1. 1
   A structural engineer works through the following algebraic expressions when checking beam calculations.

   (a)

   (a) Simplify $5a^2b \times 3ab^3$.

   [1 mark]

   • A$15a^3b^4$
   • B$15a^2b^3$
   • C$8a^3b^4$
   • D$15a^3b^3$

   (b)

   (b) Expand and simplify $(2x+3)(x-5)$.

   [1 mark]

   • A$2x^2+7x-15$
   • B$2x^2-7x-15$
   • C$2x^2-13x-15$
   • D$2x^2-7x+15$

   (c)

   (c) Factorise $x^2 - 49$ fully.

   [1 mark]

   • A$(x-49)(x+1)$
   • B$(x-7)^2$
   • C$(x-7)(x+7)$
   • Dcannot be factorised

   Total for question 1: 3 marks

2. 2
   A delivery van can carry a maximum load of $500$ kg. The van already carries fixed equipment weighing $86$ kg, and each type A box weighs $18$ kg.

   (a)

   (a) Form an inequality for the number of type A boxes, $n$, that can be loaded, and solve it to find the maximum whole number of boxes.

   [2 marks]

   (b)

   (b) The van then also carries a crate weighing $34$ kg in addition to the fixed equipment. Solve $86 + 34 + 18n \le 500$ to find the new maximum whole number of type A boxes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence is defined by the $n$th term formula $T_n = 3n^2 - n$.

   (a)

   (a) Show that the 4th term of the sequence is $44$, and find the 5th term.

   [3 marks]

   (b)

   (b) By finding the first and second differences of $T_1, T_2, T_3, T_4, T_5$, show that this sequence has a constant second difference, and state its value.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A phone's battery percentage, $P\%$, when charging from empty, is modelled by $P = 4t$ for the first $20$ minutes, where $t$ is time in minutes. After $20$ minutes the charging rate slows, and the model becomes $P = 80 + \frac{1}{3}(t - 20)$ until the battery is full.

   (a)

   (a) Find the battery percentage after $20$ minutes, and after $50$ minutes of charging, using the appropriate formula for each stage.

   [4 marks]

   (b)

   (b) Find the gradient of each stage of charging, and interpret what each gradient represents in this context.

   [4 marks]

   (c)

   (c) The manufacturer states that charging time to full is inversely proportional to charging current, $I$, so that $T = \frac{k}{I}$, and that at a current of $2$ A the time taken (using only the second-stage rate throughout) is $90$ minutes. Find $k$, and hence find the charging time at a current of $3$ A.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   A theatre uses three separate algebraic models to plan a show: ticket sales modelled by $x^2 - 5x - 24 = 0$, walk-up sales restricted by $3x - 7 > 2x + 9$, and total costs balanced by the equation $5(x-2) = 3(x+4)$.

   (a)

   (a) Solve $x^2 - 5x - 24 = 0$ by factorising and give the positive root.

   [1 mark]

   • A3
   • B$-3$
   • C24
   • D8

   (b)

   (b) Solve the inequality $3x - 7 > 2x + 9$.

   [1 mark]

   • A$x > 16$
   • B$x < 16$
   • C$x > 2$
   • D$x > -16$

   (c)

   (c) Solve $5(x-2) = 3(x+4)$.

   [1 mark]

   • A22
   • B11
   • C$-11$
   • D7

   Total for question 5: 3 marks

6. 6
   A school physics club uses the formula $R = \frac{V^2}{P}$ for resistance, and a maths challenge poses the expression $\frac{3}{x+2} + \frac{2}{x-1}$.

   (a)

   (a) Simplify $\frac{3}{x+2} + \frac{2}{x-1}$, giving your answer as a single fraction.

   [2 marks]

   (b)

   (b) Rearrange $R = \frac{V^2}{P}$ to make $V$ the subject.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Line $L_1$ passes through the points $(1, 4)$ and $(5, 12)$. Line $L_2$ has equation $y = -\frac{1}{2}x + 7$.

   (a)

   (a) Find the equation of $L_1$ in the form $y = mx+c$, and hence show that $L_1$ and $L_2$ are perpendicular.

   [3 marks]

   (b)

   (b) Find the coordinates of the point where $L_1$ and $L_2$ intersect.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   A factory's monthly output (in units) forms a sequence: $50, 62, 74, 86, ...$, where each month's output increases by a fixed amount from the previous month. Separately, the cost per unit to run the factory, $C$ (in £), is inversely proportional to the number of units produced, $u$, and when $u = 200$, $C = £15$.

   (a)

   (a) Find the $n$th term formula for the monthly output sequence, and hence find the output in month 12.

   [4 marks]

   (b)

   (b) Find the formula for $C$ in terms of $u$, and hence find the cost per unit when the factory produces the month 12 output found in part (a).

   [4 marks]

   (c)

   (c) The manager wants to know in which month the output first exceeds $300$ units. Using the $n$th term formula from part (a), form and solve an inequality to find this month.

   [5 marks]

   Total for question 8: 13 marks

End of questions