| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Rearrange to make variable subject (quadratic) |
| Difficulty | Standard +0.3 This is a straightforward optimization setup question requiring basic volume formula manipulation and surface area calculation. Part (a) involves simple algebraic rearrangement of V=2x²h=1030, and part (b) requires substituting into the standard cuboid surface area formula. Both are routine procedures with clear steps, making this slightly easier than average for A-level. |
| Spec | 1.02z Models in context: use functions in modelling1.07t Construct differential equations: in context |
Total
Question 8:
8
Total
PMT
1. (a) Given that 8 = 2k , write down the value of k. (1)
(b) Given that 4x = 82 − x , find the value of x. (4)
2. Given that (2 + √7)(4 − √7) = a + b√7, where a and b are integers,
(a) find the value of a and the value of b.
(2)
2+ 7
Given that = c + d√7 where c and d are rational numbers,
4+ 7
(b) find the value of c and the value of d.
(3)
3
3. y = 7 + 10x2.
dy
(a) Find .
dx
(2)
⌠
(b) Find y dx.
⌡
(3)
4. (a) By completing the square, find in terms of k the roots of the equation
x2 + 2kx – 7 = 0. (4)
(b) Prove that, for all values of k, the roots of x2 + 2kx – 7 = 0 are real and different.
(2)
(c) Given that k = √2, find the exact roots of the equation.
(2)
PMT
5. The straight line l has equation 4y + x = 0.
1
The straight line l has equation y = 2x − 3.
2
(a) On the same axes, sketch the graphs of l and l . Show clearly the coordinates of all
1 2
points at which the graphs meet the coordinate axes.
(3)
The lines l and l intersect at the point A.
1 2
(b) Calculate, as exact fractions, the coordinates of A.
(3)
(c) Find an equation of the line through A which is perpendicular to l . Give your answer in
1
the form ax + by + c = 0, where a, b and c are integers.
(3)
6. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays
£500. Her payments then increase by £50 each year, so that she pays £550 in the second year,
£600 in the third year, and so on.
(a) Find the amount that Anne will pay in the 40th year.
(2)
(b) Find the total amount that Anne will pay in over the 40 years.
(2)
Over the same 40 years, Brian will also pay money into the savings scheme. In the first year
he pays in £890 and his payments then increase by £d each year.
Given that Brian and Anne will pay in exactly the same amount over the 40 years,
(c) find the value of d.
(4)
PMT
7. Figure 1
C
B (8, 5)
M
A (2, –3)
The points A and B have coordinates (2, –3) and (8, 5) respectively, and AB is a chord of a
circle with centre C, as shown in Fig. 1.
(a) Find the gradient of AB.
(2)
The point M is the mid-point of AB.
(b) Find an equation for the line through C and M.
(5)
Given that the x-coordinate of C is 4,
(c) find the y-coordinate of C,
(2)
5√17
(d) show that the radius of the circle is .
4
(4)\includegraphics{figure_4}
A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2$x$ cm by $x$ cm and height $h$ cm, as shown in Fig. 4.
Given that the capacity of a carton has to be 1030 cm$^3$,
\begin{enumerate}[label=(\alph*)]
\item express $h$ in terms of $x$, [2]
\item show that the surface area, $A$ cm$^2$, of a carton is given by
$$A = 4x^2 + \frac{3090}{x}.$$ [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [5]}}