Questions - OCR MEI C2 - A-Level Maths

OCR MEI C2 2013 January Q8

Sketch the graph of $y = 3 ^ { x }$.
Solve the equation $3 ^ { 5 x - 1 } = 500000$.

OCR MEI C2 2013 January Q9

Show that the equation $\frac { \tan \theta } { \cos \theta } = 1$ may be rewritten as $\sin \theta = 1 - \sin ^ { 2 } \theta$.
Hence solve the equation $\frac { \tan \theta } { \cos \theta } = 1$ for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C2 2013 January Q10

10 Fig. 10 shows a sketch of the curve $y = x ^ { 2 } - 4 x + 3$. The point A on the curve has $x$-coordinate 4 . At point B the curve crosses the $x$-axis. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19552108-0808-4946-a937-9074d58519b2-4_768_734_500_667} \captionsetup{labelformat=empty} \caption{Fig. 10}

\end{figure}

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

OCR MEI C2 2013 January Q11

An arithmetic progression has first term $A$ and common difference $D$. The sum of its first two terms is 25 and the sum of its first four terms is 250 .
(A) Find the values of $A$ and $D$.
(B) Find the sum of the 21st to 50th terms inclusive of this sequence.
A geometric progression has first term $a$ and common ratio $r$, with $r \neq \pm 1$. The sum of its first two terms is 25 and the sum of its first four terms is 250 . Use the formula for the sum of a geometric progression to show that $\frac { r ^ { 4 } - 1 } { r ^ { 2 } - 1 } = 10$ and hence or otherwise find algebraically the possible values of $r$ and the corresponding values of $a$.

OCR MEI C2 2013 January Q12

12 The table shows population data for a country.

The data may be represented by an exponential model of growth. Using $t$ as the number of years after 1960, a suitable model is $p = a \times 10 ^ { k t }$.

Derive an equation for $\log _ { 10 } p$ in terms of $a , k$ and $t$.
Complete the table and draw the graph of $\log _ { 10 } p$ against $t$, drawing a line of best fit by eye.
Use your line of best fit to express $\log _ { 10 } p$ in terms of $t$ and hence find $p$ in terms of $t$.
According to the model, what was the population in 1960 ?
According to the model, when will the population reach 200 million?

OCR MEI C2 2011 June Q1

1 Find $\int _ { 2 } ^ { 5 } \left( 2 x ^ { 3 } + 3 \right) \mathrm { d } x$.

OCR MEI C2 2011 June Q2

2 A sequence is defined by $$\begin{aligned} u _ { 1 } & = 10
u _ { r + 1 } & = \frac { 5 } { u _ { r } ^ { 2 } } \end{aligned}$$ Calculate the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
What happens to the terms of the sequence as $r$ tends to infinity?

OCR MEI C2 2011 June Q3

3 The equation of a curve is $y = \sqrt { 1 + 2 x }$.

Calculate the gradient of the chord joining the points on the curve where $x = 4$ and $x = 4.1$. Give your answer correct to 4 decimal places.
Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when $x = 4$.

OCR MEI C2 2011 June Q4

4 The graph of $y = a b ^ { x }$ passes through the points $( 1,6 )$ and $( 2,3.6 )$. Find the values of $a$ and $b$.

OCR MEI C2 2011 June Q5

5 Find the equation of the normal to the curve $y = 8 x ^ { 4 } + 4$ at the point where $x = \frac { 1 } { 2 }$.

OCR MEI C2 2011 June Q6

6 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x } - 2$. Given also that the curve passes through the point $( 9,4 )$, find the equation of the curve.

OCR MEI C2 2011 June Q7

7 Solve the equation $\tan \theta = 2 \sin \theta$ for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C2 2011 June Q8

8 Using logarithms, rearrange $p = s t ^ { n }$ to make $n$ the subject.

OCR MEI C2 2011 June Q9

9 You are given that $$\log _ { a } x = \frac { 1 } { 2 } \log _ { a } 16 + \log _ { a } 75 - 2 \log _ { a } 5 .$$ Find the value of $x$.

OCR MEI C2 2011 June Q10

10 The $n$th term, $t _ { n }$, of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ } .$$ Express $t _ { 1 }$ and $t _ { 2 }$ in terms of $\sin \theta ^ { \circ }$.

OCR MEI C2 2011 June Q11

The standard formulae for the volume $V$ and total surface area $A$ of a solid cylinder of radius $r$ and height $h$ are $$V = \pi r ^ { 2 } h \quad \text { and } \quad A = 2 \pi r ^ { 2 } + 2 \pi r h .$$ Use these to show that, for a cylinder with $A = 200$, $$V = 100 r - \pi r ^ { 3 }$$
Find $\frac { \mathrm { d } V } { \mathrm {~d} r }$ and $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} r ^ { 2 } }$.
Use calculus to find the value of $r$ that gives a maximum value for $V$ and hence find this maximum value, giving your answers correct to 3 significant figures.

OCR MEI C2 2011 June Q12

12 Jim and Mary are each planning monthly repayments for money they want to borrow.

Jim's first payment is $\pounds 500$, and he plans to pay $\pounds 10$ less each month, so that his second payment is $\pounds 490$, his third is $\pounds 480$, and so on.
(A) Calculate his 12th payment.
(B) He plans to make 24 payments altogether. Show that he pays $\pounds 9240$ in total.
Mary's first payment is $\pounds 460$ and she plans to pay $2 \%$ less each month than the previous month, so that her second payment is $\pounds 450.80$, her third is $\pounds 441.784$, and so on.
(A) Calculate her 12th payment.
(B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
(C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
(D) How much would Mary's first payment need to be if she wishes to pay $2 \%$ less each month as before, but to pay the same in total as Jim, $\pounds 9240$, over the 24 months?

OCR MEI C2 2011 June Q13

13 Fig. 13.1 shows a greenhouse which is built against a wall. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{97ed9d1d-b9e5-47d6-a451-b14757c0e19d-4_606_828_347_358} \captionsetup{labelformat=empty} \caption{Fig. 13.1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{97ed9d1d-b9e5-47d6-a451-b14757c0e19d-4_401_350_529_1430} \captionsetup{labelformat=empty} \caption{Fig. 13.2}

\end{figure} The greenhouse is a prism of length 5.5 m . The curve AC is an arc of a circle with centre B and radius 2.1 m , as shown in Fig. 13.2. The sector angle ABC is 1.8 radians and ABD is a straight line. The curved surface of the greenhouse is covered in polythene.

Find the length of the arc AC and hence find the area of polythene required for the curved surface of the greenhouse.
Calculate the length BD .
Calculate the volume of the greenhouse.