Questions C3 (1301 questions)

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OCR C3 Q4
7 marks Moderate -0.3
  1. Given that \(x = (4t + 9)^{\frac{1}{2}}\) and \(y = 6e^{\frac{2t+1}{4}}\), find expressions for \(\frac{dx}{dt}\) and \(\frac{dy}{dx}\). [4]
  2. Hence find the value of \(\frac{dy}{dt}\) when \(t = 4\), giving your answer correct to 3 significant figures. [3]
OCR C3 Q5
8 marks Standard +0.3
  1. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(-180° < \theta < 180°\). [5]
OCR C3 Q6
9 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{1}{\sqrt{3x + 2}}\). The shaded region is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\).
  1. Find the exact area of the shaded region. [4]
  2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer. [5]
OCR C3 Q7
8 marks Standard +0.3
The curve \(y = \ln x\) is transformed to the curve \(y = \ln(\frac{1}{2}x - a)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.
  1. Give full details of the translation and stretch involved. [2]
  2. Sketch the graph of \(y = \ln(\frac{1}{2}x - a)\). [2]
  3. Sketch, on another diagram, the graph of \(y = |\ln(\frac{1}{2}x - a)|\). [2]
  4. State, in terms of \(a\), the set of values of \(x\) for which \(|\ln(\frac{1}{2}x - a)| = -\ln(\frac{1}{2}x - a)\). [2]
OCR C3 Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows the curve with equation \(y = x^8 e^{-x^2}\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve, the line \(y = 0\) and the line \(PQ\).
  1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2. [5]
  2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places. [4]
  3. Deduce an approximation to the area of region \(B\). [2]
OCR C3 Q9
12 marks Standard +0.3
Functions f and g are defined by $$f(x) = 2 \sin x \quad \text{for } -\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi,$$ $$g(x) = 4 - 2x^2 \quad \text{for } x \in \mathbb{R}.$$
  1. State the range of f and the range of g. [2]
  2. Show that gf(0.5) = 2.16, correct to 3 significant figures, and explain why fg(0.5) is not defined. [4]
  3. Find the set of values of \(x\) for which \(f^{-1}g(x)\) is not defined. [6]
OCR C3 Q1
5 marks Moderate -0.8
Differentiate each of the following with respect to \(x\).
  1. \(x^3(x + 1)^5\) [2]
  2. \(\sqrt{3x^4 + 1}\) [3]
OCR C3 Q2
5 marks Standard +0.8
Solve the inequality \(|4x - 3| < |2x + 1|\). [5]
OCR C3 Q3
7 marks Moderate -0.8
The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$
  1. Evaluate ff(169). [2]
  2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
  3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [3]
OCR C3 Q4
7 marks Moderate -0.8
The integral \(I\) is defined by $$I = \int_0^{13} (2x + 1)^{\frac{3}{2}} \, dx.$$
  1. Use integration to find the exact value of \(I\). [4]
  2. Use Simpson's rule with two strips to find an approximate value for \(I\). Give your answer correct to 3 significant figures. [3]
OCR C3 Q5
7 marks Moderate -0.3
A substance is decaying in such a way that its mass, \(m\) kg, at a time \(t\) years from now is given by the formula $$m = 240e^{-0.04t}.$$
  1. Find the time taken for the substance to halve its mass. [3]
  2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year. [4]
OCR C3 Q6
9 marks Standard +0.3
  1. Given that \(\int_0^a (6e^{2x} + x) \, dx = 42\), show that \(a = \frac{1}{2} \ln(15 - \frac{1}{6}a^2)\). [5]
  2. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 decimal places. Use a starting value of 1 and show the result of each iteration. [4]
OCR C3 Q7
9 marks Moderate -0.3
  1. Sketch the graph of \(y = \sec x\) for \(0 \leq x \leq 2\pi\). [2]
  2. Solve the equation \(\sec x = 3\) for \(0 \leq x \leq 2\pi\), giving the roots correct to 3 significant figures. [3]
  3. Solve the equation \(\sec \theta = 5 \cos \theta\) for \(0 \leq \theta \leq 2\pi\), giving the roots correct to 3 significant figures. [4]
OCR C3 Q8
11 marks Standard +0.3
