Solve the simultaneous equations 2x + y = 11 and x - y = 1.
(Total for Question 1 is 1 mark)
2
Solve the simultaneous equations 3x - 2y = 4 and x + y = 8.
(Total for Question 2 is 1 mark)
3
A line has equation y = x + 2 and a curve has equation y = x2 - 4. Which one of these pairs gives both solutions of the simultaneous equations?
A) (3, 5) and (-2, 0)
B) (-3, -1) and (2, 4)
C) (3, 5) only
D) (6, 8) and (-2, 0)
(Total for Question 3 is 1 mark)
4
Solve the simultaneous equations 4x - y = 7 and 2x + y = 11.
(Total for Question 4 is 1 mark)
5
A line has equation y = 2x - 1. Substitute this into y = x2 - 3x + 5 to form a quadratic equation in x, giving your answer in the form x2 + bx + c = 0.
(Total for Question 5 is 1 mark)
6
Solve the simultaneous equations x + 3y = 14 and x - 3y = 2.
(Total for Question 6 is 1 mark)
7
Solve the simultaneous equations 5x - 3y = 19 and 2x + 3y = 16.
(Total for Question 7 is 1 mark)
8
Solve the simultaneous equations y = x - 1 and y = x2 - 5x + 7.
(Total for Question 8 is 2 marks)
9
Solve the simultaneous equations xy = 18 and x + y = 9.
(Total for Question 9 is 2 marks)
10
Two numbers have a sum of 11. Their product is 24. Find the two numbers.
(Total for Question 10 is 2 marks)
11
Solve the simultaneous equations x2 + y2 = 25 and y = x + 1.
(Total for Question 11 is 2 marks)
12
Show that the line y = 5x + 3 does not intersect the curve y = x2 + 3x + 9.
(Total for Question 12 is 2 marks)
13
The line y = 4x + k is a tangent to the curve y = x2 + 2x + 7. Find the value of k.
(Total for Question 13 is 2 marks)
14
Solve the simultaneous equations y = 2x - 3 and y = x2 - 4x + 2.
(Total for Question 14 is 3 marks)
15
Two numbers have a sum of 17. The sum of their squares is 145. Find the two numbers.
(Total for Question 15 is 3 marks)
16
A curve has equation x2 + y2 = 45. A line has equation x - y = 3. The line intersects the curve at two points, A and B. Find the coordinates of A and B.
(Total for Question 16 is 3 marks)
17
The line y = 5x + c is a tangent to the curve y = x2 + 3x + 11. Find the value of c.
(Total for Question 17 is 3 marks)
18
A circle has equation x2 + y2 = 32. A line has equation y = x + c. Given that the line is a tangent to the circle, find the possible values of c.
(Total for Question 18 is 4 marks)
19
A line has equation y = kx + 5, where k is a constant. A curve has equation y = x2 + 3x + 9.
(a)Show that the x-coordinates of the points of intersection of the line and the curve satisfy x2 + (3 - k)x + 4 = 0.(2)
(b)Hence find the range of values of k for which the line intersects the curve at two distinct points.(3)
(Total for Question 19 is 5 marks)
Mark scheme · A5D Simultaneous Equations: Linear and Quadratic: Fluency and Exam Drill
Question 1
B1 x = 4 and y = 3 cao
Answer: x = 4, y = 3
Question 2
B1 x = 4 and y = 4 cao
Answer: x = 4, y = 4
Question 3
B1 A selected cao
Answer: A) (3, 5) and (-2, 0)
Question 4
B1 x = 3 and y = 5 cao
Answer: x = 3, y = 5
Question 5
B1 x2 - 5x + 6 = 0 cao
Answer: x2 - 5x + 6 = 0
Question 6
B1 x = 8 and y = 2 cao
Answer: x = 8, y = 2
Question 7
B1 x = 5 and y = 2 cao
Answer: x = 5, y = 2
Question 8
M1 substitute x-1 for y and rearrange to a correct 3-term quadratic equal to zero, x2 - 6x + 8 = 0
A1 both coordinate pairs correct: (2, 1) and (4, 3) cao
Answer: (2, 1) and (4, 3)
Question 9
M1 rearrange x+y=9 to y=9-x and substitute into xy=18 to form x2-9x+18=0
A1 both pairs of solutions stated: (3, 6) and (6, 3) cao
Answer: (3, 6) and (6, 3)
Question 10
M1 form and attempt to solve t2 - 11t + 24 = 0
A1 the two numbers are 3 and 8 oe
Answer: 3 and 8
Question 11
M1 substitute x+1 for y and rearrange to a correct 3-term quadratic equal to zero, x2 + x - 12 = 0
A1 both coordinate pairs correct: (-4, -3) and (3, 4) cao
Answer: (-4, -3) and (3, 4)
Question 12
M1 set 5x+3 equal to x2+3x+9 and rearrange to x2-2x+6=0, then find the discriminant b2-4ac = 4-24
A1 cso: discriminant = -20, and since -20 < 0 there are no real solutions, so the line does not intersect the curve
Answer: Discriminant = -20 < 0, so no real intersection points exist.
Question 13
M1 set 4x+k equal to x2+2x+7, rearrange to x2-2x+(7-k)=0, and set the discriminant equal to zero
A1 k = 6 cao
Answer: k = 6
Question 14
M1 set 2x-3 equal to x2-4x+2
M1 rearrange to a correct 3-term quadratic equal to zero, x2-6x+5=0, and factorise (x-1)(x-5)=0
A1 both coordinate pairs correct: (1, -1) and (5, 7) cao
Answer: (1, -1) and (5, 7)
Question 15
M1 square the sum, (x+y)2 = x2+2xy+y2 = 289
M1 substitute x2+y2=145 to find 2xy=289-145, giving xy=72
A1 form and solve t2-17t+72=0, e.g. (t-8)(t-9)=0, giving the numbers 8 and 9
Answer: 8 and 9
Question 16
M1 rearrange x-y=3 to x=y+3 and substitute: (y+3)2+y2=45
M1 expand and simplify to y2+3y-18=0, factorise (y+6)(y-3)=0
A1 both coordinate pairs correct: A(-3, -6) and B(6, 3) cao
Answer: A(-3, -6), B(6, 3)
Question 17
M1 set 5x+c equal to x2+3x+11 and rearrange to x2-2x+(11-c)=0 oe
M1 set the discriminant equal to zero (tangent condition), 4-4(11-c)=0