| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Open box from cut-corner sheet |
| Difficulty | Standard +0.3 This is a standard C2 optimization problem involving forming and differentiating a cubic function. Part (a) requires basic algebraic expansion, parts (b)-(d) are routine calculus (finding stationary points), and part (e) asks for standard second derivative test. While multi-part with 12 marks total, each step follows a well-practiced procedure with no novel insight required, making it slightly easier than average. |
| Spec | 1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
Total
Question 7:
7
Total
PhysicsAndMathsTutor.com
1.
f(x) = 4x3 + 3x2 – 2x – 6.
Find the remainder when f(x) is divided by (2x + 1). (3 marks)
2. The point A has coordinates (2, 5) and the point B has coordinates (−2, 8).
Find, in cartesian form, an equation of the circle with diameter AB. (4 marks)
3. f(x) = x3 – 19x – 30.
(a) Show that (x + 2) is a factor of f(x). (2 marks)
(b) Factorise f(x) completely. (4 marks)
3 x−4
4. Express + as a single fraction in its simplest form.
x2 +2x x2 −4
(7 marks)
5. Find, in degrees, the value of θ in the interval 0 ≤θ < 360° for which
2cos2θ − cosθ − 1 = sin2θ.
Give your answers to 1 decimal place where appropriate. (8 marks)
6. A geometric series is a + ar + ar 2 + . . .
(a) Prove that the sum of the first n terms of this series is given by
a(1−rn)
S = . (4 marks)
n
1−r
The second and fourth terms of the series are 3 and 1.08 respectively.
Given that all terms in the series are positive, find
(b) the value of r and the value of a, (5 marks)
(c) the sum to infinity of the series. (3 marks)
PhysicsAndMathsTutor.com
8. Figure 1
O
6.5 cm
A B
Figure 1 shows the sector AOB of a circle, with centre O and radius 6.5 cm, and
∠AOB = 0.8 radians.
(a) Calculate, in cm2, the area of the sector AOB. (2 marks)
(b) Show that the length of the chord AB is 5.06 cm, to 3 significant figures. (3 marks)
The segment R, shaded in Fig. 1, is enclosed by the arc AB and the straight line AB.
(c) Calculate, in cm, the perimeter of R. (2 marks)
4 Turn over
PhysicsAndMathsTutor.com
9. Figure 2
y
y = x + 1
R
B
A y = 6x – x2 − 3
O x
Figure 2 shows the line with equation y = x + 1 and the curve with equation
y = 6x – x2 – 3.
The line and the curve intersect at the points A and B, and O is the origin.
(a) Calculate the coordinates of A and the coordinates of B. (5 marks)
The shaded region R is bounded by the line and the curve.
(b) Calculate the area of R. (7 marks)
END
5 Turn over\includegraphics{figure_2}
A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side $x$ cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.
\begin{enumerate}[label=(\alph*)]
\item Show that the volume, $V$ cm$^3$, of the tray is given by
$$V = 4x(x^2 - 45x + 500)$$ [3]
\item State the range of possible values of $x$. [1]
\item Find the value of $x$ for which $V$ is a maximum. [4]
\item Hence find the maximum value of $V$. [2]
\item Justify that the value of $V$ you found in part (d) is a maximum. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [12]}}