P3: Algebra and functionsEdexcel International A Level Maths: Topic test

1. 1
   The function $\mathrm{p}$ is defined by $\mathrm{p}(x)=\dfrac{2x^2+5x-3}{x^2+x-6}$ for all real $x$ for which the expression is defined.

   (a)

   Which of the following is $\mathrm{p}(x)$ in its simplest form?

   [1 mark]

   • A$\dfrac{2x-1}{x-2}$
   • B$\dfrac{2x+1}{x-2}$
   • C$\dfrac{2x-1}{x+2}$
   • D$\dfrac{5x-3}{x-6}$

   (b)

   For which values of $x$ is $\mathrm{p}(x)$ undefined?

   [1 mark]

   • A$x=2$ only
   • B$x=-3$ and $x=2$
   • C$x=3$ and $x=-2$
   • D$x=\dfrac12$ and $x=2$

   (c)

   Solve $\mathrm{p}(x)=3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=\dfrac{x^3+2x^2-5x+4}{x+3}$, $x\in\mathbb{R}$, $x\neq-3$.

   (a)

   Which of the following is equal to $\mathrm{g}(x)$?

   [1 mark]

   • A$x^2-x-2-\dfrac{10}{x+3}$
   • B$x^2+5x+10+\dfrac{34}{x+3}$
   • C$x^2-x-2+\dfrac{10}{x+3}$
   • D$x^2+2x-5+\dfrac{4}{x+3}$

   (b)

   The constant term $4$ in the numerator of $\mathrm{g}$ is replaced by $k$. Find the value of $k$ for which $(x+3)$ is a factor of the numerator.

   [1 mark]

   • A$k=6$
   • B$k=10$
   • C$k=-10$
   • D$k=-6$

   (c)

   Solve $\mathrm{g}(x)=x^2-x-1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(x)=\sqrt{x+5}$, $x\geqslant-5$, and $\mathrm{g}(x)=x^2-3$, $x\in\mathbb{R}$.

   (a)

   State the range of $\mathrm{g}$. Find $\mathrm{fg}(x)$ and state its range.

   [3 marks]

   (b)

   Find $\mathrm{gf}(x)$, stating its domain, and hence solve $\mathrm{gf}(x)=6$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $\mathrm{h}$ and $\mathrm{k}$ are defined by $\mathrm{h}(x)=\dfrac{2x+3}{x-1}$, $x\in\mathbb{R}$, $x\neq1$, and $\mathrm{k}(x)=x^2-4x+9$, $x\geqslant a$, where $a$ is a constant.

   (a)

   Find $\mathrm{h}^{-1}(x)$ and state its domain. Hence solve $\mathrm{h}(x)=\mathrm{h}^{-1}(x)$, giving your answers in exact form.

   [6 marks]

   (b)

   (i) State the smallest value of $a$ for which $\mathrm{k}$ has an inverse. (ii) Taking this value of $a$, find $\mathrm{k}^{-1}(x)$ and state its domain. (iii) Find the exact value of $\mathrm{hk}^{-1}(14)$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=|4-3x|$, $x\in\mathbb{R}$.

   (a)

   Solve $\mathrm{f}(x)=2$.

   [1 mark]

   • A$x=\dfrac23$ only
   • B$x=2$ only
   • C$x=\dfrac23$ and $x=2$
   • D$x=-\dfrac23$ and $x=-2$

   (b)

   Which of the following is the range of $\mathrm{f}$?

   [1 mark]

   • A$\mathrm{f}(x)\geqslant0$
   • B$\mathrm{f}(x)>0$
   • C$\mathrm{f}(x)\geqslant4$
   • D$\mathrm{f}(x)\in\mathbb{R}$

   (c)

   Solve $\mathrm{f}(x)=x+2$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $y=\mathrm{f}(x)$ has a minimum point at $M(3,-2)$ and crosses the $y$-axis at $(0,7)$.

   (a)

   Find the coordinates of the minimum point on the curve $y=\mathrm{f}(2x)+1$.

   [1 mark]

   • A$(6,\,-1)$
   • B$\left(\dfrac32,\,-3\right)$
   • C$(3,\,-1)$
   • D$\left(\dfrac32,\,-1\right)$

   (b)

   Which of the following describes the stationary point of the curve $y=-2\mathrm{f}(x+1)$?

   [1 mark]

   • AA minimum at $(2,\,4)$
   • BA maximum at $(2,\,4)$
   • CA maximum at $(4,\,4)$
   • DA minimum at $(2,\,-4)$

   (c)

   The curve $y=\mathrm{f}(3x)-4$ is obtained from $y=\mathrm{f}(x)$ by a sequence of two transformations. Describe each transformation.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $\mathrm{p}$ is defined by $\mathrm{p}(x)=2-3\cos2x$, $0\leqslant x\leqslant2\pi$, where $x$ is in radians.

   (a)

   Describe a sequence of three transformations that maps the graph of $y=\cos x$ onto the graph of $y=\mathrm{p}(x)$.

   [3 marks]

   (b)

   State the range of $\mathrm{p}$ and solve $\mathrm{p}(x)=\dfrac12$, giving your answers as exact multiples of $\pi$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(x)=\dfrac{x^2+x-6}{x^2-9}$, $x\in\mathbb{R}$, $x\neq\pm3$, and $\mathrm{g}(x)=|2x-5|$, $x\in\mathbb{R}$.

   (a)

   Simplify $\mathrm{f}(x)$. Hence find the range of $\mathrm{f}$.

   [6 marks]

   (b)

   Solve $\mathrm{gf}(x)=1$.

   [6 marks]

   Total for question 8: 12 marks

End of questions