Graphs of functions, asymptotes and proportional relationshipsEdexcel A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^2(2x-1)^2$.

   (a)

   Which statement correctly describes where $C$ meets the $x$-axis?

   [1 mark]

   • AIt touches the axis at $x=0$ and at $x=\frac12$
   • BIt crosses the axis at $x=0$ and at $x=\frac12$
   • CIt touches the axis at $x=0$ and crosses it at $x=\frac12$
   • DIt touches the axis at $x=0$ and at $x=-\frac12$

   (b)

   Which statement about the shape of $C$ is correct?

   [1 mark]

   • A$y\to\infty$ as $x\to\infty$ and $y\to-\infty$ as $x\to-\infty$
   • B$y\le0$ for all values of $x$
   • C$y\ge0$ only when $x\ge0$
   • D$y\ge0$ for all $x$, and $y\to\infty$ as $x\to\pm\infty$

   (c)

   Solve $x^2(2x-1)^2=1$, and hence state how many times the line $y=1$ meets $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $H$ has equation $y=\frac{3}{x-2}+1$.

   (a)

   What are the equations of the asymptotes of $H$?

   [1 mark]

   • A$x=-2$ and $y=1$
   • B$x=2$ and $y=1$
   • C$x=2$ and $y=3$
   • D$x=1$ and $y=2$

   (b)

   Where does $H$ meet the coordinate axes?

   [1 mark]

   • A$(-1,0)$ and $\left(0,\frac32\right)$
   • B$(1,0)$ and $\left(0,-\frac12\right)$
   • C$(-1,0)$ and $\left(0,-\frac12\right)$
   • D$(5,0)$ and $\left(0,-\frac12\right)$

   (c)

   Find the coordinates of the points where $H$ meets the line $y=x+1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A fixed mass of gas is kept at constant temperature. Its pressure $P$ kPa is inversely proportional to its volume $V$ litres, and $P=150$ when $V=2$.

   (a)

   Find an equation for $P$ in terms of $V$, and hence find the pressure when $V=5$.

   [3 marks]

   (b)

   A student plots $P$ against $\frac1V$. State the shape of this graph and its gradient. Hence find the volume at which the pressure is $240$ kPa.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   (i) State the coordinates of the points where $C$ meets the $x$-axis, saying whether $C$ crosses or touches the axis at each.
   (ii) State what happens to $y$ as $x\to\infty$ and as $x\to-\infty$.
   (iii) Find the coordinates of all the points where $C$ and $L$ intersect.

   [6 marks]

   (b)

   A second curve $K$ has equation $y=\frac{2}{x+1}-3$.
   (i) State the equations of the asymptotes of $K$.
   (ii) Find the coordinates of the points where $K$ meets the axes.
   (iii) Explain why $K$ never meets the line $y=-3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions