Questions — OCR MEI (4456 questions)

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OCR MEI C3 2011 January Q2
4 marks Moderate -0.8
Solve the inequality \(|2x + 1| \geqslant 4\). [4]
OCR MEI C3 2011 January Q3
5 marks Standard +0.3
The area of a circular stain is growing at a rate of \(1 \text{ mm}^2\) per second. Find the rate of increase of its radius at an instant when its radius is \(2\) mm. [5]
OCR MEI C3 2011 January Q4
3 marks Easy -1.2
Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}
OCR MEI C3 2011 January Q5
8 marks Standard +0.3
  1. On a single set of axes, sketch the curves \(y = e^x - 1\) and \(y = 2e^{-x}\). [3]
  2. Find the exact coordinates of the point of intersection of these curves. [5]
OCR MEI C3 2011 January Q6
4 marks Standard +0.3
A curve is defined by the equation \((x + y)^2 = 4x\). The point \((1, 1)\) lies on this curve. By differentiating implicitly, show that \(\frac{dy}{dx} = \frac{2}{x + y} - 1\). Hence verify that the curve has a stationary point at \((1, 1)\). [4]
OCR MEI C3 2011 January Q7
8 marks Standard +0.3
Fig. 7 shows the curve \(y = f(x)\), where \(f(x) = 1 + 2 \arctan x\), \(x \in \mathbb{R}\). The scales on the \(x\)- and \(y\)-axes are the same. \includegraphics{figure_7}
  1. Find the range of f, giving your answer in terms of \(\pi\). [3]
  2. Find \(f^{-1}(x)\), and add a sketch of the curve \(y = f^{-1}(x)\) to the copy of Fig. 7. [5]
OCR MEI C3 2011 January Q8
18 marks Standard +0.3
  1. Use the substitution \(u = 1 + x\) to show that $$\int_0^1 \frac{x^3}{1 + x} dx = \int_a^b \left( u^2 - 3u + 3 - \frac{1}{u} \right) du,$$ where \(a\) and \(b\) are to be found. Hence evaluate \(\int_0^1 \frac{x^3}{1 + x} dx\), giving your answer in exact form. [7] Fig. 8 shows the curve \(y = x^2 \ln(1 + x)\). \includegraphics{figure_8}
  2. Find \(\frac{dy}{dx}\). Verify that the origin is a stationary point of the curve. [5]
  3. Using integration by parts, and the result of part (i), find the exact area enclosed by the curve \(y = x^2 \ln(1 + x)\), the \(x\)-axis and the line \(x = 1\). [6]
OCR MEI C3 2011 January Q9
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = \frac{1}{\cos^2 x}\), \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\), together with its asymptotes \(x = \frac{1}{2}\pi\) and \(x = -\frac{1}{2}\pi\). \includegraphics{figure_9}
  1. Use the quotient rule to show that the derivative of \(\frac{\sin x}{\cos x}\) is \(\frac{1}{\cos^2 x}\). [3]
  2. Find the area bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\pi\). [3]
The function \(g(x)\) is defined by \(g(x) = \frac{1}{2}f(x + \frac{1}{4}\pi)\).
  1. Verify that the curves \(y = f(x)\) and \(y = g(x)\) cross at \((0, 1)\). [3]
  2. State a sequence of two transformations such that the curve \(y = f(x)\) is mapped to the curve \(y = g(x)\). On the copy of Fig. 9, sketch the curve \(y = g(x)\), indicating clearly the coordinates of the minimum point and the equations of the asymptotes to the curve. [8]
  3. Use your result from part (ii) to write down the area bounded by the curve \(y = g(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = -\frac{1}{4}\pi\). [1]
OCR MEI C3 2012 January Q1
3 marks Moderate -0.3
Differentiate \(x^2 \tan 2x\). [3]
OCR MEI C3 2012 January Q2
4 marks Moderate -0.8
The functions \(\text{f}(x)\) and \(\text{g}(x)\) are defined as follows. $$\text{f}(x) = \ln x, \quad x > 0$$ $$\text{g}(x) = 1 + x^2, \quad x \in \mathbb{R}$$ Write down the functions \(\text{fg}(x)\) and \(\text{gf}(x)\), and state whether these functions are odd, even or neither. [4]
OCR MEI C3 2012 January Q3
5 marks Standard +0.3
Show that \(\int_0^{\frac{\pi}{2}} x \cos \frac{1}{2} x \, dx = \frac{\sqrt{2}}{2} \pi + 2\sqrt{2} - 4\). [5]
OCR MEI C3 2012 January Q4
2 marks Standard +0.8
Prove or disprove the following statement: 'No cube of an integer has 2 as its units digit.' [2]
OCR MEI C3 2012 January Q5
6 marks Moderate -0.3
Each of the graphs of \(y = \text{f}(x)\) and \(y = \text{g}(x)\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for \(\text{f}(x)\) and \(\text{g}(x)\).
  1. \includegraphics{figure_5i} [3]
  2. \includegraphics{figure_5ii} [3]
OCR MEI C3 2012 January Q6
8 marks Standard +0.3
Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius \(r\) metres of the oil slick \(t\) hours after the start of the leak is modelled by the equation $$r = 20(1 - e^{-0.2t}).$$
  1. Find the radius of the slick when \(t = 2\), and the rate at which the radius is increasing at this time. [4]
  2. Find the rate at which the area of the slick is increasing when \(t = 2\). [4]
OCR MEI C3 2012 January Q7
8 marks Standard +0.8
Fig. 7 shows the curve \(x^3 + y^3 = 3xy\). The point P is a turning point of the curve. \includegraphics{figure_7}
  1. Show that \(\frac{dy}{dx} = \frac{y - x^2}{y^2 - x}\). [4]
  2. Hence find the exact \(x\)-coordinate of P. [4]
OCR MEI C3 2012 January Q8
18 marks Standard +0.3
Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x-2}}\), together with the lines \(y = x\) and \(x = 11\). The curve meets these lines at P and Q respectively. R is the point \((11, 11)\). \includegraphics{figure_8}
  1. Verify that the \(x\)-coordinate of P is 3. [2]
  2. Show that, for the curve, \(\frac{dy}{dx} = \frac{x-4}{2(x-2)^{\frac{3}{2}}}\). Hence find the gradient of the curve at P. Use the result to show that the curve is not symmetrical about \(y = x\). [7]
  3. Using the substitution \(u = x - 2\), show that \(\int_3^{11} \frac{x}{\sqrt{x-2}} \, dx = 25\frac{1}{3}\). Hence find the area of the region PQR bounded by the curve and the lines \(y = x\) and \(x = 11\). [9]
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]