| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.3 This is a straightforward geometric series question testing standard formulas (sum to infinity, nth term, sum of n terms). Part (a) is a 'show that' using S∞ = a/(1-r), parts (b-c) apply standard formulas, and part (d) requires minor algebraic manipulation with the constraint that n is odd. All techniques are routine C2 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
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Question 7:
7
Total
PhysicsAndMathsTutor.com
1. Given that p = log 16, express in terms of p,
q
(a) log 2,
q
(2)
(b) log (8q).
q
(4)
2. The expansion of (2 – px)6 in ascending powers of x, as far as the term in x2, is
64 + Ax + 135x2.
Given that p > 0, find the value of p and the value of A.
(7)
3. A circle C has equation
x2 + y2 – 6x + 8y – 75 = 0.
(a) Write down the coordinates of the centre of C, and calculate the radius of C.
(3)
A second circle has centre at the point (15, 12) and radius 10.
(b) Sketch both circles on a single diagram and find the coordinates of the point where they
touch.
(4)
4. (a) Sketch, for 0 ≤ x ≤ 360°, the graph of y = sin (x + 30°).
(2)
(b) Write down the coordinates of the points at which the graph meets the axes.
(3)
(c) Solve, for 0 ≤ x < 360°, the equation
sin (x + 30°) = −1 .
2
(3)
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5. Figure 1
B C
3 cm
A
The shape of a badge is a sector ABC of a circle with centre A and radius AB, as shown in
Fig 1. The triangle ABC is equilateral and has a perpendicular height 3 cm.
(a) Find, in surd form, the length AB.
(2)
(b) Find, in terms of π, the area of the badge.
(2)
2 3
(c) Prove that the perimeter of the badge is (π+6) cm.
3
(2)
6. f(x) = 6x3 + px2 + qx + 8, where p and q are constants.
Given that f(x) is exactly divisible by (2x − 1), and also that when f(x) is divided by (x − 1) the
remainder is −7,
(a) find the value of p and the value of q.
(6)
(b) Hence factorise f(x) completely.
(3)
3 Turn over
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7. A geometric series has first term 1200. Its sum to infinity is 960.
(a) Show that the common ratio of the series is −1 .
4
(3)
(a) Find, to 3 decimal places, the difference between the ninth and tenth terms of the series.
(3)
(c) Write down an expression for the sum of the first n terms of the series.
(2)
Given that n is odd,
(d) prove that the sum of the first n terms of the series is
960(1 + 0.25n).
(2)
8. A circle C has centre (3, 4) and radius 3√2. A straight line l has equation y = x + 3.
(a) Write down an equation of the circle C.
(2)
(b) Calculate the exact coordinates of the two points where the line l intersects C, giving your
answers in surds.
(5)
(c) Find the distance between these two points.
(2)
4 Turn over
RA geometric series has first term $1200$. Its sum to infinity is $960$.
\begin{enumerate}[label=(\alph*)]
\item Show that the common ratio of the series is $-\frac{1}{4}$. [3]
\item Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
\item Write down an expression for the sum of the first $n$ terms of the series. [2]
\end{enumerate}
Given that $n$ is odd,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item prove that the sum of the first $n$ terms of the series is
$$960(1 + 0.25^n).$$ [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [10]}}