Questions - OCR MEI - A-Level Maths

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OCR MEI C3 2013 January Q9

19 marks Standard +0.3

Fig. 9 shows the line $y = x$ and the curve $y = f(x)$, where $f(x) = \frac{1}{2}(e^x - 1)$. The line and the curve intersect at the origin and at the point P$(a, a)$. \includegraphics{figure_9}

Show that $e^a = 1 + 2a$. [1]
Show that the area of the region enclosed by the curve, the $x$-axis and the line $x = a$ is $\frac{1}{2}a$. Hence find, in terms of $a$, the area enclosed by the curve and the line $y = x$. [6]
Show that the inverse function of f$(x)$ is g$(x)$, where g$(x) = \ln(1 + 2x)$. Add a sketch of $y = g(x)$ to the copy of Fig. 9. [5]
Find the derivatives of f$(x)$ and g$(x)$. Hence verify that $g'(a) = \frac{1}{f'(a)}$. Give a geometrical interpretation of this result. [7]

Differentiate $\frac{\ln x}{x^2}$, simplifying your answer. [4]
Using integration by parts, show that $\int \frac{\ln x}{x^2} \, dx = -\frac{1}{x}(1 + \ln x) + c$. [4]

OCR MEI C3 2011 June Q4

6 marks Moderate -0.3

The height $h$ metres of a tree after $t$ years is modelled by the equation $$h = a - be^{-kt},$$ where $a$, $b$ and $k$ are positive constants.

Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of $a$ and $b$. [3]
Given also that the tree grows to a height of 6 metres in 8 years, find the value of $k$, giving your answer correct to 2 decimal places. [3]

OCR MEI C3 2011 June Q5

5 marks Standard +0.3

Given that $y = x^2\sqrt{1 + 4x}$, show that $\frac{dy}{dx} = \frac{2x(5x + 1)}{\sqrt{1 + 4x}}$. [5]

OCR MEI C3 2011 June Q6

6 marks Standard +0.3

A curve is defined by the equation $\sin 2x + \cos y = \sqrt{3}$.

Verify that the point P $(\frac{\pi}{6}, \frac{\pi}{6})$ lies on the curve. [1]
Find $\frac{dy}{dx}$ in terms of $x$ and $y$. Hence find the gradient of the curve at the point P. [5]

OCR MEI C3 2011 June Q7

4 marks Standard +0.3

Multiply out $(3^n + 1)(3^n - 1)$. [1]
Hence prove that if $n$ is a positive integer then $3^{2n} - 1$ is divisible by 8. [3]

OCR MEI C3 2011 June Q8

18 marks Standard +0.3

\includegraphics{figure_8} Fig. 8 shows the curve $y = f(x)$, where $f(x) = \frac{1}{e^x + e^{-x} + 2}$.

Show algebraically that $f(x)$ is an even function, and state how this property relates to the curve $y = f(x)$. [3]
Find $f'(x)$. [3]
Show that $f(x) = \frac{e^x}{(e^x + 1)^2}$. [2]
Hence, using the substitution $u = e^x + 1$, or otherwise, find the exact area enclosed by the curve $y = f(x)$, the $x$-axis, and the lines $x = 0$ and $x = 1$. [5]
Show that there is only one point of intersection of the curves $y = f(x)$ and $y = \frac{1}{4}e^x$, and find its coordinates. [5]

OCR MEI C3 2011 June Q9

18 marks Standard +0.3

Fig. 9 shows the curve $y = f(x)$. The endpoints of the curve are P $(-\pi, 1)$ and Q $(\pi, 3)$, and $f(x) = a + \sin bx$, where $a$ and $b$ are constants. \includegraphics{figure_9}

Using Fig. 9, show that $a = 2$ and $b = \frac{1}{2}$. [3]
Find the gradient of the curve $y = f(x)$ at the point $(0, 2)$. Show that there is no point on the curve at which the gradient is greater than this. [5]
Find $f^{-1}(x)$, and state its domain and range. Write down the gradient of $y = f^{-1}(x)$ at the point $(2, 0)$. [6]
Find the area enclosed by the curve $y = f(x)$, the $x$-axis, the $y$-axis and the line $x = \pi$. [4]

OCR MEI C3 2014 June Q1

3 marks Moderate -0.8

Evaluate $\int_0^{\frac{\pi}{4}} (1 - \sin 3x) \, dx$, giving your answer in exact form. [3]

OCR MEI C3 2014 June Q2

5 marks Standard +0.3

Find the exact gradient of the curve $y = \ln(1 - \cos 2x)$ at the point with $x$-coordinate $\frac{1}{4}\pi$. [5]

OCR MEI C3 2014 June Q3

4 marks Standard +0.3

Solve the equation $|3 - 2x| = 4|x|$. [4]

OCR MEI C3 2014 June Q4

7 marks Standard +0.3

Fig. 4 shows the curve $y = f(x)$, where $$f(x) = a + \cos bx, \quad 0 \leq x \leq 2\pi,$$ and $a$ and $b$ are positive constants. The curve has stationary points at $(0, 3)$ and $(2\pi, 1)$. \includegraphics{figure_4}

Find $a$ and $b$. [2]
Find $f^{-1}(x)$, and state its domain and range. [5]

OCR MEI C3 2014 June Q5

5 marks Standard +0.3

A spherical balloon of radius $r$ cm has volume $V$ cm$^3$, where $V = \frac{4}{3}\pi r^3$. The balloon is inflated at a constant rate of 10 cm$^3$ s$^{-1}$. Find the rate of increase of $r$ when $r = 8$. [5]

OCR MEI C3 2014 June Q6

8 marks Moderate -0.3

The value $£V$ of a car $t$ years after it is new is modelled by the equation $V = Ae^{-kt}$, where $A$ and $k$ are positive constants which depend on the make and model of the car.

Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 e^{-0.2t}.$$ Calculate how much value, to the nearest £100, this car has lost after 1 year. [2]
At the same time as Brian buys his car, Kate buys a new hatchback for £15000. Her car loses £2000 of its value in the first year. Show that, for Kate's car, $k = 0.143$ correct to 3 significant figures. [3]
Find how long it is before Brian's and Kate's cars have the same value. [3]

OCR MEI C3 2014 June Q7

4 marks Standard +0.3

Either prove or disprove each of the following statements.

'If $m$ and $n$ are consecutive odd numbers, then at least one of $m$ and $n$ is a prime number.' [2]
'If $m$ and $n$ are consecutive even numbers, then $mn$ is divisible by 8.' [2]

OCR MEI C3 2014 June Q8

18 marks Standard +0.3

Fig. 8 shows the curve $y = f(x)$, where $f(x) = \frac{x}{\sqrt{2 + x^2}}$. \includegraphics{figure_8}

Show algebraically that $f(x)$ is an odd function. Interpret this result geometrically. [3]
Show that $f'(x) = \frac{2}{(2 + x^2)^{\frac{3}{2}}}$. Hence find the exact gradient of the curve at the origin. [5]
Find the exact area of the region bounded by the curve, the $x$-axis and the line $x = 1$. [4]
1. Show that if $y = \frac{x}{\sqrt{2 + x^2}}$, then $\frac{1}{y^2} = \frac{2}{x^2} + 1$. [2]
2. Differentiate $\frac{1}{y^2} = \frac{2}{x^2} + 1$ implicitly to show that $\frac{dy}{dx} = \frac{2y^3}{x^3}$. Explain why this expression cannot be used to find the gradient of the curve at the origin. [4]