| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Moderate -0.3 This is a straightforward C1 integration and differentiation question requiring standard techniques: integrate to find y(x) using a given point, verify a point lies on the curve, find a tangent equation, and solve for where gradients are equal. All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums |
Total
Question 7:
7
Total
PMT
1. (a) Solve the inequality
3x – 8 > x + 13.
(2)
(b) Solve the inequality
x2 – 5x – 14 > 0.
(3)
1
2. Given that 2x = and 2y = 4√2,
2
(a) find the exact value of x and the exact value of y,
(3)
(b) calculate the exact value of 2y − x .
(2)
3. (a) Prove, by completing the square, that the roots of the equation x2 + 2kx + c = 0, where k
and c are constants, are −k ± √(k2 − c).
(4)
The equation x2 + 2kx ± 81 = 0 has equal roots.
(b) Find the possible values of k.
(2)
4. In the first month after opening, a mobile phone shop sold 280 phones. A model for future
trading assumes that sales will increase by x phones per month for the next 35 months, so that
(280 + x) phones will be sold in the second month, (280 + 2x) in the third month, and so on.
Using this model with x = 5, calculate
(a) (i) the number of phones sold in the 36th month,
(2)
(ii) the total number of phones sold over the 36 months.
(2)
The shop sets a sales target of 17000 phones to be sold over the 36 months.
Using the same model,
(b) find the least value of x required to achieve this target.
(4)
PMT
5. Figure 1
y
Q
y2 = 4(x – 2)
2x – 3y = 12
O A x
P
Figure 1 shows the curve with equation y2 = 4(x – 2) and the line with equation 2x – 3y = 12.
The curve crosses the x-axis at the point A, and the line intersects the curve at the points P and
Q.
(a) Write down the coordinates of A.
(1)
(b) Find, using algebra, the coordinates of P and Q.
(6)
(c) Show that ∠PAQ is a right angle.
(4)
PMT
6. Figure 2
y C
D
B
O A x
The points A (3, 0) and B (0, 4) are two vertices of the rectangle ABCD, as shown in Fig. 2.
(a) Write down the gradient of AB and hence the gradient of BC.
(3)
The point C has coordinates (8, k), where k is a positive constant.
(b) Find the length of BC in terms of k.
(2)
Given that the length of BC is 10 and using your answer to part (b),
(c) find the value of k,
(4)
(d) find the coordinates of D.
(2)
PMT
7. The curve C has equation y = f(x). Given that
dy
= 3x2 – 20x + 29
dx
and that C passes through the point P(2, 6),
(a) find y in terms of x.
(4)
(b) Verify that C passes through the point (4, 0).
(2)
(c) Find an equation of the tangent to C at P.
(3)
The tangent to C at the point Q is parallel to the tangent at P.
(d) Calculate the exact x-coordinate of Q.
(5)
ENDThe curve $C$ has equation $y = f(x)$. Given that
$$\frac{dy}{dx} = 3x^2 - 20x + 29$$
and that $C$ passes through the point $P(2, 6)$,
\begin{enumerate}[label=(\alph*)]
\item find $y$ in terms of $x$.
[4]
\item Verify that $C$ passes through the point $(4, 0)$.
[2]
\item Find an equation of the tangent to $C$ at $P$.
[3]
\end{enumerate}
The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Calculate the exact $x$-coordinate of $Q$.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [14]}}