Mathematical language, notation and argumentAQA A-Level Maths: Subtopic test

1. 1
   Let $A=\{x\in\mathbb{R}:x^2<9\}$ and $B=\{x\in\mathbb{R}:x\ge1\}$. The universal set is $\mathbb{R}$ and $A'$ denotes the complement of $A$.

   (a)

   Which inequality describes $A\cap B$?

   [1 mark]

   • A$-3<x<3$
   • B$1\le x<3$
   • C$x\ge1$
   • D$-3<x\le1$

   (b)

   Which inequality describes $A\cup B$?

   [1 mark]

   • A$1\le x<3$
   • B$x\ge1$
   • C$-3<x<3$
   • D$x>-3$

   (c)

   Write down $A'\cap B$ as an inequality, showing how you obtain it.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=x^2-4x+7$ for $x\in\mathbb{R}$, $x\ge2$.

   (a)

   Which statement gives the range of $f$?

   [1 mark]

   • A$f(x)\ge3$
   • B$f(x)\ge7$
   • C$f(x)\ge-3$
   • D$f(x)\in\mathbb{R}$

   (b)

   Which is the solution of $f(x)=12$?

   [1 mark]

   • A$x=-1$ or $x=5$
   • B$x=-1$
   • C$x=5$
   • D$x=1$ or $x=-5$

   (c)

   The domain of $f$ is changed to $x\ge4$. Find the range of $f$ for this new domain, justifying your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student solves $\sqrt{x+2}=x$ and writes: Line 1: $x+2=x^2$. Line 2: $x^2-x-2=0$. Line 3: $(x-2)(x+1)=0$. Line 4: so $x=2$ or $x=-1$.

   (a)

   Show that $x=-1$ is not a solution of the original equation, and explain how it arose.

   [3 marks]

   (b)

   Rewrite the student's argument using the correct connecting symbols and complete it with a valid conclusion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Events $A$ and $B$ are such that $P(A)=0.5$, $P(B)=0.4$ and $P(A\cup B)=0.7$.

   (a)

   (i) Find $P(A\cap B)$. (ii) Find $P(A'\cap B')$. (iii) Find $P(A\cap B')$.

   [6 marks]

   (b)

   A student claims that $A$ and $B$ are mutually exclusive because $P(A)+P(B)=0.9<1$. (i) Using set notation, explain why the claim is wrong. (ii) Write down, in set notation, the event that exactly one of $A$ and $B$ occurs and find its probability. (iii) Comment on the student's reasoning.

   [6 marks]

   Total for question 4: 12 marks

End of questions