| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Multiple Region or Composite Area |
| Difficulty | Standard +0.3 This is a standard C2 integration question requiring factorisation, finding roots, differentiation for gradient, and calculating area between curve and x-axis. All techniques are routine for this level, though part (d) requires careful setup of definite integrals. The 7 marks for part (d) reflect computational work rather than conceptual difficulty. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
Total
| Answer | Marks |
|---|---|
| in ter | a circle with centre O and radius 5 m. |
| Answer | Marks |
|---|---|
| are straight lines and the size of ∠AOD is θ rad | ians. |
| Answer | Marks | Guidance |
|---|---|---|
| er | ms of θ , an expression for the area of the flower | b |
Question 7:
7
Total
PhysicsAndMathsTutor.com
1. f(x) ≡ ax3 + bx2 – 7x + 14, where a and b are constants.
Given that when f(x) is divided by (x – 1) the remainder is 9,
(a) write down an equation connecting a and b. (2 marks)
Given also that (x + 2) is a factor of f(x),
(b) find the values of a and b. (4 marks)
2. (i) Differentiate with respect to x
x2 +2x
2x3 + √x + . (5 marks)
x2
(ii) Evaluate
4
⌠ x 1
+ dx. (5 marks)
⌡ 2 x2
1
3. (a) An arithmetic series has first term a and common difference d. Prove that the sum of the
first n terms of the series is
1 n[2a + (n – 1)d]. (4 marks)
2
A company made a profit of £54000 in the year 2001. A model for future performance assumes
that yearly profits will increase in an arithmetic sequence with common difference £d. This
model predicts total profits of £619200 for the 9 years 2001 to 2009 inclusive.
(b) Find the value of d. (4 marks)
Using your value of d,
(c) find the predicted profit for the year 2011. (2 marks)
An alternative model assumes that the company’s yearly profits will increase in a geometric
sequence with common ratio 1.06. Using this alternative model and again taking the profit in
2001 to be £54000,
(d) find the predicted profit for the year 2011.
(3 marks)
PhysicsAndMathsTutor.com
4. (a) Write down formulae for sin (A + B) and sin (A − B).
Using X = A + B and Y = A − B, prove that
X +Y X −Y
Sin X + sin Y = 2 sin cos .
2 2
(4 marks)
(b) Hence, or otherwise, solve, for 0 ≤ θ < 360,
sin 40° + sin 20° = 0.
(5 marks)
3 Turn over
C
d with perimeter ABCD.
ians.
bed. (3
Figure 1 shows
AD is an arc of
BC is an arc of
OAB and ODC
(a) Find, in ter
Given that the
rc of
rc of
ODC
in ter | a circle with centre O and radius 5 m.
a circle with centre O and radius 7 m.
are straight lines and the size of ∠AOD is θ rad | ians.
bed.
er | ms of θ , an expression for the area of the flower | b
D
R\includegraphics{figure_2}
Figure 2 shows part of the curve C with equation y = f(x), where
$$f(x) = x^3 - 6x^2 + 5x.$$
The curve crosses the x-axis at the origin O and at the points A and B.
(a) Factorise f(x) completely [3 marks]
(b) Write down the x-coordinates of the points A and B. [1 marks]
(c) Find the gradient of C at A. [3 marks]
The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
(d) Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]
\hfill \mbox{\textit{Edexcel C2 Q7 [14]}}