Questions - OCR MEI - A-Level Maths

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Find $\frac{dy}{dx}$ and show that $\frac{d^2y}{dx^2} = -\frac{24}{x^4}$. [3]
Hence find the coordinates of the stationary point on the curve. Verify that the stationary point is a maximum. [5]
Find the equation of the normal to the curve when $x = -1$. Give your answer in the form $ax + by + c = 0$. [5]

OCR MEI C2 2014 June Q12

10 marks Moderate -0.3

Oskar is designing a building. Fig. 12 shows his design for the end wall and the curve of the roof. The units for $x$ and $y$ are metres. \includegraphics{figure_12}

Use the trapezium rule with 5 strips to estimate the area of the end wall of the building. [4]
Oskar now uses the equation $y = -0.001x^3 - 0.025x^2 + 0.6x + 9$, for $0 \leq x \leq 15$, to model the curve of the roof.
1. Calculate the difference between the height of the roof when $x = 12$ given by this model and the data shown in Fig. 12. [2]
2. Use integration to find the area of the end wall given by this model. [4]

OCR MEI C2 2014 June Q13

13 marks Moderate -0.3

The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960.

Year	1965	1970	1975	1980	1985	1990	1995	2000	2005	2010
Number of years since 1960 $(t)$	5	10	15	20	25	30	35	40	45	50
Reduction in thickness since 1960 $(h$ m$)$	0.7	1.0	1.7	2.3	3.6	4.7	6.0	8.2	12	15.9

An exponential model may be used for these data, assuming that the relationship between $h$ and $t$ is of the form $h = a \times 10^{bt}$, where $a$ and $b$ are constants to be determined.

Show that this relationship may be expressed in the form $\log_{10} h = mt + c$, stating the values of $m$ and $c$ in terms of $a$ and $b$. [2]
Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of $\log_{10} h$ against $t$, drawing by eye a line of best fit. [4]
Use your graph to find $h$ in terms of $t$ for this model. [4]
Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model. [2]
Give one reason why this model will not be suitable in the long term. [1]

OCR MEI C2 2016 June Q1

5 marks Easy -1.3

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ when $y = 6\sqrt{x}$. [2]
Find $\int \frac{12}{x^2} \mathrm{d}x$. [3]

OCR MEI C2 2016 June Q2

3 marks Moderate -0.8

A sequence is defined as follows. $u_1 = a$, where $a > 0$ To obtain $u_{r+1}$

find the remainder when $u_r$ is divided by 3,
multiply the remainder by 5,
the result is $u_{r+1}$.

Find $\sum_{r=2}^4 u_r$ in each of the following cases.

$a = 5$
$a = 6$ [3]

OCR MEI C2 2016 June Q3

5 marks Standard +0.3

An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12. Calculate the difference between the tenth terms of the AP and the GP. [5]

OCR MEI C2 2016 June Q4

5 marks Moderate -0.3

\includegraphics{figure_4} Fig. 4 shows triangle ABC, where AB = 7.2 cm, AC = 5.6 cm and angle BAC = 68°. Calculate the size of angle ACB. [5]

OCR MEI C2 2016 June Q5

4 marks Easy -1.3

Fig. 5 shows the graph of a sine function. \includegraphics{figure_5} State the equation of this curve. [2]
Sketch the graph of $y = \sin x - 3$ for $0° \leq x \leq 450°$. [2]

OCR MEI C2 2016 June Q6

4 marks Moderate -0.3

A sector of a circle has radius $r$ cm and sector angle $\theta$ radians. It is divided into two regions, A and B. Region A is an isosceles triangle with the equal sides being of length $a$ cm, as shown in Fig. 6. \includegraphics{figure_6}

Express the area of B in terms of $a$, $r$ and $\theta$. [2]
Given that $r = 12$ and $\theta = 0.8$, find the value of $a$ for which the areas of A and B are equal. Give your answer correct to 3 significant figures. [2]

OCR MEI C2 2016 June Q7

5 marks Moderate -0.8

Show that, when $x$ is an acute angle, $\tan x \sqrt{1 - \sin^2 x} = \sin x$. [2]
Solve $4 \sin^2 y = \sin y$ for $0° \leq y \leq 360°$. [3]

OCR MEI C2 2016 June Q8

5 marks Moderate -0.8

Simplify $\log_a 1 - \log_a (a^m)^3$. [2]
Use logarithms to solve the equation $3^{2x+1} = 1000$. Give your answer correct to 3 significant figures. [3]

OCR MEI C2 2016 June Q9

11 marks Standard +0.3

Fig. 9 shows the cross-section of a straight, horizontal tunnel. The $x$-axis from 0 to 6 represents the floor of the tunnel. \includegraphics{figure_9} With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.

$x$	0	1	2	3	4	5	6
$y$	0	4.0	4.9	5.0	4.9	4.0	0

The length of the tunnel is 50 m.

Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel. [4]
An engineer models the height of the roof of the tunnel using the curve $y = \frac{x}{81}(108x - 54x^2 + 12x^3 - x^4)$. This curve is symmetrical about $x = 3$.
1. Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel. [2]
2. Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel. [5]

OCR MEI C2 2016 June Q10

13 marks Moderate -0.8

Calculate the gradient of the chord of the curve $y = x^2 - 2x$ joining the points at which the values of $x$ are 5 and 5.1. [2]
Given that $\mathrm{f}(x) = x^2 - 2x$, find and simplify $\frac{\mathrm{f}(5 + h) - \mathrm{f}(5)}{h}$. [4]
Use your result in part (ii) to find the gradient of the curve $y = x^2 - 2x$ at the point where $x = 5$, showing your reasoning. [2]
Find the equation of the tangent to the curve $y = x^2 - 2x$ at the point where $x = 5$. Find the area of the triangle formed by this tangent and the coordinate axes. [5]

OCR MEI C2 2016 June Q11

12 marks Moderate -0.3

There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.

Week	1	2	3	4	5	6	7	8	9	10
Number of flu viruses	7	10	24	32	40	38	63	96	234	480

These data may be modelled by an equation of the form $y = a \times 10^{bt}$, where $y$ is the number of flu viruses detected in week $t$ of the epidemic, and $a$ and $b$ are constants to be determined.

Explain why this model leads to a straight-line graph of $\log_{10} y$ against $t$. State the gradient and intercept of this graph in terms of $a$ and $b$. [3]
Complete the values of $\log_{10} y$ in the table, draw the graph of $\log_{10} y$ against $t$, and draw by eye a line of best fit for the data. Hence determine the values of $a$ and $b$ and the equation for $y$ in terms of $t$ for this model. [8]

During the decline of the epidemic, an appropriate model was $$y = 921 \times 10^{-0.137w},$$ where $y$ is the number of flu viruses detected in week $w$ of the decline.