The modulus functionEdexcel International A Level Maths: Subtopic test

1. 1
   Consider the function $f(x)=|2x-1|$, $x\in\mathbb{R}$, and the line $y=x+5$.

   (a)

   Which gives all solutions of $|2x-1|=x+5$?

   [1 mark]

   • A$x=6$ and $x=-\frac43$
   • B$x=6$ only
   • C$x=6$ and $x=-4$
   • D$x=6$ and $x=2$

   (b)

   Which gives all solutions of $|2x-1|>x+5$?

   [1 mark]

   • A$-\frac43<x<6$
   • B$x>6$
   • C$x<-\frac43$ or $x>6$
   • D$x<-\frac43$

   (c)

   State the coordinates of the vertex of the graph of $y=f(x)$ and the coordinates of its $y$-intercept.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=x^2-4x+3$, $x\in\mathbb{R}$.

   (a)

   How many real solutions does $|f(x)|=\frac12$ have?

   [1 mark]

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

   (b)

   Which are the coordinates of the local maximum point of $y=|f(x)|$?

   [1 mark]

   • A$(2,-1)$
   • B$(2,1)$
   • C$(0,3)$
   • D$(4,3)$

   (c)

   Solve $|f(x)|=3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $|3x-2|=2x+1$ and the inequality $|3x-2|<2x+1$.

   (a)

   Solve the equation $|3x-2|=2x+1$.

   [3 marks]

   (b)

   Hence solve the inequality $|3x-2|<2x+1$, explaining your reasoning.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $p(x)=x^2-x-6$, $x\in\mathbb{R}$.

   (a)

   (i) Factorise $p(x)$ and state the $x$-intercepts and the $y$-intercept of $y=p(x)$.
   (ii) Find the coordinates of the local maximum point of $y=|p(x)|$.
   (iii) Find the values of $k$ for which $|p(x)|=k$ has exactly four solutions.

   [6 marks]

   (b)

   (i) State the $x$-intercepts and the $y$-intercept of $y=p(|x|)$.
   (ii) Solve $|p(x)|=x+2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions