Questions — OCR C3 (339 questions)

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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]
OCR C3 2010 January Q6
7 marks Standard +0.3
Given that $$\int_0^{\ln 4} (ke^{3x} + (k - 2)e^{-\frac{x}{3}}) \, dx = 185,$$ find the value of the constant \(k\). [7]
OCR C3 2010 January Q7
7 marks Standard +0.3
  1. Leaking oil is forming a circular patch on the surface of the sea. The area of the patch is increasing at a rate of 250 square metres per hour. Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 1900 square metres. Give your answer correct to 2 significant figures. [4]
  2. The mass of a substance is decreasing exponentially. Its mass now is 150 grams and its mass, \(m\) grams, at a time \(t\) years from now is given by $$m = 150e^{-kt},$$ where \(k\) is a positive constant. Find, in terms of \(k\), the number of years from now at which the mass will be decreasing at a rate of 3 grams per year. [3]
OCR C3 2010 January Q8
11 marks Standard +0.8
  1. The curve \(y = \sqrt{x}\) can be transformed to the curve \(y = \sqrt{2x + 3}\) by means of a stretch parallel to the \(y\)-axis followed by a translation. State the scale factor of the stretch and give details of the translation. [3]
  2. It is given that \(N\) is a positive integer. By sketching on a single diagram the graphs of \(y = \sqrt{2x + 3}\) and \(y = \frac{N}{x}\), show that the equation $$\sqrt{2x + 3} = \frac{N}{x}$$ has exactly one real root. [3]
  3. A sequence \(x_1, x_2, x_3, \ldots\) has the property that $$x_{n+1} = N^{\frac{1}{2}}(2x_n + 3)^{-\frac{1}{4}}.$$ For certain values of \(x_1\) and \(N\), it is given that the sequence converges to the root of the equation $$\sqrt{2x + 3} = \frac{N}{x}.$$
    1. Find the value of the integer \(N\) for which the sequence converges to the value 1.9037 (correct to 4 decimal places). [2]
    2. Find the value of the integer \(N\) for which, correct to 4 decimal places, \(x_3 = 2.6022\) and \(x_4 = 2.6282\). [3]
OCR C3 2010 January Q9
12 marks Challenging +1.2
The value of \(\tan 10°\) is denoted by \(p\). Find, in terms of \(p\), the value of
  1. \(\tan 55°\), [3]
  2. \(\tan 5°\), [4]
  3. \(\tan \theta\), where \(\theta\) satisfies the equation \(3 \sin(\theta + 10°) = 7 \cos(\theta - 10°)\). [5]
OCR C3 2013 January Q1
6 marks Moderate -0.8
For each of the following curves, find the gradient at the point with \(x\)-coordinate 2.
  1. \(y = \frac{3x}{2x + 1}\) [3]
  2. \(y = \sqrt{4x^2 + 9}\) [3]
OCR C3 2013 January Q2
5 marks Moderate -0.3
The acute angle \(A\) is such that \(\tan A = 2\).
  1. Find the exact value of \(\cosec A\). [2]
  2. The angle \(B\) is such that \(\tan (A + B) = 3\). Using an appropriate identity, find the exact value of \(\tan B\). [3]
OCR C3 2013 January Q3
7 marks Standard +0.8
  1. Given that \(|t| = 3\), find the possible values of \(|2t - 1|\). [3]
  2. Solve the inequality \(|x - t^2| > |x + 3\sqrt{2}|\). [4]
OCR C3 2013 January Q4
6 marks Moderate -0.3
The mass, \(m\) grams, of a substance is increasing exponentially so that the mass at time \(t\) hours is given by $$m = 250e^{0.02t}.$$
  1. Find the time taken for the mass to increase to twice its initial value, and deduce the time taken for the mass to increase to 8 times its initial value. [3]
  2. Find the rate at which the mass is increasing at the instant when the mass is 400 grams. [3]
OCR C3 2013 January Q5
9 marks Standard +0.3
\includegraphics{figure_5} The diagram shows the curve \(y = \frac{6}{\sqrt{3x + 1}}\). The shaded region is bounded by the curve and the lines \(x = 2\), \(x = 9\) and \(y = 0\).
  1. Show that the area of the shaded region is \(4\sqrt{7}\) square units. [4]
  2. The shaded region is rotated completely about the \(x\)-axis. Show that the volume of the solid produced can be written in the form \(k\ln 2\), where the exact value of the constant \(k\) is to be determined. [5]
OCR C3 2013 January Q6
11 marks Standard +0.3
  1. By sketching the curves \(y = \ln x\) and \(y = 8 - 2x^2\) on a single diagram, show that the equation $$\ln x = 8 - 2x^2$$ has exactly one real root. [3]
  2. Explain how your diagram shows that the root is between 1 and 2. [1]
  3. Use the iterative formula $$x_{n+1} = \sqrt{4 - \frac{1}{2}\ln x_n},$$ with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places. [4]
  4. The curves \(y = \ln x\) and \(y = 8 - 2x^2\) are each translated by 2 units in the positive \(x\)-direction and then stretched by scale factor 4 in the \(y\)-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places. [3]