.

(1)
(d) Find the area of triangle ABC.
(2)
 Straight line graphs

15. The line L1 has equation 2y − 3x − k = 0, where k is a constant.

Given that the point A(1, 4) lies on L1 , find

(a) the value of k,

(1)
(b) the gradient of L1 .
(2)

The line L2 passes through A and is perpendicular to L1 .

(c) Find an equation of L2 giving your answer in the form ax + by + c = 0, where a, b and c
are integers.
(4)

The line L2 crosses the x-axis at the point B.

(d) Find the coordinates of B.

(2)
(e) Find the exact length of AB.
(2)

16. The points P and Q have coordinates (–1, 6) and (9, 0) respectively.

The line l is perpendicular to PQ and passes through the mid-point of PQ.

Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are
integers.
(5)