1. 1
   A straight line passes through the points $(1, 4)$ and $(5, 12)$.

   (a)

   What is the gradient of this line?

   [1 mark]

   • A$2$
   • B$0.5$
   • C$8$
   • D$4$

   (b)

   A different line passes through the points $(0, 7)$ and $(3, 7)$. What is the gradient of this line?

   [1 mark]

   • AUndefined
   • B$7$
   • C$0$
   • D$3$

   (c)

   A third line passes through the points $(2, 3)$ and $(2, 9)$. What is the gradient of this line?

   [1 mark]

   • A$0$
   • BUndefined
   • C$6$
   • D$3$

   Total for question 1: 3 marks

2. 2
   The tangent to a curve at the point P(3, 10) has been drawn, and it also passes through the point $(5, 16)$.

   (a)

   Find the gradient of this tangent.

   [2 marks]

   (b)

   State what the gradient found in part (a) represents in the context of the curve at point P.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve passes through the point $(2, 16)$. A nearby point on the curve is $(2.1, 16.83)$.

   (a)

   Use these two points to estimate the gradient of the curve at $x = 2$, giving your answer to 2 decimal places.

   [3 marks]

   (b)

   Explain why using a point closer to $x = 2$ (for example, $x = 2.01$) instead of $x = 2.1$ would generally give a more accurate estimate of the true gradient of the curve at $x = 2$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   The temperature, in degrees Celsius, of a cooling cup of coffee is recorded over time. At $t = 4$ minutes, the temperature is 62 degrees Celsius. At $t = 4.5$ minutes, the temperature is 57.5 degrees Celsius. At $t = 6$ minutes, the temperature is 50 degrees Celsius.

   (a)

   Use these values to estimate the rate of change of temperature (gradient), in degrees Celsius per minute, at $t = 4$, and interpret the sign of your answer.

   [4 marks]

   (b)

   The tangent to the temperature-time graph at $t = 5$ is estimated using the points $(4, 62)$ and $(6, 50)$. Find the gradient of this tangent, and hence estimate the rate at which the coffee is cooling at $t = 5$.

   [4 marks]

   (c)

   Compare your estimated rates of cooling from part (a) (at $t = 4$) and part (b) (at $t = 5$), and explain what this comparison tells you about how the rate of cooling is changing over time.

   [5 marks]

   Total for question 4: 13 marks

End of questions