Further algebra: inequalities and graphsAQA A-Level Further Maths: Topic test

1. 1
   Let $\mathrm{f}(x)=(x-1)(x+3)(x-5)$.

   (a)

   Which set of values of $x$ satisfies $\mathrm{f}(x)<0$?

   [1 mark]

   • A$-3<x<1$ or $x>5$
   • B$x<-3$ or $1<x<5$
   • C$-3<x<5$
   • D$x<-3$ or $x>5$

   (b)

   Which set of values of $x$ satisfies $x\,\mathrm{f}(x)\geq0$?

   [1 mark]

   • A$x\leq-3$, or $0\leq x\leq1$, or $x\geq5$
   • B$-3\leq x\leq0$ or $1\leq x\leq5$
   • C$x\leq-3$ or $x\geq5$
   • D$x\leq-3$, or $0<x<1$, or $x\geq5$

   (c)

   Solve $\dfrac{\mathrm{f}(x)}{x-1}\geq0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\dfrac{4-x}{2x-3}$.

   (a)

   Which pair of equations gives the asymptotes of $C$?

   [1 mark]

   • A$x=\frac32$ and $y=-1$
   • B$x=-\frac32$ and $y=\frac12$
   • C$x=\frac32$ and $y=-\frac12$
   • D$x=3$ and $y=-\frac12$

   (b)

   At which point does $C$ cross the $y$-axis?

   [1 mark]

   • A$\left(0,\frac43\right)$
   • B$(0,-4)$
   • C$(4,0)$
   • D$\left(0,-\frac43\right)$

   (c)

   Solve the inequality $\dfrac{4-x}{2x-3}>1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The hyperbola $H$ has equation $\dfrac{x^2}{4}-\dfrac{y^2}{25}=1$.

   (a)

   Write down the coordinates of the points where $H$ crosses the $x$-axis and the equations of the asymptotes of $H$.

   [3 marks]

   (b)

   The hyperbola $H$ is translated by the vector $\begin{pmatrix}3\\-1\end{pmatrix}$. Find an equation of the image curve and the equations of its asymptotes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(x)=|2x-5|$ and $\mathrm{g}(x)=|x+4|$ for real $x$. The curve $C$ has equation $y=\sqrt{x}$ for $x\geq0$.

   (a)

   Solve $\mathrm{f}(x)<\mathrm{g}(x)$ and hence find the sum of all the integers that satisfy $\mathrm{f}(x)<\mathrm{g}(x)$.

   [6 marks]

   (b)

   The curve $C$ is rotated through $90^\circ$ anticlockwise about the origin, and the image is then enlarged by scale factor $3$ with centre the origin. Find the Cartesian equation of the final image of $C$, stating the range of values of $x$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $\mathrm{f}(x)=x^2-2x-8$ for real $x$.

   (a)

   How many real solutions does the equation $|\mathrm{f}(x)|=5$ have?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (b)

   The graph of $y=\dfrac{1}{\mathrm{f}(x)}$ has a stationary point. Which statement describes it?

   [1 mark]

   • AA local minimum at $(1,-9)$
   • BA local maximum at $\left(1,-\frac19\right)$
   • CA local minimum at $\left(1,-\frac19\right)$
   • DA local maximum at $(1,-9)$

   (c)

   Solve $|\mathrm{f}(x)|=8$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=\dfrac{2x^2-x+3}{x+1}$.

   (a)

   Which equation gives the oblique asymptote of $C$?

   [1 mark]

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

   (b)

   Which interval contains no value of $y$ taken by $C$?

   [1 mark]

   • A$y<-5-4\sqrt3$ or $y>-5+4\sqrt3$
   • B$-5-4\sqrt3\leq y\leq-5+4\sqrt3$
   • C$-5-4\sqrt3<y<-5+4\sqrt3$
   • D$3-4\sqrt2<y<3+4\sqrt2$

   (c)

   Show that $C$ does not intersect its oblique asymptote.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has equation $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=x^2+6x+5$.

   (a)

   The curve $C$ is stretched by scale factor $\frac12$ parallel to the $x$-axis. Find an equation of the image and the coordinates of its minimum point.

   [3 marks]

   (b)

   The curve $C$ is reflected in the line $y=-x$. Find an equation of the image and the coordinates of its vertex.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The ellipse $E$ has equation $\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$. For real $x\neq1$, $\mathrm{h}(x)=\left|\dfrac{2x+1}{x-1}\right|$.

   (a)

   The ellipse $E$ is rotated through $90^\circ$ anticlockwise about the origin and the image is then translated by $\begin{pmatrix}2\\1\end{pmatrix}$. Find an equation of the final image, and state its centre and its semi-axes.

   [6 marks]

   (b)

   Solve the inequality $\mathrm{h}(x)<x$.

   [6 marks]

   Total for question 8: 12 marks

End of questions