Questions — OCR MEI (4456 questions)

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OCR MEI C3 2016 June Q7
4 marks Standard +0.8
You are given that \(n\) is a positive integer. By expressing \(x^{2n} - 1\) as a product of two factors, prove that \(2^{2n} - 1\) is divisible by 3. [4]
OCR MEI C3 2016 June Q8
18 marks Standard +0.8
Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x+4}}\) and the line \(x = 5\). The curve has an asymptote \(l\). The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q. \includegraphics{figure_8}
  1. Show that for this curve \(\frac{dy}{dx} = \frac{x + 8}{2(x + 4)^{\frac{3}{2}}}\). [5]
  2. Find the coordinates of the point P. [4]
  3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\). [9]
OCR MEI C3 2016 June Q9
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = e^{2x} + k e^{-2x}\) and \(k\) is a constant greater than 1. The curve crosses the \(y\)-axis at P and has a turning point Q. \includegraphics{figure_9}
  1. Find the \(y\)-coordinate of P in terms of \(k\). [1]
  2. Show that the \(x\)-coordinate of Q is \(\frac{1}{4}\ln k\), and find the \(y\)-coordinate in its simplest form. [5]
  3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\ln k\). Give your answer in the form \(ak + b\). [4]
The function \(g(x)\) is defined by \(g(x) = f(x + \frac{1}{4}\ln k)\).
    1. Show that \(g(x) = \sqrt{k}(e^{2x} + e^{-2x})\). [3]
    2. Hence show that \(g(x)\) is an even function. [2]
    3. Deduce, with reasons, a geometrical property of the curve \(y = f(x)\). [3]
OCR MEI C3 Q1
5 marks Moderate -0.3
You are given that \(y^2 = 4x + 7\).
  1. Use implicit differentiation to find \(\frac{dy}{dx}\) in terms of \(y\). [2]
  2. Make \(x\) the subject of the equation. Find \(\frac{dx}{dy}\) and hence show that in this case \(\frac{dx}{dy} = \frac{1}{\frac{dx}{dy}}\). [3]
OCR MEI C3 Q2
4 marks Moderate -0.8
  1. Expand \((e^x + e^{-x})^2\). [1]
  2. Hence find \(\int (e^x + e^{-x})^2 dx\). [3]
OCR MEI C3 Q3
6 marks Moderate -0.8
  1. Sketch the graph of \(y = |3x - 6|\). [2]
  2. Solve the equation \(|3x - 6| = x + 4\) and illustrate your answer on your graph. [4]
OCR MEI C3 Q4
4 marks Standard +0.3
Find \(\int x \sin 3x dx\). [4]
OCR MEI C3 Q5
4 marks Moderate -0.3
Make \(x\) the subject of \(t = \ln \sqrt{\frac{5}{(x-3)}}\). [4]
OCR MEI C3 Q6
7 marks Standard +0.3
The function f(x) is defined as \(f(x) = \frac{\ln x}{x}\). The graph of the function is shown in Fig. 6. \includegraphics{figure_6}
  1. Give the coordinates of the point, P, where the curve crosses the \(x\)-axis. [1]
  2. Use calculus to find the coordinates of the stationary point, Q, and show that it is a maximum. [6]
OCR MEI C3 Q7
6 marks Moderate -0.3
An oil slick is circular with radius \(r\) km and area \(A\) km\(^2\). The radius increases with time at a rate given by \(\frac{dr}{dt} = 0.5\), in kilometres per hour.
  1. Show that \(\frac{dA}{dt} = \pi r\). [4]
  2. Find the rate of increase of the area of the slick at a time when the radius is 6 km. [2]
OCR MEI C3 Q8
18 marks Standard +0.3
Fig. 8 shows the graph of \(y = x\sqrt{1 + x}\). The point P on the curve is on the \(x\)-axis. \includegraphics{figure_8}
  1. Write down the coordinates of P. [1]
  2. Show that \(\frac{dy}{dx} = \frac{3x + 2}{2\sqrt{1 + x}}\). [4]
  3. Hence find the coordinates of the turning point on the curve. What can you say about the gradient of the curve at P? [4]
  4. By using a suitable substitution, show that \(\int_0^0 x\sqrt{1 + x} dx = \int_0^1 \left(u^{\frac{3}{2}} - u^{\frac{1}{2}}\right) du\). Evaluate this integral, giving your answer in an exact form. What does this value represent? [7]
  5. Use your answer to part (ii) to differentiate \(y = x\sqrt{1 + x} \sin 2x\) with respect to \(x\). (You need not simplify your result.) [2]
OCR MEI C3 Q9
18 marks Standard +0.3
The functions f(x) and g(x) are defined by $$f(x) = x^2, \quad g(x) = 2x - 1,$$ for all real values of \(x\).
