1. 1
   A maths tutor sets a starter exercise using the expression $P=4x^2-3x$ and the index expression $2a^3\times3a^2$, to check substitution and index-law skills before a test.

   (a)

   What is the value of $P=4x^2-3x$ when $x=-2$?

   [1 mark]

   • A22
   • B10
   • C$-22$
   • D14

   (b)

   Simplify $2a^3\times3a^2$.

   [1 mark]

   • A$6a^6$
   • B$5a^5$
   • C$6a^5$
   • D$6a^{-1}$

   (c)

   Evaluate $(3a^2)^0$, for $a\neq0$.

   [1 mark]

   • A0
   • B1
   • C3
   • D9

   Total for question 1: 3 marks

2. 2
   A bicycle hire company charges a fixed deposit plus an hourly rate. For one hire lasting $x$ hours, the total cost in dollars satisfies $4x+9=45$. A separate car park charge, in dollars, for $y$ half-hour blocks satisfies $7(y-4)=3y+20$.

   (a)

   Solve $4x+9=45$ for $x$.

   [2 marks]

   (b)

   Solve $7(y-4)=3y+20$ for $y$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A café sells two sizes of smoothie, small ($s$ dollars) and large ($l$ dollars). One order of 2 small and 3 large smoothies costs $23.50 total, giving $2s+3l=23.5$. Another order of 4 small and 1 large smoothie costs $18.50 total, giving $4s+l=18.5$.

   (a)

   Solve the simultaneous equations to find $s$, the price of a small smoothie in dollars.

   [3 marks]

   (b)

   Hence find $l$, the price of a large smoothie, and verify your solution satisfies $2s+3l=23.5$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A rectangular vegetable plot has length $(x+7)$ metres and width $x$ metres. The area of the plot is 60 square metres.

   (a)

   Form an equation in $x$ for the area of the plot, and show that it can be written as $x^2+7x-60=0$.

   [4 marks]

   (b)

   Solve $x^2+7x-60=0$ by factorisation to find the width of the plot, $x$ (a positive value).

   [4 marks]

   (c)

   The farmer increases the width by 20% but keeps the length the same. Calculate the percentage increase in the area of the plot.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   A tutor sets three quick-fire algebra questions on expanding and factorising, based on the expressions $(x+8)(x-5)$, $8xy+12y^2$ and $6x^2-5x-6$.

   (a)

   Expand and simplify $(x+8)(x-5)$.

   [1 mark]

   • A$x^2-3x-40$
   • B$x^2+3x-13$
   • C$x^2+3x+40$
   • D$x^2+3x-40$

   (b)

   Factorise fully $8xy+12y^2$.

   [1 mark]

   • A$4y(2x+3y)$
   • B$4(2xy+3y^2)$
   • C$2y(4x+6y)$
   • D$4y(2x+3y^2)$

   (c)

   Factorise $6x^2-5x-6$.

   [1 mark]

   • A$(2x+3)(3x-2)$
   • B$(2x+2)(3x-3)$
   • C$(2x-3)(3x+2)$
   • D$(x-3)(6x+2)$

   Total for question 5: 3 marks

6. 6
   A cylindrical water tank has total surface area $S=2\pi r^2+2\pi rh$. A pendulum's period is given by $T=2\pi\sqrt{\frac{l}{g}}$.

   (a)

   Rearrange the formula $S=2\pi r^2+2\pi rh$ to make $h$ the subject.

   [2 marks]

   (b)

   Rearrange the formula $T=2\pi\sqrt{\frac{l}{g}}$ to make $l$ the subject.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A teacher sets a proof task using $n$ to represent any integer.

   (a)

   Prove that the sum of any two consecutive odd numbers is always a multiple of 4.

   [3 marks]

   (b)

   Prove that $(n+3)^2-(n-3)^2$ is always a multiple of 12 for any integer $n$.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   The temperature, in degrees Celsius, inside a greenhouse over a 10-hour period is modelled by $C(t)=-t^2+8t+5$, where $t$ is the number of hours after sunrise.

   (a)

   Write $C(t)=-t^2+8t+5$ in the form $a(t+b)^2+c$.

   [4 marks]

   (b)

   Hence write down the maximum temperature in the greenhouse during this period, and the number of hours after sunrise at which it occurs.

   [4 marks]

   (c)

   Using your completed-square form, find the values of $t$ for which the greenhouse temperature is at least 16°C.

   [5 marks]

   Total for question 8: 13 marks

End of questions