| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape area |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing standard circle geometry and sector formulas. Part (a) requires equating two areas (rectangle + semicircle = sector) and algebraic manipulation, but the steps are routine. Parts (b)-(d) involve direct application of perimeter formulas with minimal problem-solving. The question is slightly easier than average due to its predictable structure and standard techniques, though the multi-part nature and algebraic manipulation in part (a) prevent it from being trivially easy. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
Total
Question 7:
7
Total
PhysicsAndMathsTutor.com
1. (a) Using the factor theorem, show that (x + 3) is a factor of
x3 – 3x2 – 10x + 24. (2 marks)
(b) Factorise x3 – 3x2 – 10x + 24 completely. (4 marks)
2. f(n) = n3 + pn2 + 11n + 9, where p is a constant.
(a) Given that f(n) has a remainder of 3 when it is divided by (n + 2), prove that p = 6.
(2 marks)
(b) Show that f(n) can be written in the form (n + 2)(n + q)(n + r) + 3, where q and r are
integers to be found. (3 marks)
(c) Hence show that f(n) is divisible by 3 for all positive integer values of n. (2 marks)
3. Find the values of θ, to 1 decimal place, in the interval −180 ≤ θ < 180 for which
2 sin2 θ ° − 2 sin θ ° = cos2 θ °. (8 marks)
4. Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per
year. Hence, after x years, the total value of an initial £1 investment is £y, where
y = 1.04x.
(a) Sketch the graph of y = 1.04x, x ≥ 0. (2 marks)
(b) Calculate, to the nearest £, the total value of an initial £800 investment after 10 years.
(2 marks)
(c) Use logarithms to find the number of years it takes to double the total value of any initial
investment. (3 marks)
PhysicsAndMathsTutor.com
5. The curve C with equation y = p + qex, where p and q are constants, passes through the point
(0, 2). At the point P (ln 2, p + 2q on C, the gradient is 5.
(a) Find the value of p and the value of q. (5 marks)
The normal to C at P crosses the x-axis at L and the y-axis at M.
(b) Show that the area of ∆OLM, where O is the origin, is approximately 53.8 (5 marks)
6. Figure 3
y
C P
R
O A x
Figure 3 shows part of the curve C with equation
y = 3 x2 – 1 x3.
2 4
The curve C touches the x-axis at the origin and passes through the point A(p, 0).
(a) Show that p = 6. (1 marks)
(b) Find an equation of the tangent to C at A. (4 marks)
The curve C has a maximum at the point P.
(c) Find the x-coordinate of P. (2 marks)
The shaded region R, in Fig. 3, is bounded by C and the x-axis.
(d) Find the area of R. (4 marks)
3 Turn over
Figure 1
O
Q
Shape Y
PhysicsAndMathsTutor.com
n
x
8. f(x) = 1+ , k, n ∈ ℕ, n > 2.
k
Given that the coefficient of x3 is twice the coefficient of x2 in the binomial expansion of f(x),
(a) prove that n = 6k + 2. (3 marks)
Given also that the coefficients of x4 and x5 are equal and non-zero,
(b) form another equation in n and k and hence show that k = 2 and n = 14. (4 marks)
Using these values of k and n,
(c) expand f(x) in ascending powers of x, up to and including the term in x5. Give each
coefficient as an exact fraction in its lowest terms (4 marks)
END
5 Turn over\includegraphics{figure_1}
Figure 1 shows the cross-sections of two drawer handles.
Shape $X$ is a rectangle $ABCD$ joined to a semicircle with $BC$ as diameter. The length $AB = d$ cm and $BC = 2d$ cm.
Shape $Y$ is a sector $OPQ$ of a circle with centre $O$ and radius $2d$ cm.
Angle $POQ$ is $\theta$ radians.
Given that the areas of the shapes $X$ and $Y$ are equal,
\begin{enumerate}[label=(\alph*)]
\item prove that $\theta = 1 + \frac{1}{2}\pi$. [5]
\end{enumerate}
Using this value of $\theta$, and given that $d = 3$, find in terms of $\pi$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item the perimeter of shape $X$, [2]
\item the perimeter of shape $Y$. [3]
\item Hence find the difference, in mm, between the perimeters of shapes $X$ and $Y$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [12]}}