Simultaneous equationsAQA A-Level Maths: Subtopic test

1. 1
   The straight lines $x+2y=7$ and $3x-y=7$ meet at the point $P$. The first line crosses the $y$-axis at $A$ and the second line crosses the $y$-axis at $B$.

   (a)

   Find the coordinates of $P$.

   [1 mark]

   • A$(2,3)$
   • B$(3,2)$
   • C$(1,3)$
   • D$(1,-4)$

   (b)

   Find the length $AB$.

   [1 mark]

   • A$\frac{21}{2}$
   • B$\frac72$
   • C$7$
   • D$\frac{14}{3}$

   (c)

   Find the area of triangle $ABP$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $y=x+1$ and the curve $y=x^{2}-5$ intersect at two points.

   (a)

   Which equation has the $x$-coordinates of the intersection points as its solutions?

   [1 mark]

   • A$x^2+x-6=0$
   • B$x^2-x+6=0$
   • C$x^2-x-6=0$
   • D$x^2-x-4=0$

   (b)

   Find the coordinates of the intersection point that has a positive $x$-coordinate.

   [1 mark]

   • A$(-2,-1)$
   • B$(3,10)$
   • C$(4,3)$
   • D$(3,4)$

   (c)

   Find the exact distance between the two points of intersection.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $2x+y=k$, where $k$ is a constant, and the curve $C$ with equation $x^{2}+y^{2}=20$.

   (a)

   Show that the $x$-coordinates of any points of intersection satisfy $5x^2-4kx+k^2-20=0$.

   [3 marks]

   (b)

   Given that the line is a tangent to the curve, find the possible values of $k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rectangle has a perimeter of 34 cm. Its length is $x$ cm and its width is $y$ cm.

   (a)

   The diagonal of the rectangle is 13 cm. Form two simultaneous equations in $x$ and $y$ and solve them to find the length and width of the rectangle.

   [6 marks]

   (b)

   Show that no rectangle with a perimeter of 34 cm can have a diagonal of length 12 cm.

   [6 marks]

   Total for question 4: 12 marks

End of questions