1. 1
   The polynomial $p$ is defined for $x\in\mathbb{R}$ by $p(x)=(x+2)(x-1)(x-4)$.

   (a)

   Find the set of values of $x$ for which $p(x)>0$.

   [1 mark]

   • A$x<-2$ or $1<x<4$
   • B$-2<x<4$
   • C$-2<x<1$ or $x>4$
   • D$x>4$

   (b)

   Find the number of positive integers $x$ that satisfy $p(x)\le0$.

   [1 mark]

   • A2
   • B4
   • C3
   • D5

   (c)

   Solve the inequality $p(x-3)<0$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find the $x$-coordinates of the points where the graphs of $y=f(x)$ and $y=g(x)$ meet.

   [1 mark]

   • A$-1$ and $4$
   • B$1$ and $-4$
   • C$-1$ and $3$
   • D$3$ and $4$

   (b)

   Solve $g(x)\ge f(x)$.

   [1 mark]

   • A$x\le-1$ or $x\ge4$
   • B$-1<x<4$
   • C$-1\le x\le3$
   • D$-1\le x\le4$

   (c)

   Find the number of integers $n$ for which $f(n)\le g(n)$ and $f(n)<0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An open box is made from a square sheet of card of side $12$ cm by cutting a square of side $x$ cm from each corner and folding up the sides. The volume of the box is $V(x)=x(12-2x)^{2}$ cm$^{3}$, for $0<x<6$.

   (a)

   Show that $V(x)\ge100$ is equivalent to $(x-1)(x^{2}-11x+25)\ge0$.

   [3 marks]

   (b)

   Hence find, in exact form, the set of values of $x$ for which the volume of the box is at least $100$ cm$^{3}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined by $f(x)=\dfrac{x}{x-2}$, for $x\in\mathbb{R}$, $x\neq2$, and $g(x)=x-4$, for $x\in\mathbb{R}$.

   (a)

   Solve the inequality $f(x)\le g(x)$ analytically, giving your answer in exact form.

   [6 marks]

   (b)

   (i) A student multiplies both sides of $f(x)\le g(x)$ by $(x-2)$ and obtains the answer $x\le\frac{7-\sqrt{17}}{2}$ or $x\ge\frac{7+\sqrt{17}}{2}$. Explain the error in the student's method, and verify that one value of $x$ in the student's answer does not satisfy $f(x)\le g(x)$.\n(ii) Use technology to solve $f(x)>\mathrm{e}^{x-3}$ for $x>2$. Give the boundary value correct to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions