Questions C3 (1301 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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OCR C3 Q8
10 marks Standard +0.8
  1. Sketch on the same diagram the graphs of $$y = \sin^{-1} x, \quad -1 \leq x \leq 1$$ and $$y = \cos^{-1} (2x), \quad -\frac{1}{2} \leq x \leq \frac{1}{2}.$$ [3]
Given that the graphs intersect at the point with coordinates \((a, b)\),
  1. show that \(\tan b = \frac{1}{2}\), [3]
  2. find the value of \(a\) in the form \(k\sqrt{5}\). [4]
OCR C3 Q9
12 marks Standard +0.3
$$\text{f}(x) = e^{3x + 1} - 2, \quad x \in \mathbb{R}.$$
  1. State the range of f. [1]
The curve \(y = \text{f}(x)\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).
  1. Find the exact coordinates of \(P\) and \(Q\). [3]
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 3ex + e - 2.$$ [4]
  3. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\). [4]
OCR C3 Q1
4 marks Moderate -0.5
Show that $$\int_1^7 \frac{2}{4x-1} \, dx = \ln 3.$$ [4]
OCR C3 Q2
5 marks Standard +0.8
Find the set of values of \(x\) such that $$|3x + 1| \leq |x - 2|.$$ [5]
OCR C3 Q3
6 marks Standard +0.8
Find all values of \(\theta\) in the interval \(-180 < \theta < 180\) for which $$\tan^2 \theta^\circ + \sec \theta^\circ = 1.$$ [6]
OCR C3 Q4
6 marks Moderate -0.8
Solve each equation, giving your answers in exact form.
  1. \(\mathrm{e}^{4x-3} = 2\) [2]
  2. \(\ln(2y - 1) = 1 + \ln(3 - y)\) [4]
OCR C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\cosec \theta - \sin \theta > 0\) for all values of \(\theta\) in the interval \(0 < \theta < \pi\)" is false. [2]
  2. Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\cosec \theta - \sin \theta = 2,$$ giving your answers to 2 decimal places. [5]
OCR C3 Q6
8 marks Standard +0.2
The curve \(C\) has the equation \(y = x^2 - 5x + 2\ln \frac{x}{3}\), \(x > 0\).
  1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3x + 5y + 21 = 0.$$ [5]
  2. Find the \(x\)-coordinates of the stationary points of \(C\). [3]
OCR C3 Q7
11 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve \(y = \text{f}(x)\) which has a maximum point at \((-45, 7)\) and a minimum point at \((135, -1)\).
  1. Showing the coordinates of any stationary points, sketch the curve with equation \(y = 1 + 2\text{f}(x)\). [3]
Given that $$\text{f}(x) = A + 2\sqrt{2} \cos x° - 2\sqrt{2} \sin x°, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$ where \(A\) is a constant,
  1. show that f\((x)\) can be expressed in the form $$\text{f}(x) = A + R \cos (x + \alpha)°,$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
  2. state the value of \(A\), [1]
  3. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \text{f}(x)\) crosses the \(x\)-axis. [4]
OCR C3 Q8
12 marks Standard +0.2
The function f is defined by $$\text{f}(x) \equiv 3 - x^2, \quad x \in \mathbb{R}, \quad x \geq 0.$$
  1. State the range of f. [1]
  2. Sketch the graphs of \(y = \text{f}(x)\) and \(y = \text{f}^{-1}(x)\) on the same diagram. [3]
  3. Find an expression for f\(^{-1}(x)\) and state its domain. [3]
The function g is defined by $$\text{g}(x) \equiv \frac{8}{3-x}, \quad x \in \mathbb{R}, \quad x \neq 3.$$
  1. Evaluate fg\((-3)\). [2]
  2. Solve the equation $$\text{f}^{-1}(x) = \text{g}(x).$$ [3]
OCR C3 Q9
13 marks Standard +0.3
A curve has the equation \(y = (2x + 3)\mathrm{e}^{-x}\).
  1. Find the exact coordinates of the stationary point of the curve. [4]
The curve crosses the \(y\)-axis at the point \(P\).
  1. Find an equation for the normal to the curve at \(P\). [2]
The normal to the curve at \(P\) meets the curve again at \(Q\).
  1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
  2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
  3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]
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]
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]