Using graphs to solve equationsAQA A-Level Maths: Subtopic test

1. 1
   A student draws the curve $y=x^2-4x-5$ and the straight line $y=2x-5$ on the same axes.

   (a)

   Which pair of values gives the $x$-coordinates of the points where the curve and the line intersect?

   [1 mark]

   • A$x=-1$ and $x=5$
   • B$x=0$ and $x=3$
   • C$x=0$ and $x=6$
   • D$x=-6$ and $x=0$

   (b)

   Find the coordinates of the point of intersection with the larger $x$-coordinate.

   [1 mark]

   • A$(6,7)$
   • B$(6,-5)$
   • C$(7,6)$
   • D$(5,0)$

   (c)

   The student now wants to solve $x^2-4x-3=0$ using the same curve. Find the equation of the straight line that should be drawn.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The graph of $y=x^3-3x$ is drawn, with a horizontal line $y=k$ on the same axes. The curve has a maximum at $(-1,2)$ and a minimum at $(1,-2)$.

   (a)

   How many distinct real solutions does $x^3-3x=0$ have?

   [1 mark]

   • A$1$
   • B$2$
   • C$0$
   • D$3$

   (b)

   For which positive value of $k$ does the line $y=k$ meet the curve at exactly two distinct points?

   [1 mark]

   • A$k=0$
   • B$k=2$
   • C$k=3$
   • D$k=1$

   (c)

   State the values of $k$ for which the equation $x^3-3x=k$ has exactly one real solution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=x^2-5x+2$ and a line has equation $y=x-7$.

   (a)

   Show that the $x$-coordinates of any points where the line meets the curve satisfy $x^2-6x+9=0$, and explain what this tells you about the line.

   [3 marks]

   (b)

   The line $y=x+c$ does not meet the curve. Find the range of possible values of $c$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^2-2x-8$ and the line $L$ has equation $y=2x-3$.

   (a)

   (i) Find the coordinates of the points where $C$ meets the $x$-axis.
   (ii) Find the coordinates of the points where $L$ meets $C$.

   [6 marks]

   (b)

   By considering the graph of $y=x^2-2x-8$ and a suitable straight line in each case, solve (i) $x^2-2x-3=0$ and (ii) $x^2-3x-10=0$. State the equation of each line used.

   [6 marks]

   Total for question 4: 12 marks

End of questions