  1. State the ranges of f(x) and g(x). Explain why f(x) has no inverse. [3]
  2. Find an expression for the inverse function g\(^{-1}\)(x) in terms of \(x\). Sketch the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\) on the same axes. [4]
  3. Find expressions for gf(x) and fg(x). [2]
  4. Solve the equation gf(x) = fg(x). Sketch the graphs of \(y = gf(x)\) and \(y = fg(x)\) on the same axes to illustrate your answer. [4]
  5. Show that the equation f(x + a) = g\(^{-1}\)(x) has no solution if \(a > \frac{1}{4}\). [5]
OCR MEI C3 Q1
18 marks Standard +0.3
Fig. 9 shows the curve \(y = \frac{x^2}{3x - 1}\). P is a turning point, and the curve has a vertical asymptote \(x = a\). \includegraphics{figure_1}
  1. Write down the value of \(a\). [1]
  2. Show that \(\frac{dy}{dx} = \frac{x(3x - 2)}{(3x - 1)^2}\) [3]
  3. Find the exact coordinates of the turning point P. Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point. [7]
  4. Using the substitution \(u = 3x - 1\), show that \(\int \frac{x^2}{3x - 1} dx = \frac{1}{27} \int \left( u + 2 + \frac{1}{u} \right) du\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac{2}{3}\) and \(x = 1\). [7]
OCR MEI C3 Q2
4 marks Moderate -0.3
Differentiate \(\sqrt{1 + 6x^2}\). [4]
OCR MEI C3 Q3
6 marks Standard +0.3
Show that the curve \(y = x^2 \ln x\) has a stationary point when \(x = \frac{1}{\sqrt{e}}\). [6]
OCR MEI C3 Q4
8 marks Moderate -0.3
The equation of a curve is \(y = \frac{x^2}{2x + 1}\).
  1. Show that \(\frac{dy}{dx} = \frac{2x(x + 1)}{(2x + 1)^2}\). [4]
  2. Find the coordinates of the stationary points of the curve. You need not determine their nature. [4]
OCR MEI C3 Q5
4 marks Moderate -0.3
  1. Differentiate \(\sqrt{1 + 2x}\).
  2. Show that the derivative of \(\ln(1 - e^{-x})\) is \(\frac{1}{e^x - 1}\). [4]
OCR MEI C3 Q6
18 marks Standard +0.3
The function \(\text{f}(x) = \frac{\sin x}{2 - \cos x}\) has domain \(-\pi \leqslant x \leqslant \pi\). Fig. 8 shows the graph of \(y = \text{f}(x)\) for \(0 \leqslant x \leqslant \pi\). \includegraphics{figure_6}
  1. Find \(\text{f}(-x)\) in terms of \(\text{f}(x)\). Hence sketch the graph of \(y = \text{f}(x)\) for the complete domain \(-\pi \leqslant x \leqslant \pi\). [3]
  2. Show that \(\text{f}'(x) = \frac{2\cos x - 1}{(2 - \cos x)^2}\). Hence find the exact coordinates of the turning point P. State the range of the function \(\text{f}(x)\), giving your answer exactly. [8]
  3. Using the substitution \(u = 2 - \cos x\) or otherwise, find the exact value of \(\int_0^\pi \frac{\sin x}{2 - \cos x} dx\). [4]
  4. Sketch the graph of \(y = \text{f}(2x)\). [1]
  5. Using your answers to parts (iii) and (iv), write down the exact value of \(\int_0^{\frac{\pi}{2}} \frac{\sin 2x}{2 - \cos 2x} dx\). [2]
OCR MEI C3 Q7
7 marks Standard +0.3
Fig. 3 shows the curve defined by the equation \(y = \arcsin(x - 1)\), for \(0 \leqslant x \leqslant 2\). \includegraphics{figure_7}
  1. Find \(x\) in terms of \(y\), and show that \(\frac{dx}{dy} = \cos y\). [3]
  2. Hence find the exact gradient of the curve at the point where \(x = 1.5\). [4]
OCR MEI C3 Q8
7 marks Standard +0.8
A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve. [7]
OCR MEI C3 Q1
4 marks Moderate -0.8
  1. Show algebraically that the function \(\text{f}(x) = \frac{2x}{1-x^2}\) is odd. [2] Fig. 7 shows the curve \(y = \text{f}(x)\) for \(0 \leq x < 4\), together with the asymptote \(x = 1\). \includegraphics{figure_7}
  2. Use the copy of Fig. 7 to complete the curve for \(-4 \leq x \leq 4\). [2]
OCR MEI C3 Q2
4 marks Moderate -0.8
The functions f(x) and 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 fg(x) and gf(x), and state whether these functions are odd, even or neither. [4]
OCR MEI C3 Q3
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 f(x) and g(x).
  1. \includegraphics{figure_3i} [3]
  2. \includegraphics{figure_3ii} [3]
OCR MEI C3 Q4
6 marks Moderate -0.3
Fig. 4 shows the curve \(y = \text{f}(x)\), where \(\text{f}(x) = \sqrt{1 - 9x^2}\), \(-a < x < a\). \includegraphics{figure_4}
  1. Find the value of \(a\). [2]
  2. Write down the range of f(x). [1]
  3. Sketch the curve \(y = \text{f}(\frac{1}{3}x) - 1\). [3]
OCR MEI C3 Q5
4 marks Moderate -0.8
You are given that f(x) and g(x) are odd functions, defined for \(x \in \mathbb{R}\).
  1. Given that s(x) = f(x) + g(x), prove that s(x) is an odd function. [2]
  2. Given that p(x) = f(x)g(x), determine whether p(x) is odd, even or neither. [2]