Inequalities with polynomials and rational expressionsAQA A-Level Further Maths: Subtopic test

1. 1
   Let $p(x)=x^3-7x+6$.

   (a)

   Which of the following is the complete set of solutions of $p(x)=0$?

   [1 mark]

   • A$1,\;2,\;3$
   • B$1,\;2,\;-3$
   • C$-1,\;-2,\;3$
   • D$-1,\;2,\;3$

   (b)

   Which of the following is the solution set of $p(x)>0$?

   [1 mark]

   • A$x<-3$ or $1<x<2$
   • B$-3<x<2$
   • C$x>2$
   • D$-3<x<1$ or $x>2$

   (c)

   Solve $x^3+6<7x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $q(x)=x^4-5x^2+4$.

   (a)

   Which of the following is the complete set of solutions of $q(x)=0$?

   [1 mark]

   • A$-2,\;-1,\;1,\;2$
   • B$-4,\;-1,\;1,\;4$
   • C$1,\;4$
   • D$-2,\;2$

   (b)

   Which of the following is the solution set of $q(x)<0$?

   [1 mark]

   • A$x<-2$ or $-1<x<1$ or $x>2$
   • B$-1<x<1$
   • C$-2<x<-1$ or $1<x<2$
   • D$-2<x<2$

   (c)

   Find the set of values of $x$ for which $\sqrt{q(x)}$ is real.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cubic curve $C_1$ has equation $y=x^3-x$ and a parabola $C_2$ has equation $y=x^2-1$.

   (a)

   Find the coordinates of the points where $C_1$ and $C_2$ intersect.

   [3 marks]

   (b)

   Find the set of values of $x$ for which $C_1$ lies above $C_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $A(x)=\left|\dfrac{x+2}{x-1}\right|$ for real $x\ne1$.

   (a)

   (i) Explain why $A(x)<x+1$ has no solutions with $x\le-1$.
   (ii) For $x>-1$, show that $A(x)<x+1$ is equivalent to $\left(x^2-x-3\right)\left(x^2+x+1\right)>0$.
   (iii) Show that $x^2+x+1>0$ for all real $x$.

   [6 marks]

   (b)

   Hence solve the inequality $\left|\dfrac{x+2}{x-1}\right|\ge x+1$, giving exact values.

   [6 marks]

   Total for question 4: 12 marks

End of questions