1. 1
   A company's weekly online sales, in units, form an arithmetic sequence over several consecutive weeks. In week 1, sales are 42 units, and in week 4, sales are 72 units.

   (a)

   What is the common difference, $d$, of this sequence?

   [1 mark]

   • A24
   • B10
   • C30
   • D15

   (b)

   Which expression gives the $n$th term of this sequence?

   [1 mark]

   • A$10n+32$
   • B$10n+42$
   • C$42n+10$
   • D$10n+22$

   (c)

   In which week do sales first exceed 150 units?

   [1 mark]

   • A11
   • B13
   • C12
   • D10

   Total for question 1: 3 marks

2. 2
   A straight line has equation $3x+2y=18$. A second line is parallel to this line and passes through the point $(4,-1)$.

   (a)

   Find the gradient and the $y$-intercept of the line $3x+2y=18$.

   [2 marks]

   (b)

   Find the equation of the line parallel to $3x+2y=18$ that passes through $(4,-1)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Functions $f$ and $g$ are defined by $f(x)=3x+1$ and $g(x)=\frac{x-4}{2}$.

   (a)

   Find $fg(x)$, simplifying fully.

   [3 marks]

   (b)

   Find $f^{-1}(x)$, the inverse function of $f$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A drone's height above the ground, in metres, at time $t$ seconds after launch is given by $h=t^3-6t^2+9t+2$, for $0\leq t\leq5$.

   (a)

   Find $\frac{dh}{dt}$, and hence find the drone's vertical velocity at $t=1$ second.

   [4 marks]

   (b)

   Find the values of $t$ at which the drone is momentarily stationary (where $\frac{dh}{dt}=0$), for $0\leq t\leq5$.

   [4 marks]

   (c)

   By considering the gradient of $h$ just before and just after $t=1$, determine whether the drone reaches a maximum or minimum height at $t=1$, then calculate this height.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   Points $A(-3,4)$ and $B(5,0)$ are given.

   (a)

   Find the midpoint of $AB$.

   [1 mark]

   • A$(2,4)$
   • B$(1,-2)$
   • C$(4,-2)$
   • D$(1,2)$

   (b)

   Find the length of $AB$, giving your answer as a surd in its simplest form.

   [1 mark]

   • A$\sqrt{68}$
   • B$4\sqrt5$
   • C$8$
   • D$4\sqrt6$

   (c)

   Find the gradient of the line $AB$.

   [1 mark]

   • A$-\frac{1}{2}$
   • B$\frac{1}{2}$
   • C$-2$
   • D$2$

   Total for question 5: 3 marks

6. 6
   Let $f(x)=x^2+2x$. The graph of $y=f(x)$ is transformed to give the graph of $y=f(x-3)$, and separately to give $y=2f(x)$.

   (a)

   Find an expression for $f(x-3)$, simplifying fully.

   [2 marks]

   (b)

   Find an expression for $2f(x)$, simplifying fully, and state the transformation that maps $y=f(x)$ onto $y=2f(x)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A shop's staffing plan restricts the number of morning workers, $x$, and afternoon workers, $y$, by the inequalities $x\geq2$, $y\geq1$ and $2x+y\leq14$.

   (a)

   Find the coordinates of the three vertices of the region satisfying all three inequalities.

   [3 marks]

   (b)

   The shop wants to maximise $P=3x+2y$, a measure of staff coverage, within this region. Evaluate $P$ at each vertex found in part (a), and state the maximum value of $P$ and where it occurs.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   Curve $C$ has equation $y=\frac{24}{x}$ and line $L$ has equation $y=-x+11$.

   (a)

   Find the coordinates of the two points where curve $C$ and line $L$ intersect.

   [4 marks]

   (b)

   Using the two points found in part (a), estimate the gradient of the chord joining them, and state what this represents for curve $C$ between $x=3$ and $x=8$.

   [4 marks]

   (c)

   The gradient function of $C: y=\frac{24}{x}$ is $\frac{dy}{dx}=-\frac{24}{x^2}$. Calculate the exact gradient of $C$ at $x=3$ and at $x=8$, and explain why the chord gradient found in part (b) lies between these two values.

   [5 marks]

   Total for question 8: 13 marks

End of questions