Sketching graphs of functionsEdexcel International A Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=\frac{3}{x}$, for $x\neq0$.

   (a)

   Which of the following are the equations of the asymptotes of $C$?

   [1 mark]

   • A$x=3$ and $y=3$
   • B$x=0$ and $y=3$
   • C$y=0$ only
   • D$x=0$ and $y=0$

   (b)

   In which quadrants does $C$ lie?

   [1 mark]

   • Afirst and second
   • Bsecond and fourth
   • Cfirst and third
   • Dthird and fourth

   (c)

   Find the coordinates of the points where $C$ meets the line $y=x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $D$ has equation $y=x^2(x-4)$.

   (a)

   Which of the following correctly describes the curve $D$ at the origin?

   [1 mark]

   • Ait crosses the $x$-axis, changing from negative to positive
   • Bit touches the $x$-axis at a maximum point
   • Cit touches the $x$-axis at a minimum point
   • Dit has a vertical asymptote

   (b)

   Which of the following describes the behaviour of $D$ for large $|x|$?

   [1 mark]

   • A$y\to+\infty$ as $x\to+\infty$ and $y\to-\infty$ as $x\to-\infty$
   • B$y\to+\infty$ as $x\to\pm\infty$
   • C$y\to-\infty$ as $x\to+\infty$ and $y\to+\infty$ as $x\to-\infty$
   • D$y\to-\infty$ as $x\to\pm\infty$

   (c)

   Use the shape of $D$ to explain why the equation $x^2(x-4)=5$ has exactly one real root.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For $0^\circ\le x\le360^\circ$, consider the curves $y=\sin x$ and $y=\cos x$.

   (a)

   Explain, using the shape of the two graphs, how many solutions there are to $\sin x=\cos x$ in this interval, and find them.

   [3 marks]

   (b)

   Find the set of values of $x$ in this interval for which $\sin x>\cos x$, and explain why the end values are not included.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_1$ has equation $y=x(x-2)(x+2)$ and the curve $C_2$ has equation $y=\frac{2}{x}$, for $x\neq0$.

   (a)

   (i) State the coordinates of the points where $C_1$ meets the $x$-axis, and describe the behaviour of $C_1$ as $x\to\pm\infty$.
   (ii) State the equations of the asymptotes of $C_2$ and the quadrants in which it lies.
   (iii) Explain why $C_1$ and $C_2$ do not meet for $0<x<2$.

   [6 marks]

   (b)

   Show that the $x$-coordinates of the points where $C_1$ and $C_2$ meet satisfy $x^4-4x^2-2=0$, and hence find the coordinates of these points, giving your values to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions