1. 1
   A robotics club is setting three warm-up puzzles for a competition. Puzzle 1 asks students to simplify $3(2x - 5) - 4(x - 1)$. Puzzle 2 asks students to solve the inequality $5x - 3 \le 2x + 12$. Puzzle 3 uses the sequence with $n$th term $T(n) = 4n - 7$.

   (a)

   What is the simplified form of $3(2x - 5) - 4(x - 1)$?

   [1 mark]

   • A$2x - 11$
   • B$2x - 19$
   • C$2x - 1$
   • D$10x - 19$

   (b)

   What is the solution to $5x - 3 \le 2x + 12$?

   [1 mark]

   • A$x \geq 5$
   • B$x \leq 5$
   • C$x \leq 3$
   • D$x \leq 15$

   (c)

   What is the 12th term of the sequence with $n$th term $T(n) = 4n - 7$?

   [1 mark]

   • A37
   • B45
   • C41
   • D48

   Total for question 1: 3 marks

2. 2
   A gardener is designing a rectangular allotment. The perimeter, in metres, is given by $2(3x + 4) + 2(x + 1)$, where $x$ is a length in metres. The gardener also uses the formula $V = \frac{1}{3}\pi r^2 h$ to estimate the volume of a conical compost heap of radius $r$ and height $h$.

   (a)

   The perimeter of the allotment is 58 metres. Form and solve an equation to find the value of $x$.

   [2 marks]

   (b)

   Make $h$ the subject of the formula $V = \frac{1}{3}\pi r^2 h$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Line $L$ has equation $y = 2x - 3$. Curve $C$ has equation $y = x^2 - 4x + 5$.

   (a)

   State the gradient of line $L$, and find the coordinates of the point where it crosses the $y$-axis.

   [3 marks]

   (b)

   Find the coordinates of the point(s) where line $L$ intersects curve $C$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   An engineering firm is working on three separate tasks for a client: simplifying a cost ratio expressed as an algebraic fraction, $\frac{x^2 - 9}{x^2 + x - 6}$; verifying the identity $(2n+1)^2 - (2n-1)^2$ used in a quality-control check, for any integer $n$; and analysing the motion of a delivery drone, whose speed in m/s after $t$ seconds is $v = 2.5t$ for $0 \le t \le 8$, after which it travels at a constant speed for a further 6 seconds.

   (a)

   Simplify fully $\frac{x^2 - 9}{x^2 + x - 6}$.

   [4 marks]

   (b)

   Show that $(2n+1)^2 - (2n-1)^2$ is always a multiple of 8, for any integer $n$.

   [4 marks]

   (c)

   State what the gradient of the drone's speed-time graph represents during the first 8 seconds and find its value. Hence find the total distance travelled by the drone over the whole 14 seconds.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   A puzzle-design website publishes three warm-up questions each week. This week's ask students to simplify $4(3x - 2) - 2(5x - 6)$, solve the inequality $7x + 4 > 3x - 8$, and evaluate the sequence with $n$th term $U(n) = 3n^2 - 5$.

   (a)

   What is the simplified form of $4(3x - 2) - 2(5x - 6)$?

   [1 mark]

   • A$2x - 4$
   • B$2x + 20$
   • C$10x + 4$
   • D$2x + 4$

   (b)

   What is the solution to $7x + 4 > 3x - 8$?

   [1 mark]

   • A$x > -3$
   • B$x < -3$
   • C$x > -12$
   • D$x > -1$

   (c)

   What is $U(4)$ for the sequence with $n$th term $U(n) = 3n^2 - 5$?

   [1 mark]

   • A39
   • B43
   • C47
   • D48

   Total for question 5: 3 marks

6. 6
   A carpenter is building a rectangular picture frame. The perimeter, in cm, is given by $2(4x - 1) + 2(2x + 5)$, where $x$ is a length in cm. The carpenter also uses the formula $s = ut + \frac{1}{2}at^2$ to model the distance, $s$, a sliding sawblade travels.

   (a)

   The perimeter of the frame is 104 cm. Form and solve an equation to find the value of $x$.

   [2 marks]

   (b)

   Make $a$ the subject of the formula $s = ut + \frac{1}{2}at^2$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Line $M$ has equation $y = -x + 7$. Curve $D$ has equation $y = x^2 - 5x + 10$.

   (a)

   State the gradient of line $M$, and find the coordinates of the point where it crosses the $x$-axis.

   [3 marks]

   (b)

   Find the coordinates of the point(s) where line $M$ intersects curve $D$.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   A cyclist's speed, in m/s, increases uniformly from rest, so that after $t$ seconds her speed is $v = 1.5t$ for $0 \le t \le 10$, after which she travels at a constant speed for a further 20 seconds. A sequence generator separately produces terms using $W(n) = 5n - 2$, and a fairground ride requires that the number of tickets sold, $x$, satisfies both $3x + 10 \le 100$ and $x \ge 15$.

   (a)

   State what the gradient of the cyclist's speed-time graph represents during the first 10 seconds and find its value. Hence find the total distance she travels over the whole 30 seconds.

   [4 marks]

   (b)

   Determine whether 143 is a term in the sequence with $n$th term $W(n) = 5n - 2$. You must show your working.

   [4 marks]

   (c)

   The number of tickets sold, $x$, satisfies $3x + 10 \le 100$ and $x \ge 15$. Find the set of possible whole-number values of $x$, and state how many possible values there are.

   [5 marks]

   Total for question 8: 13 marks

End of questions