• A Level Maths

Demo

• A Level
• Simultaneous Equations

1. Solve the following simultaneous equations by elimination:

1. $2x-y=6$ and $4x+3y=22$
2. $3x-2y=-6$ and $6x+3y=2$

1. $x = 4$ and $y = 2$
2. $x = -\frac{2}{3}$ and $y = 2$

2. Solve the following simultaneous equations by substitution:

1. $x+3y=11$ and $4x-7y=6$
2. $3x-y=7$ and $10x+3y=-2$

1. $x = 5$ and $y = 2$
2. $x = 1$ and $y = -4$

3. Solve the following simultaneous equations graphically:

1. $y=3x-5$ and $y=3-x$
2. $y=2x-7$ and $y=8-3x$

1. $x = 2$ and $y = 1$
2. $x = 3$ and $y = -1$

4. $3x+ky=8$ and $x-2ky=5$ are simultaneous equations where $k$ is a constant. Given that $y=0.5$, find $k$.

$-2$

5. $2x-py=5$ and $4x+5y+q=0$ are simultaneous equations, where $p$ and $p$ are constants. The solution is $x=q$ and $y=-1$. Find $p$ and $q$.

$p = 3$ and $q = 1$

6. Solve:

1. $x+y=11$ and $xy=30$
2. $2x+2y=7$ and $x^2-4y^2=8$

1. $x = 5, 6$, $y = 6, 5$
2. $x = 3, \frac{19}{3}$, $y = \frac{1}{2}, -\frac{17}{6}$

7. Solve:

1. $x-y=6$ and $xy=4$
2. $2x+3y=13$ and $x^2+y^2=78$

1. $x = 3\pm\sqrt{13}$, $y = -3\pm\sqrt{13}$
2. $x = 2\pm3\sqrt{5}$ and $y = 3\mp2\sqrt{5}$

8. Solve the simultaneous equations $y=2-4x$ and $3x^2+xy+11=0$, giving your answers in the form $a\pm b\sqrt{3}$, where $a$ and $b$ are integers.

$x = 1\pm2\sqrt{3}$, $y = -2\mp8\sqrt{3}$

9. $(1,p)$ is a solution to the simultaneous equations $y=kx-5$ and $4x^2-xy=6$, where $k$ and $p$ are constants.

1. Find $k$ and $p$.
2. Find the second pair of solutions to these equations.

1. $p = -2$ and $k = 3$
2. $(-6,-23)$

10. Given that $y-x=k$ and $x^2+y^2=4$ and that there is exactly one pair of solutions to these simultaneous equations, find $k$.

$\pm2\sqrt{2}$

11. Given the simultaneous equations $2x-y=1$ and $x^2+4ky+5k=0$ have one pair of solutions, find $k$ and hence solve the simultaneous equations.

$k = \dfrac{1}{16}$, solution $(-\frac{1}{4}, -\frac{3}{2})$

12. A boy throws a ball in the air. The position of the ball, $p$ is given by $p=3x-\frac{1}{2}x^2$. Determine if the ball will hit the ceiling, which has a height given by $h = 0.5x + 4$.

No

13. Find the coordinates of the points where the circle $(x-1)^2+(y-3)^2=45$ meets the $x$-axis.

$(-5, 0)$ and $(7, 0)$

14. A circle $(x-3)^2+(y-5)^2=34$ meets a line $y=x+4$ at two points. Find the coordinates of those two points.

$(-2,2)$ and $(6,10)$

15. Determine if the line $x-y-10=0$ meets the circle $x^2-4x+y^2=21$.

Substitute in $y = x + 10$ to get $2x^2 +16x + 79 = 0$, which has a negative discriminant so no intersections.

16. Show that the line $x+y=11$ meets the circle with equation $x^2+(y-3)^2=32$ at only one point. Find that point.

$(4,7)$

17. The circle $(x-p)^2+(y-6)^2=20$ meets the line $x+y=a$ at the point $(3,10)$. Find the value of $a$, and hence work out the two possible values of $p$.

$a=13$, $p = 1, 5$

18. The line with equation $y=kx$ intersects the circle with equation $x^2-10x+y^2-12y+57=0$ at two distinct points. Find the range of possible values for $k$.

$\dfrac{30-2\sqrt{57}}{21} < k < \dfrac{30+2\sqrt{57}}{21}$

19. The line with equation $y=4x-1$ does not intersect the circle with equation $x^2+2x+y^2=k$. Find the range of possible values for $k$.

$k < \dfrac{8}{17}$

20. The line with equation $y=2x+5$ meets the circle with equation $x^2+kx+y^2=4$ at exactly one point. Find two possible values of $k$.

$k = -20\pm\sqrt{420}$

21. The circle with equation $(x-4)^2+(y+7)^2=50$ meets the straight line with equation $x-y-5=0$ at two points, $A$ and $B$.

1. Show that the perpendicular bisector of $AB$ passes through the centre of the circle.
2. Find the area of triangle $OAB$.

1. $y = -x-3$. At $x = 4$, $y = -7$
2. $20$

22. A circle, $C_1$, has centre $(8,4)$ and radius $8$. A second circle, $C_2$ has centre $(16,4)$ and radius $4$. The two circles intersect. Find the exact coordinates of the points of intersection.

$(15,4\pm\sqrt{15})$

23. Solve the following simultaneous equations $2x^2-3xy-2y^2=12$ and $2x-3y=4$.

$x = 3.5$, $y = 1$ or
$x = 5$, $y = 2$

24. A rectangle has side lengths $a$ and $ab$. Its area is 1.5 and the perimeter is 7. Find $a$ and $b$.

$a=\frac{1}{2}$, $b = 6$ or
$a = 3$, $b = \frac{1}{6}$