| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Moderate -0.3 This is a straightforward C1 circle question requiring midpoint formula, gradient calculations, perpendicular lines, and simultaneous equations. All techniques are standard and well-practiced at this level. The multi-part structure guides students through each step, making it slightly easier than average but not trivial due to the algebraic manipulation required in parts (c) and (d). |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
Total
Question 7:
7
Total
PMT
1. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7.
(3)
142
(b) Hence, or otherwise, evaluate ∑(7r +2).
r=1
(3)
2. Solve the simultaneous equations
x – 3y + 1 = 0,
x2 – 3xy + y2 = 11.
(7)
3. The first three terms of an arithmetic series are p, 5p – 8, and 3p + 8 respectively.
(a) Show that p = 4.
(2)
(b) Find the value of the 40th term of this series.
(3)
4. f(x) = x2 – kx + 9, where k is a constant.
(a) Find the set of values of k for which the equation f(x) = 0 has no real solutions.
(4)
Given that k = 4,
(b) express f(x) in the form (x – p)2 + q, where p and q are constants to be found,
(3)
dy 1
5. =5+ .
dx x2
(a) Use integration to find y in terms of x.
(3)
(b) Given that y = 7 when x = 1, find the value of y at x = 2.
(4)
PMT
6. A container made from thin metal is in the shape of a right circular cylinder with height h cm
and base radius r cm. The container has no lid. When full of water, the container holds 500 cm3
of water.
Show that the exterior surface area, A cm2, of the container is given by
1000
A = πr 2 + .
r
(4)
7. Figure 1
B
A
The points A(−3, −2) and B(8 , 4) are at the ends of a diameter of the circle shown in Fig. 1.
(a) Find the coordinates of the centre of the circle.
(2)
(b) Find an equation of the diameter AB, giving your answer in the form ax + by + c = 0,
where a, b and c are integers.
(4)
(c) Find an equation of tangent to the circle at B.
(3)
The line l passes through A and the origin.
(d) Find the coordinates of the point at which l intersects the tangent to the circle at B, giving
your answer as exact fractions.
(4)
END\includegraphics{figure_1}
The points $A(-3, -2)$ and $B(8, 4)$ are at the ends of a diameter of the circle shown in Fig. 1.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the centre of the circle. [2]
\item Find an equation of the diameter $AB$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers. [4]
\item Find an equation of tangent to the circle at $B$. [3]
\end{enumerate}
The line $l$ passes through $A$ and the origin.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the coordinates of the point at which $l$ intersects the tangent to the circle at $B$, giving your answer as exact fractions. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [13]}}