Easy -1.2 This is a straightforward application of the trapezium rule with explicit instructions (6 strips) and a simple function. Part (a) is basic graph sketching of a standard transformed trig function. The calculation is routine and mechanical with no problem-solving required, making it easier than average for A-level.
6. (a) Sketch the graph of \(y = 1 + \cos x , \quad 0 \leqslant x \leqslant 2 \pi\)
Show on your sketch the coordinates of the points where your graph meets the coordinate axes.
(b) Use the trapezium rule, with 6 strips of equal width, to find an approximate value for
$$\int _ { 0 } ^ { 2 \pi } ( 1 + \cos x ) d x$$
At least one complete cycle. Condone minimal curvature but not linear graphs. Look for minimum value (not a cusp) between two maxima; tolerant of curves without zero gradients at ends.
Correct shape and position
A1
Complete cycle with correct shape and position. Maximum at \(2\pi\) should appear at same height as \(x = 0\). Tolerant of curvature slips.
\((0, 2)\) and \((\pi, 0)\) marked
B1 (3 marks)
Only for range \(0\) to \(2\pi\). Must have sketch. Ignore degree references. Accept \(2\) and \(\pi\) as intercepts on axes but NOT \((2,0)\) and \((0,\pi)\).
Part (b):
Answer
Marks
Strip width \(= \frac{\pi}{3}\)
B1
Attempts correct \(y\) values; sight of \(0, \frac{1}{2}, \frac{3}{2}\) and \(2\)
## Question 6:
### Part (a):
One cosine cycle drawn | M1 | At least one complete cycle. Condone minimal curvature but not linear graphs. Look for minimum value (not a cusp) between two maxima; tolerant of curves without zero gradients at ends.
Correct shape and position | A1 | Complete cycle with correct shape and position. Maximum at $2\pi$ should appear at same height as $x = 0$. Tolerant of curvature slips.
$(0, 2)$ and $(\pi, 0)$ marked | B1 (3 marks) | Only for range $0$ to $2\pi$. Must have sketch. Ignore degree references. Accept $2$ and $\pi$ as intercepts on axes but NOT $(2,0)$ and $(0,\pi)$.
### Part (b):
Strip width $= \frac{\pi}{3}$ | B1 |
Attempts correct $y$ values; sight of $0, \frac{1}{2}, \frac{3}{2}$ and $2$ | M1 |
Area $\approx \frac{1}{2} \times \frac{\pi}{3}\left\{2 + 2 + 2(1.5 + 0.5 + 0 + 0.5 + 1.5)\right\} = 2\pi$ or awrt $6.28$ | M1, A1 (4 marks) |
6. (a) Sketch the graph of $y = 1 + \cos x , \quad 0 \leqslant x \leqslant 2 \pi$
Show on your sketch the coordinates of the points where your graph meets the coordinate axes.\\
(b) Use the trapezium rule, with 6 strips of equal width, to find an approximate value for
$$\int _ { 0 } ^ { 2 \pi } ( 1 + \cos x ) d x$$
\hfill \mbox{\textit{Edexcel C12 2019 Q6 [7]}}