1. 1
   The function $f$ is defined by $f(x)=x^{2}-2x$. The point $P$ on the curve $y=f(x)$ has $x$-coordinate 3, and the point $Q$ on the same curve has $x$-coordinate $3+h$, where $h\neq0$.

   (a)

   Find the gradient of the chord $PQ$ when $h=0.1$.

   [1 mark]

   • A$0.41$
   • B$4$
   • C$4.01$
   • D$4.1$

   (b)

   The gradient of the chord $PQ$ is calculated for values of $h$ that get closer and closer to 0. Write down the value that these chord gradients approach.

   [1 mark]

   • A$3$
   • B$0$
   • C$4$
   • D$6$

   (c)

   Show that the gradient of the chord $PQ$ is $4+h$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cup of coffee is left to cool. Its temperature, $T$ °C, $t$ minutes after it is poured, is recorded. The coffee is at 90 °C when $t=0$ and at 62.5 °C when $t=10$. The average rate of change of $T$ is calculated over short time intervals that start or end at $t=10$. From $t=10$ to $t=10+h$, the average rate of change is $-2.0707$ °C per minute when $h=1$, $-2.1176$ when $h=0.1$ and $-2.1223$ when $h=0.01$. From $t=10-h$ to $t=10$, it is $-2.1768$ when $h=1$, $-2.1282$ when $h=0.1$ and $-2.1234$ when $h=0.01$.

   (a)

   Estimate the instantaneous rate of change of the temperature of the coffee when $t=10$, giving your answer to three significant figures.

   [1 mark]

   • A$-2.07$
   • B$-2.12$
   • C$-2.18$
   • D$2.12$

   (b)

   Which statement correctly interprets the value of $\frac{\mathrm{d}T}{\mathrm{d}t}$ at $t=10$?

   [1 mark]

   • AAt $t=10$ the temperature is falling at about 2.12 °C per minute.
   • BAt $t=10$ the temperature of the coffee is about 2.12 °C.
   • CIn the first 10 minutes the temperature falls by about 2.12 °C.
   • DIt takes about 2.12 minutes for the temperature to fall by 1 °C.

   (c)

   Find the average rate of change of the temperature of the coffee between $t=0$ and $t=10$. Give your answer with units.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^{3}$. The point $A(2,\,8)$ lies on $C$, and the point $B$ on $C$ has $x$-coordinate $2+h$, where $h>0$.

   (a)

   Show that the gradient of the chord $AB$ is $12+6h+h^{2}$.

   [3 marks]

   (b)

   Hence write down the gradient of $C$ at $A$, and find the set of values of $h$ for which the gradient of the chord $AB$ differs from the gradient of $C$ at $A$ by less than 0.61.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A stone is dropped from rest from the top of a sea cliff that is 78.4 m high. The distance, $s$ metres, that the stone has fallen $t$ seconds after it is released is modelled by $s=4.9t^{2}$, until the stone reaches the sea.

   (a)

   (i) Find the average speed of the stone during the first 2 seconds of its fall.
   (ii) Show that the average speed of the stone between $t=2$ and $t=2+h$ is $19.6+4.9h$ m s$^{-1}$, where $h>0$.
   (iii) Hence deduce the speed of the stone when $t=2$.

   [6 marks]

   (b)

   (i) Find the time taken for the stone to reach the sea.
   (ii) By considering the average speed between times $t$ and $t+h$, show that $\frac{\mathrm{d}s}{\mathrm{d}t}=9.8t$.
   (iii) Hence find the speed of the stone as it reaches the sea.

   [6 marks]

   Total for question 4: 12 marks

End of questions