Question 2 - A-Level Maths

OCR MEI C2 — Question 2 11 marks

Exam Board	OCR MEI
Module	C2 (Core Mathematics 2)
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Tangents, normals and gradients
Type	Normal meets curve/axis — further geometry
Difficulty	Standard +0.3 This is a standard C2 calculus question combining differentiation (finding normal), coordinate geometry, and definite integration for area. Part (i) involves routine differentiation, finding gradient of normal, and line equation - all textbook techniques. Part (ii) requires setting up and evaluating a definite integral with some care about the region, but follows standard methods. The 11 marks reflect multiple steps rather than conceptual difficulty. Slightly above average due to the integration of multiple techniques and the geometric visualization required, but well within expected C2 scope.
Spec	1.07i Differentiate x^n: for rational n and sums 1.07m Tangents and normals: gradient and equations 1.08e Area between curve and x-axis: using definite integrals

Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_2}

Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C (16, 0). [6]
Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]

Show mark scheme Show mark scheme source

Question 2:

Answer	Marks	Guidance
2	(i)	at A y = 3

dy

2x4

dx

dy

their  244

dx

grad of normal = –1/

their 4

y  3 = (1/ ) × (x  4) oe isw

4 substitution of y = 0 and completion to

given result with at least 1 correct interim

Answer	Marks
step www	B1

B1

M1*

M1dep*

A1

Answer	Marks
[6]	must follow from attempt at differentiation
or substitution of x = 16 to obtain y = 0	correct interim step may occur before

substitution

Answer	Marks	Guidance
2	(ii)	at B, x = 3

x3 4x2

F[x]  3x

3 2

F[ 4]  F[their 3]

area of triangle = 18 soi

1 area of region = 19 oe isw

Answer	Marks
3	B1

M1*

dep

B1

A1

Answer	Marks
[5]	may be embedded

condone one error, must be three terms,

ignore + c

dependent on integration attempted

Answer	Marks
19.3 or better	may be embedded in final answer

Question 2:
2 | (i) | at A y = 3
dy
2x4
dx
dy
their  244
dx
grad of normal = –1/
their 4
y  3 = (1/ ) × (x  4) oe isw
4
substitution of y = 0 and completion to
given result with at least 1 correct interim
step www | B1
B1
M1*
M1dep*
A1
A1
[6] | must follow from attempt at differentiation
or substitution of x = 16 to obtain y = 0 | correct interim step may occur before
substitution
2 | (ii) | at B, x = 3
x3 4x2
F[x]  3x
3 2
F[ 4]  F[their 3]
area of triangle = 18 soi
1
area of region = 19 oe isw
3 | B1
M1*
M1*
dep
B1
A1
[5] | may be embedded
condone one error, must be three terms,
ignore + c
dependent on integration attempted
19.3 or better | may be embedded in final answer

Show LaTeX source

Fig. 10 shows a sketch of the curve $y = x^2 - 4x + 3$. The point A on the curve has $x$-coordinate 4. At point B the curve crosses the $x$-axis.

\includegraphics{figure_2}

\begin{enumerate}[label=(\roman*)]
\item Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the $x$-axis at C (16, 0). [6]

\item Find the area of the region ABC bounded by the curve, the normal at A and the $x$-axis. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C2  Q2 [11]}}

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Q1 13 Q2 11 Q3 12 Q4 12 Q5 5