Algebra: inequalities and polynomialsAQA A-Level Maths: Topic test

1. 1
   Consider the inequality $\dfrac{x+3}{2}-\dfrac{2x-1}{3}\ge1$.

   (a)

   Which set of values of $x$ satisfies the inequality?

   [1 mark]

   • A$x\ge5$
   • B$x\le5$
   • C$x\le1$
   • D$x\ge1$

   (b)

   Which set of values of $x$ satisfies both this inequality and $x^{2}>9$?

   [1 mark]

   • A$-3<x\le5$
   • B$3<x\le5$
   • C$x<-3$ or $x>3$
   • D$x<-3$ or $3<x\le5$

   (c)

   Find the set of values of $x$ that satisfy both the inequality and $4-x<10$, using set notation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A region $R$ of the $xy$-plane is defined by the inequalities $y\le8-x^{2}$ and $y>x+2$.

   (a)

   Which set of values of $x$ describes the points that lie in $R$?

   [1 mark]

   • A$-3<x<2$
   • B$-2<x<3$
   • C$x<-3$ or $x>2$
   • D$-3\le x\le2$

   (b)

   Which of the following points lies in $R$?

   [1 mark]

   • A$(0,1)$
   • B$(2,4)$
   • C$(0,5)$
   • D$(-3,2)$

   (c)

   Find the greatest value of $y$ for any point in $R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=(2x-1)(x+4)^{2}$.

   (a)

   Expand and simplify $f(x)$.

   [3 marks]

   (b)

   Use algebraic division to find the quotient and remainder when $f(x)$ is divided by $(x-1)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cubic $g(x)=2x^{3}+px^{2}-8x+q$, where $p$ and $q$ are constants, has $(x-1)$ and $(2x+1)$ as factors.

   (a)

   Use the factor theorem to find the values of $p$ and $q$.

   [6 marks]

   (b)

   Given that $p=13$ and $q=-7$, express $g(x)$ as a product of three linear factors, and find the coordinates of all the points where the graph of $y=g(x)$ crosses the axes.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $E(x)=\dfrac{2x^{2}-x-6}{x^{2}-4}$.

   (a)

   Which expression is equal to $E(x)$ for all $x$ for which $E(x)$ is defined?

   [1 mark]

   • A$\dfrac{2x-3}{x+2}$
   • B$\dfrac{2x+3}{x-2}$
   • C$\dfrac{x-6}{-4}$
   • D$\dfrac{2x+3}{x+2}$

   (b)

   For which values of $x$ is $E(x)$ undefined?

   [1 mark]

   • A$x=2$ and $x=-2$
   • B$x=-2$ only
   • C$x=-\dfrac32$ only
   • D$x=2$ only

   (c)

   Solve $E(x)=3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $h(x)=x^{3}-6x^{2}+11x-6$.

   (a)

   Which of the following is a factor of $h(x)$?

   [1 mark]

   • A$x+1$
   • B$x+3$
   • C$x-1$
   • D$x-4$

   (b)

   What is the quotient when $h(x)$ is divided by $(x-1)$?

   [1 mark]

   • A$x^{2}+5x+6$
   • B$x^{2}-5x+6$
   • C$x^{2}-5x-6$
   • D$x^{2}-6x+11$

   (c)

   Hence solve $h(x)=0$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A school librarian buys $x$ hardback books at £8 each and $y$ paperback books at £5 each. She has a budget of at most £120, she must buy at least 4 hardbacks, and she must buy no more than 20 books in total.

   (a)

   Write down three inequalities that $x$ and $y$ must satisfy.

   [3 marks]

   (b)

   The librarian buys exactly 20 books. Find the greatest number of hardbacks she can buy, and state the number of paperbacks and the total cost in that case.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $f(x)=x^{3}-3x^{2}-10x+24$ and $g(x)=x^{2}-x-12$.

   (a)

   Show that $(x-2)$ is a factor of $f(x)$, and factorise $f(x)$ completely.

   [6 marks]

   (b)

   Simplify $\dfrac{f(x)}{g(x)}$, stating the values of $x$ for which it is undefined. Hence solve $\dfrac{f(x)}{g(x)}=2$, explaining your answer.

   [6 marks]

   Total for question 8: 12 marks

End of questions