Simultaneous equations

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Simultaneous equations
Question Paper 1
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 1
Algebra
Simultaneous equations
Question Paper 1
Time Allowed:
88 minutes
Score:
/73
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
1
The line AB has equation 4x – 3y = 7.
(a)
(i)
Find the gradient of AB.
(2)
(ii)
Find an equation of the straight line that is parallel to AB and which passes through
the point C(3, – 5), giving your answer in the form px + qy = r, where p, q and r are
integers.
(3)
(b)
The line AB intersects the line with equation 3x – 2y = 4 at the point D. Find the
coordinates of D.
(3)
(c)
The point E with coordinates (k – 2, 2k – 3) lies on the line AB. Find the value of the
constant k.
(2)
(Total 10 marks)
2
(a)
(i)
Express 4 – 10x – x2 in the form p – (x + q)2
(2)
(ii)
Hence write down the equation of the line of symmetry of the curve with
equation y = 4 – 10x – x2.
(1)
(b)
The curve C has equation y = 4 – 10x – x2 and the line L has equation
y = k(4x – 13), where k is a constant.
(i)
Show that the x-coordinates of any points of intersection of the curve C with the line
L satisfy the equation
x2 + 2(2k + 5)x – (13k + 4) = 0
(1)
(ii)
Given that the curve C and the line L intersect in two distinct points, show that
4k2 + 33k + 29 > 0
(3)
(iii)
Solve the inequality 4k2 + 33k + 29 > 0.
(4)
(Total 11 marks)
3
The line AB has equation 3x + 5y = 11.
(a)
(i)
Find the gradient of AB.
(2)
Page 1 of 4
(ii)
The point A has coordinates (2, 1). Find an equation of the line which passes through
the point A and which is perpendicular to AB.
(3)
(b)
The line AB intersects the line with equation 2x + 3y = 8 at the point C. Find the
coordinates of C.
(3)
(Total 8 marks)
4
The curve C has equation y = k(x2 + 3), where k is a constant.
The line L has equation y = 2x + 2.
(a)
Show that the x-coordinates of any points of intersection of the curve C with the line L
satisfy the equation
kx2 – 2x + 3k – 2 = 0
(1)
(b)
The curve C and the line L intersect in two distinct points.
(i)
Show that
3k2 – 2k – 1 < 0
(4)
(ii)
Hence find the possible values of k.
(4)
(Total 9 marks)
5
The straight line L has equation y = 3x – 1 and the curve C has equation
y = (x + 3)(x – 1)
(a)
Sketch on the same axes the line L and the curve C, showing the values of the intercepts
on the x-axis and the y-axis.
(5)
(b)
Show that the x-coordinates of the points of intersection of L and C satisfy the equation
x2 – x – 2 = 0.
(2)
Page 2 of 4
(c)
Hence find the coordinates of the points of intersection of L and C.
(4)
(Total 11 marks)
6
(a)
(i)
Express x2 – 4x + 9 in the form (x – p)2 + q, where p and q are integers.
(2)
(ii)
Hence, or otherwise, state the coordinates of the minimum point of the curve with
equation y = x2 – 4x + 9.
(2)
(b)
The line L has equation y + 2x = 12 and the curve C has equation y = x2 – 4x + 9.
(i)
Show that the x-coordinates of the points of intersection of L and C satisfy the
equation
x2 – 2x – 3 = 0
(1)
(ii)
Hence find the coordinates of the points of intersection of L and C.
(4)
(Total 9 marks)
7
The quadratic equation x2 + (m + 4)x + (4m + 1) = 0, where m is a constant, has equal roots.
(a)
Show that m2 – 8m + 12 = 0.
(3)
(b)
Hence find the possible values of m.
(2)
(Totsl 5 marks)
8
The point A has coordinates (1, 7) and the point B has coordinates (5, 1).
(a)
(i)
Find the gradient of the line AB.
(2)
(ii)
Hence, or otherwise, show that the line AB has equation 3x + 2y = 17.
(2)
Page 3 of 4
(b)
The line AB intersects the line with equation x – 4y = 8 at the point C. Find the
coordinates of C.
(3)
(c)
Find an equation of the line through A which is perpendicular to AB.
(3)
(Total 10 marks)
Page 4 of 4

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