Questions — OCR MEI C3 (386 questions)

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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]
OCR MEI C3 Q6
5 marks Easy -1.2
  1. State the algebraic condition for the function f(x) to be an even function. What geometrical property does the graph of an even function have? [2]
  2. State whether the following functions are odd, even or neither. (A) \(\text{f}(x) = x^2 - 3\) (B) \(\text{g}(x) = \sin x + \cos x\) (C) \(\text{h}(x) = \frac{1}{x + x^3}\) [3]
OCR MEI C3 Q7
18 marks Standard +0.8
Fig. 8 shows part of the curve \(y = \text{f}(x)\), where \(\text{f}(x) = e^{-\frac{1}{5}x} \sin x\), for all \(x\). \includegraphics{figure_8}
  1. Sketch the graphs of (A) \(y = \text{f}(2x)\), (B) \(y = \text{f}(x + \pi)\). [4]
  2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P. [6]
  3. Show that \(\text{f}(x + \pi) = -e^{-\frac{1}{5}\pi}\text{f}(x)\). Hence, using the substitution \(u = x - \pi\), show that $$\int_{\pi}^{2\pi} \text{f}(x)\,dx = -e^{-\frac{1}{5}\pi} \int_{0}^{\pi} \text{f}(u)\,du.$$ Interpret this result graphically. [You should not attempt to integrate f(x).] [8]
OCR MEI C3 Q1
6 marks Moderate -0.3
  1. The function f(x) is defined by $$f(x) = \frac{1-x}{1+x}, x \neq -1.$$ Show that f(f(x)) = x. Hence write down \(f^{-1}(x)\). [3]
  2. The function g(x) is defined for all real x by $$g(x) = \frac{1-x^2}{1+x^2}.$$ Prove that g(x) is even. Interpret this result in terms of the graph of \(y = g(x)\). [3]
OCR MEI C3 Q2
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\), where $$f(x) = (e^x - 2)^2 - 1, x \in \mathbb{R}.$$ The curve crosses the x-axis at O and P, and has a turning point at Q. \includegraphics{figure_9}
  1. Find the exact x-coordinate of P. [2]
  2. Show that the x-coordinate of Q is \(\ln 2\) and find its y-coordinate. [4]
  3. Find the exact area of the region enclosed by the curve and the x-axis. [5]
The domain of f(x) is now restricted to \(x \geqslant \ln 2\).
  1. Find the inverse function \(f^{-1}(x)\). Write down its domain and range, and sketch its graph on the copy of Fig. 9. [7]
OCR MEI C3 Q3
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 Q4
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 Q5
4 marks Moderate -0.3
Write down the conditions for f(x) to be an odd function and for g(x) to be an even function. Hence prove that, if f(x) is odd and g(x) is even, then the composite function gf(x) is even. [4]
OCR MEI C3 Q6
6 marks Standard +0.3
The function f(x) is defined by $$f(x) = 1 + 2\sin 3x, \quad -\frac{\pi}{6} \leqslant x \leqslant \frac{\pi}{6}.$$ You are given that this function has an inverse, \(f^{-1}(x)\). Find \(f^{-1}(x)\) and its domain. [6]
OCR MEI C3 Q7
3 marks Moderate -0.5
Given that \(f(x) = \frac{1}{2}\ln(x - 1)\) and \(g(x) = 1 + e^{2x}\), show that g(x) is the inverse of f(x). [3]
OCR MEI C3 Q8
3 marks Moderate -0.8
Sketch the curve \(y = 2\arccos x\) for \(-1 \leqslant x \leqslant 1\). [3]
OCR MEI C3 Q1
5 marks Moderate -0.3
Find the exact value of \(\int_0^2 \sqrt{1+4x} \, dx\), showing your working. [5]
OCR MEI C3 Q2
18 marks Standard +0.3
Fig. 8 shows the line \(y = x\) and parts of the curves \(y = f(x)\) and \(y = g(x)\), where $$f(x) = e^{x-1}, \quad g(x) = 1 + \ln x.$$ The curves intersect the axes at the points A and B, as shown. The curves and the line \(y = x\) meet at the point C. \includegraphics{figure_8}
  1. Find the exact coordinates of A and B. Verify that the coordinates of C are \((1, 1)\). [5]
  2. Prove algebraically that \(g(x)\) is the inverse of \(f(x)\). [2]
  3. Evaluate \(\int_0^1 f(x) \, dx\), giving your answer in terms of \(e\). [3]
  4. Use integration by parts to find \(\int \ln x \, dx\). Hence show that \(\int_{e^{-1}}^1 g(x) \, dx = \frac{1}{e}\). [6]
  5. Find the area of the region enclosed by the lines OA and OB, and the arcs AC and BC. [2]
OCR MEI C3 Q3
19 marks Standard +0.3
A curve is defined by the equation \(y = 2x \ln(1 + x)\).
  1. Find \(\frac{dy}{dx}\) and hence verify that the origin is a stationary point of the curve. [4]
  2. Find \(\frac{d^2y}{dx^2}\) and use this to verify that the origin is a minimum point. [5]
  3. Using the substitution \(u = 1 + x\), show that \(\int \frac{x^2}{1+x} \, dx = \int \left(u - 2 + \frac{1}{u}\right) du\). Hence evaluate \(\int_0^1 \frac{x^2}{1+x} \, dx\), giving your answer in an exact form. [6]
  4. Using integration by parts and your answer to part (iii), evaluate \(\int_0^1 2x \ln(1 + x) \, dx\). [4]
OCR MEI C3 Q4
4 marks Moderate -0.3
Find \(\int xe^{3x} \, dx\). [4]
OCR MEI C3 Q5
4 marks Moderate -0.3
Show that \(\int_1^4 \frac{x}{x^2 + 2} \, dx = \frac{1}{2} \ln 6\). [4]
OCR MEI C3 Q1
4 marks Standard +0.3
Either prove or disprove each of the following statements.
  1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.' [2]
  2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]