  1. Given that \(y = \frac{4 \ln x - 3}{4 \ln x + 3}\), show that \(\frac{dy}{dx} = \frac{24}{x(4 \ln x + 3)^2}\). [3]
  2. Find the exact value of the gradient of the curve \(y = \frac{4 \ln x - 3}{4 \ln x + 3}\) at the point where it crosses the \(x\)-axis. [4]
  3. \includegraphics{figure_8iii} The diagram shows part of the curve with equation $$y = \frac{2}{x^2(4 \ln x + 3)}.$$ The region shaded in the diagram is bounded by the curve and the lines \(x = 1\), \(x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis. [4]
OCR C3 Q9
12 marks Standard +0.8
  1. Prove the identity $$\tan(\theta + 60°) \tan(\theta - 60°) \equiv \frac{\tan^2 \theta - 3}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Solve, for \(0° < \theta < 180°\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = 4 \sec^2 \theta - 3,$$ giving your answers correct to the nearest \(0.1°\). [5]
  3. Show that, for all values of the constant \(k\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = k^2$$ has two roots in the interval \(0° < \theta < 180°\). [3]
OCR C3 Q1
5 marks Moderate -0.8
Functions f and g are defined for all real values of \(x\) by $$f(x) = x^3 + 4 \quad \text{and} \quad g(x) = 2x - 5.$$ Evaluate
  1. fg(1), [2]
  2. \(f^{-1}(12)\). [3]
OCR C3 Q2
6 marks Standard +0.3
The sequence defined by $$x_1 = 3, \quad x_{n+1} = \sqrt{31 - \frac{5}{2}x_n}$$ converges to the number \(\alpha\).
  1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration. [3]
  2. Find an equation of the form \(ax^3 + bx + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root. [3]
OCR C3 Q3
7 marks Moderate -0.3
  1. Solve, for \(0° < \alpha < 180°\), the equation \(\sec \frac{1}{2}\alpha = 4\). [3]
  2. Solve, for \(0° < \beta < 180°\), the equation \(\tan \beta = 7 \cot \beta\). [4]
OCR C3 Q4
6 marks Standard +0.3
Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = (h^6 + 16)^{\frac{1}{2}} - 4.$$
  1. Find the value of \(\frac{dV}{dh}\) when \(h = 2\). [3]
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures. [3]
OCR C3 Q5
8 marks Moderate -0.8
  1. Find \(\int (3x + 7)^9 \, dx\). [3]
  2. \includegraphics{figure_5b} The diagram shows the curve \(y = \frac{1}{2\sqrt{x}}\). The shaded region is bounded by the curve and the lines \(x = 3\), \(x = 6\) and \(y = 0\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced, simplifying your answer. [5]
OCR C3 2010 January Q1
3 marks Moderate -0.8
Find \(\int \frac{10}{(2x - 7)^2} \, dx\). [3]
OCR C3 2010 January Q2
8 marks Standard +0.3
The angle \(\theta\) is such that \(0° < \theta < 90°\).
  1. Given that \(\theta\) satisfies the equation \(6 \sin 2\theta = 5 \cos \theta\), find the exact value of \(\sin \theta\). [3]
  2. Given instead that \(\theta\) satisfies the equation \(8 \cos \theta \cosec^2 \theta = 3\), find the exact value of \(\cos \theta\). [5]
OCR C3 2010 January Q3
7 marks Moderate -0.3
  1. Find, in simplified form, the exact value of \(\int_{10}^{20} \frac{60}{x} \, dx\). [2]
  2. Use Simpson's rule with two strips to find an approximation to \(\int_{10}^{20} \frac{60}{x} \, dx\). [3]
  3. Use your answers to parts (i) and (ii) to show that \(\ln 2 \approx \frac{25}{36}\). [2]
OCR C3 2010 January Q4
8 marks Moderate -0.8
\includegraphics{figure_4} The function \(f\) is defined for all real values of \(x\) by $$f(x) = 2 - \sqrt{x + 1}.$$ The diagram shows the graph of \(y = f(x)\).
  1. Evaluate \(f(-126)\). [2]
  2. Find the set of values of \(x\) for which \(f(x) = |f(x)|\). [2]
  3. Find an expression for \(f^{-1}(x)\). [3]
  4. State how the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) are related geometrically. [1]
OCR C3 2010 January Q5
9 marks Moderate -0.3
The equation of a curve is \(y = (x^2 + 1)^8\).
  1. Find an expression for \(\frac{dy}{dx}\) and hence show that the only stationary point on the curve is the point for which \(x = 0\). [4]
  2. Find an expression for \(\frac{d^2y}{dx^2}\) and hence find the value of \(\frac{d^2y}{dx^2}\) at the stationary point. [5]