1. 1
   The gradient of a curve is given by $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^{2}-4x+3$. The curve passes through the point $(1,\,5)$.

   (a)

   Find $\int\left(6x^{2}-4x+3\right)\mathrm{d}x$.

   [1 mark]

   • A$12x-4+C$
   • B$2x^{3}-2x^{2}+3x+C$
   • C$6x^{3}-4x^{2}+3x+C$
   • D$2x^{3}-4x^{2}+3x+C$

   (b)

   Find the value of the constant of integration in the equation of the curve.

   [1 mark]

   • A$0$
   • B$5$
   • C$8$
   • D$2$

   (c)

   Find the value of $y$ when $x=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=12-3x^{2}$. The graph of $y=f(x)$ meets the $x$-axis at $x=-2$ and $x=2$, and $f(x)>0$ for $-2<x<2$. The region $R$ is enclosed by the graph of $y=f(x)$ and the $x$-axis.

   (a)

   Which expression gives the area of $R$?

   [1 mark]

   • A$\int_{-2}^{2}\left(12-3x^{2}\right)\mathrm{d}x$
   • B$\int_{0}^{12}\left(12-3x^{2}\right)\mathrm{d}x$
   • C$\int_{-4}^{4}\left(12-3x^{2}\right)\mathrm{d}x$
   • D$\int_{-2}^{2}\left(3x^{2}-12\right)\mathrm{d}x$

   (b)

   Find the area of $R$.

   [1 mark]

   • A$16$
   • B$0$
   • C$32$
   • D$48$

   (c)

   Given that $0<k\le2$ and $\int_{0}^{k}f(x)\,\mathrm{d}x=11$, find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The gradient of a curve $C$ is given by $\frac{\mathrm{d}y}{\mathrm{d}x}=4x^{3}-6x+\frac{2}{x^{2}}$, for $x\neq0$.

   (a)

   Find $\int\left(4x^{3}-6x+\frac{2}{x^{2}}\right)\mathrm{d}x$.

   [3 marks]

   (b)

   The curve $C$ passes through the point $(1,\,4)$. Find the equation of $C$ and hence find the value of $y$ when $x=2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Water flows into a tank at a rate of $r(t)=12t-t^{2}$ litres per minute, where $t$ is the time in minutes, for $0\le t\le12$. Note that $r(t)>0$ for $0<t<12$. At $t=0$ the tank contains 50 litres of water. No water leaves the tank.

   (a)

   (i) Write down an integral that gives the volume of water that flows into the tank during the first 12 minutes, and find this volume.
   (ii) Find an expression for $V(t)$, the volume of water in the tank, in litres, at time $t$.

   [6 marks]

   (b)

   You may use a calculator in this part.
   (i) Find the time at which the tank contains 200 litres of water.
   (ii) Determine whether more water flows into the tank during the middle four minutes, $4\le t\le8$, than during the other eight minutes combined. Justify your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions