Questions — OCR MEI C3 (386 questions)

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OCR MEI C3 2012 January Q9
18 marks Challenging +1.2
Fig. 9 shows the curves \(y = \text{f}(x)\) and \(y = \text{g}(x)\). The function \(y = \text{f}(x)\) is given by $$\text{f}(x) = \ln \left( \frac{2x}{1+x} \right), \quad x > 0.$$ The curve \(y = \text{f}(x)\) crosses the \(x\)-axis at P, and the line \(x = 2\) at Q. \includegraphics{figure_9}
  1. Verify that the \(x\)-coordinate of P is 1. Find the exact \(y\)-coordinate of Q. [2]
  2. Find the gradient of the curve at P. [Hint: use \(\frac{a}{b} = \ln a - \ln b\).] [4]
The function \(\text{g}(x)\) is given by $$\text{g}(x) = \frac{e^x}{2-e^x}, \quad x < \ln 2.$$ The curve \(y = \text{g}(x)\) crosses the \(y\)-axis at the point R.
  1. Show that \(\text{g}(x)\) is the inverse function of \(\text{f}(x)\). Write down the gradient of \(y = \text{g}(x)\) at R. [5]
  2. Show, using the substitution \(u = 2 - e^x\) or otherwise, that \(\int_0^{\ln \frac{4}{3}} \text{g}(x) dx = \ln \frac{3}{2}\). Using this result, show that the exact area of the shaded region shown in Fig. 9 is \(\ln \frac{32}{27}\). [Hint: consider its reflection in \(y = x\).] [7]
OCR MEI C3 2013 January Q1
6 marks Moderate -0.3
  1. Given that \(y = e^{-x} \sin 2x\), find \(\frac{dy}{dx}\). [3]
  2. Hence show that the curve \(y = e^{-x} \sin 2x\) has a stationary point when \(x = \frac{1}{2} \arctan 2\). [3]
OCR MEI C3 2013 January Q2
6 marks Moderate -0.3
A curve has equation \(x^2 + 2y^2 = 4x\).
  1. By differentiating implicitly, find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [3]
  2. Hence find the exact coordinates of the stationary points of the curve. [You need not determine their nature.] [3]
OCR MEI C3 2013 January Q3
2 marks Easy -1.2
Express \(1 < x < 3\) in the form \(|x - a| < b\), where \(a\) and \(b\) are to be determined. [2]
OCR MEI C3 2013 January Q4
8 marks Standard +0.3
The temperature \(\theta\) °C of water in a container after \(t\) minutes is modelled by the equation $$\theta = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants. The initial and long-term temperatures of the water are 15°C and 100°C respectively. After 1 minute, the temperature is 30°C.
  1. Find \(a\), \(b\) and \(k\). [6]
  2. Find how long it takes for the temperature to reach 80°C. [2]
OCR MEI C3 2013 January Q5
5 marks Moderate -0.8
The driving force \(F\) newtons and velocity \(v\) km s\(^{-1}\) of a car at time \(t\) seconds are related by the equation \(F = \frac{25}{v}\).
  1. Find \(\frac{dF}{dv}\). [2]
  2. Find \(\frac{dF}{dt}\) when \(v = 50\) and \(\frac{dv}{dt} = 1.5\). [3]
OCR MEI C3 2013 January Q6
5 marks Standard +0.3
Evaluate \(\int_0^3 x(x + 1)^{-\frac{1}{2}} dx\), giving your answer as an exact fraction. [5]
OCR MEI C3 2013 January Q7
4 marks Moderate -0.8
  1. Disprove the following statement: \(3^n + 2\) is prime for all integers \(n \geq 0\). [2]
  2. Prove that no number of the form \(3^n\) (where \(n\) is a positive integer) has 5 as its final digit. [2]
OCR MEI C3 2013 January Q8
17 marks Standard +0.3
Fig. 8 shows parts of the curves \(y = f(x)\) and \(y = g(x)\), where \(f(x) = \tan x\) and \(g(x) = 1 + f(x - \frac{1}{4}\pi)\). \includegraphics{figure_8}
  1. Describe a sequence of two transformations which maps the curve \(y = f(x)\) to the curve \(y = g(x)\). [4]
It can be shown that \(g(x) = \frac{2\sin x}{\sin x + \cos x}\).
  1. Show that \(g'(x) = \frac{2}{(\sin x + \cos x)^2}\). Hence verify that the gradient of \(y = g(x)\) at the point \((\frac{1}{4}\pi, 1)\) is the same as that of \(y = f(x)\) at the origin. [7]
  2. By writing \(\tan x = \frac{\sin x}{\cos x}\) and using the substitution \(u = \cos x\), show that \(\int_0^{\frac{1}{4}\pi} f(x)dx = \int_{\frac{1}{\sqrt{2}}}^1 \frac{1}{u}du\). Evaluate this integral exactly. [4]
  3. Hence find the exact area of the region enclosed by the curve \(y = g(x)\), the \(x\)-axis and the lines \(x = \frac{1}{4}\pi\) and \(x = \frac{1}{2}\pi\). [2]
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}
  1. Show that \(e^a = 1 + 2a\). [1]
  2. 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]
  3. 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]
  4. 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]
OCR MEI C3 2011 June Q1
4 marks Moderate -0.8
Solve the equation \(|2x - 1| = |x|\). [4]
OCR MEI C3 2011 June Q2
3 marks Moderate -0.8
Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function \(gf(x)\), expressing your answer as simply as possible. [3]
OCR MEI C3 2011 June Q3
8 marks Moderate -0.3
  1. Differentiate \(\frac{\ln x}{x^2}\), simplifying your answer. [4]
  2. 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.
  1. 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]
  2. 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 \(\sin 2x + \cos y = \sqrt{3}\).
  1. Verify that the point P \((\frac{\pi}{6}, \frac{\pi}{6})\) lies on the curve. [1]
  2. 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
  1. Multiply out \((3^n + 1)(3^n - 1)\). [1]
  2. 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 \(f(x) = \frac{1}{e^x + e^{-x} + 2}\).
  1. Show algebraically that \(f(x)\) is an even function, and state how this property relates to the curve \(y = f(x)\). [3]
  2. Find \(f'(x)\). [3]
  3. Show that \(f(x) = \frac{e^x}{(e^x + 1)^2}\). [2]
  4. 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]
  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}
  1. Using Fig. 9, show that \(a = 2\) and \(b = \frac{1}{2}\). [3]
  2. 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]
  3. 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]
  4. 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}
  1. Find \(a\) and \(b\). [2]
  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.
  1. 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]
  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]
  3. Find how long it is before Brian's and Kate's cars have the same value. [3]