Further algebra, functions and proofAQA A-Level Maths: Topic test

1. 1
   The function $\mathrm{f}$ is defined for all real $x$ by $\mathrm{f}(x)=|2x+3|-5$.

   (a)

   What is the minimum value of $\mathrm{f}(x)$?

   [1 mark]

   • A$-5$
   • B$-3$
   • C$0$
   • D$5$

   (b)

   Which of the following gives all the solutions of $\mathrm{f}(x)=0$?

   [1 mark]

   • A$x=1$ only
   • B$x=-1$ or $x=4$
   • C$x=1$ or $x=-4$
   • D$x=-1$ or $x=-4$

   (c)

   Solve $\mathrm{f}(x)>0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(x)=\dfrac{3}{x+2}$ for $x\in\mathbb{R}$, $x\neq-2$, and $\mathrm{g}(x)=4x-1$ for $x\in\mathbb{R}$.

   (a)

   Which expression gives $\mathrm{fg}(x)$?

   [1 mark]

   • A$\dfrac{12}{x+2}-1$
   • B$\dfrac{3}{4x+1}$
   • C$\dfrac{3}{4x+3}$
   • D$\dfrac{3(4x-1)}{x+2}$

   (b)

   Which expression gives $\mathrm{f}^{-1}(x)$?

   [1 mark]

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

   (c)

   State the domain of $\mathrm{f}^{-1}$, giving a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{f}(x)=\dfrac{8x+5}{(2x-1)(x+4)}$ and $\mathrm{g}(x)=\dfrac{8-5x}{(x+2)(x-1)^2}$, defined for $x>1$.

   (a)

   Express $\mathrm{f}(x)$ in the form $\dfrac{A}{2x-1}+\dfrac{B}{x+4}$, where $A$ and $B$ are constants.

   [3 marks]

   (b)

   Express $\mathrm{g}(x)$ in the form $\dfrac{P}{x+2}+\dfrac{Q}{x-1}+\dfrac{R}{(x-1)^2}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $\mathrm{p}$ and $\mathrm{q}$ are defined for all real $x$ by $\mathrm{p}(x)=|x-4|$ and $\mathrm{q}(x)=2x+1$.

   (a)

   (i) Find expressions for $\mathrm{pq}(x)$ and $\mathrm{qp}(x)$. (ii) Solve $\mathrm{pq}(x)=\mathrm{qp}(x)$.

   [6 marks]

   (b)

   (i) Prove by contradiction that $\mathrm{qp}(x)=0$ has no real solution. (ii) Explain why $\mathrm{qp}$ has no inverse as defined, then find $(\mathrm{qp})^{-1}(x)$ when the domain of $\mathrm{qp}$ is restricted to $x\ge4$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A student wants to prove by contradiction the claim: there are no integers $m$ and $n$ such that $14m+21n=1$.

   (a)

   Which is the correct first line of the proof?

   [1 mark]

   • AAssume that $14m+21n\neq1$ for some integers $m$ and $n$
   • BAssume that $m$ and $n$ are both odd
   • CAssume that $14m+21n=1$ has no integer solutions
   • DAssume that there are integers $m$ and $n$ with $14m+21n=1$

   (b)

   After writing $7(2m+3n)=1$, which statement completes the contradiction?

   [1 mark]

   • A$2m+3n=7$, which is allowed
   • B$2m+3n=\frac17$, which is not an integer, but $m$ and $n$ are integers
   • C$14m+21n$ is even, but $1$ is odd
   • D$m=\frac{1}{14}$ and $n=0$

   (c)

   Use a similar method to prove that there are no integers $m$ and $n$ such that $6m+9n=2$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the equation $|3x-1|=|x+5|$.

   (a)

   Which gives all solutions of the equation?

   [1 mark]

   • A$x=3$ only
   • B$x=-3$ or $x=1$
   • C$x=-1$ or $x=3$
   • D$x=1$ or $x=-3$

   (b)

   Which gives the solution of $|3x-1|<|x+5|$?

   [1 mark]

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

   (c)

   Solve $|3x-1|=2x+7$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(x)=\ln(2x-1)$ for $x>\frac12$ and $\mathrm{g}(x)=\mathrm{e}^{x}+3$ for $x\in\mathbb{R}$.

   (a)

   Find an expression for $\mathrm{gf}(x)$ and state its range.

   [3 marks]

   (b)

   Find $\mathrm{f}^{-1}(x)$ and state its domain and range.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(x)=\dfrac{4}{(x-1)(x+3)}$ for $x>1$ and $\mathrm{g}(x)=2x+3$ for $x\in\mathbb{R}$.

   (a)

   (i) Express $\mathrm{f}(x)$ in partial fractions. (ii) Hence prove by contradiction that $\mathrm{f}(x)=0$ has no solution for $x>1$.

   [6 marks]

   (b)

   Find $\mathrm{fg}(x)$ and state its domain. Hence express $\mathrm{fg}(x)$ in the form $\dfrac{A}{x+1}+\dfrac{B}{x+3}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions