1. 1
   A toy company designs three number machines. Machine 1 takes an input $x$, doubles it, then adds 5. Machine 2 takes the output of Machine 1 and multiplies the whole result by 3. Machine 3 is used separately to collect like terms in an expression the engineers write down: $4x^2 + 3x - x^2 + 5x$.

   (a)

   Find the output of the first machine when the input is $x = 4$.

   [1 mark]

   • A9
   • B13
   • C14
   • D20

   (b)

   Which expression is equivalent to $3(2x+5)$, the result of passing Machine 1's output through Machine 2?

   [1 mark]

   • A$6x+5$
   • B$6x+15$
   • C$5x+15$
   • D$6x+8$

   (c)

   Machine 3 collects like terms in $4x^2 + 3x - x^2 + 5x$. What is the simplified result?

   [1 mark]

   • A$3x^2+2x$
   • B$3x^2+8x$
   • C$4x^2+8x$
   • D$5x^2+8x$

   Total for question 1: 3 marks

2. 2
   A market trader values his stock of boxes using the formula $V = 3n^2 - 2n$, where $n$ is the number of boxes and $V$ is the value in dollars.

   (a)

   Evaluate $V$ when $n = 5$.

   [2 marks]

   (b)

   The trader doubles his stock and adds a further $4n$ dollars of value. Simplify $2(3n^2-2n) + 4n$ fully by collecting like terms.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A laboratory records bacteria growth using powers of a base value. A technician must simplify expressions combining these powers using the index laws $a^m \times a^n = a^{m+n}$, $a^m \div a^n = a^{m-n}$ and $(a^m)^n = a^{mn}$.

   (a)

   Simplify fully $\frac{a^7 \times a^3}{a^4}$, showing each index law used.

   [3 marks]

   (b)

   Given that $b^0 = 1$, simplify $\frac{b^5 \times b^{-2}}{b^0}$, giving your answer in the form $b^n$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A fitness instructor tracks a client's heart rate recovery after exercise using the formula $H = 180-2t$, where $H$ is the heart rate in beats per minute and $t$ is the number of minutes since exercise stopped. A second client's recovery is given by $170-3t$. A third client's heart rate decay factor involves powers of a constant $k$.

   (a)

   Evaluate $H$ when $t = 12$, and state whether the heart rate has dropped below 150 beats per minute, showing full working.

   [4 marks]

   (b)

   The combined recovery expression for the first two clients is $(180-2t)+(170-3t)$. Simplify this fully by collecting like terms.

   [4 marks]

   (c)

   A third client's heart rate decay factor is given by $k^3 \times k^{-5} \div k^{-4}$, where $k$ represents a constant related to fitness level. Simplify this expression fully, showing each index law step, and state the final power of $k$.

   [5 marks]

   Total for question 4: 13 marks

End of questions