Questions C3 (1301 questions)

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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]
OCR C3 2013 January Q7
8 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with equation $$x = (y + 4)\ln (2y + 3).$$ The curve crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [3]
  2. Find the gradient of the curve at each of the points \(A\) and \(B\), giving each answer correct to 2 decimal places. [5]
OCR C3 2013 January Q8
10 marks Standard +0.3
The functions f and g are defined for all real values of \(x\) by $$\text{f}(x) = x^2 + 4ax + a^2 \text{ and } \text{g}(x) = 4x - 2a,$$ where \(a\) is a positive constant.
  1. Find the range of f in terms of \(a\). [4]
  2. Given that fg(3) = 69, find the value of \(a\) and hence find the value of \(x\) such that \(\text{g}^{-1}(x) = x\). [6]
OCR C3 2013 January Q9
10 marks Standard +0.8
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
  3. It is given that there are two values of \(\theta\), where \(-90° < \theta < 90°\), satisfying the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \frac{2}{3}\theta - \sin \frac{2}{3}\theta) = k,$$ where \(k\) is a constant. Find the set of possible values of \(k\). [3]
OCR C3 2009 June Q1
3 marks Easy -1.8
\includegraphics{figure_1} Each diagram above shows part of a curve, the equation of which is one of the following: $$y = \sin^{-1} x, \quad y = \cos^{-1} x, \quad y = \tan^{-1} x, \quad y = \sec x, \quad y = \cosec x, \quad y = \cot x.$$ State which equation corresponds to
  1. Fig. 1, [1]
  2. Fig. 2, [1]
  3. Fig. 3. [1]
OCR C3 2009 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} The diagram shows the curve with equation \(y = (2x - 3)^2\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis. [5]
OCR C3 2009 June Q3
6 marks Standard +0.3
The angles \(\alpha\) and \(\beta\) are such that $$\tan \alpha = m + 2 \quad \text{and} \quad \tan \beta = m,$$ where \(m\) is a constant.
  1. Given that \(\sec^2 \alpha - \sec^2 \beta = 16\), find the value of \(m\). [3]
  2. Hence find the exact value of \(\tan(\alpha + \beta)\). [3]
OCR C3 2009 June Q4
9 marks Standard +0.3
It is given that \(\int_a^{3a} (e^{5x} + e^x) dx = 100\), where \(a\) is a positive constant.
  1. Show that \(a = \frac{1}{5}\ln(300 + 3e^a - 2e^{3a})\). [5]
  2. Use an iterative process, based on the equation in part (i), to find the value of \(a\) correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process. [4]
OCR C3 2009 June Q5
10 marks Moderate -0.8
The functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which
  1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
  2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
  3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]
OCR C3 2009 June Q6
7 marks Standard +0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(x = (37 + 10y - 2y^2)^{\frac{1}{2}}\).
  1. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
  2. Hence find the equation of the tangent to the curve at the point \((7, 3)\), giving your answer in the form \(y = mx + c\). [5]
OCR C3 2009 June Q7
10 marks Standard +0.3
  1. Express \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence
    1. solve, for \(0° < \theta < 360°\), the equation \(8 \sin \theta - 6 \cos \theta = 9\), [4]
    2. find the greatest possible value of $$32 \sin x - 24 \cos x - (16 \sin y - 12 \cos y)$$ as the angles \(x\) and \(y\) vary. [3]
OCR C3 2009 June Q8
10 marks Standard +0.3
\includegraphics{figure_4} The diagram shows the curves \(y = \ln x\) and \(y = 2 \ln(x - 6)\). The curves meet at the point \(P\) which has \(x\)-coordinate \(a\). The shaded region is bounded by the curve \(y = 2 \ln(x - 6)\) and the lines \(x = a\) and \(y = 0\).
  1. Give details of the pair of transformations which transforms the curve \(y = \ln x\) to the curve \(y = 2 \ln(x - 6)\). [3]
  2. Solve an equation to find the value of \(a\). [4]
  3. Use Simpson's rule with two strips to find an approximation to the area of the shaded region. [3]
OCR C3 2009 June Q9
12 marks Challenging +1.2
  1. Show that, for all non-zero values of the constant \(k\), the curve $$y = \frac{kx^2 - 1}{kx^2 + 1}$$ has exactly one stationary point. [5]
  2. Show that, for all non-zero values of the constant \(m\), the curve $$y = e^{mx}(x^2 + mx)$$ has exactly two stationary points. [7]
OCR C3 2010 June Q1
6 marks Easy -1.2
Find \(\frac{dy}{dx}\) in each of the following cases:
  1. \(y = x^3 e^{2x}\), [2]
  2. \(y = \ln(3 + 2x^2)\), [2]
  3. \(y = \frac{x}{2x + 1}\). [2]
OCR C3 2010 June Q2
4 marks Moderate -0.3
The transformations R, S and T are defined as follows. \begin{align} \text{R} &: \text{ reflection in the } x\text{-axis}
\text{S} &: \text{ stretch in the } x\text{-direction with scale factor 3}
\text{T} &: \text{ translation in the positive } x\text{-direction by 4 units} \end{align}
  1. The curve \(y = \ln x\) is transformed by R followed by T. Find the equation of the resulting curve. [2]
  2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]
OCR C3 2010 June Q3
6 marks Standard +0.3
  1. Express the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\) in the form \(a \sin^2 \theta + b \sin \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Hence solve, for \(-180° < \theta < 180°\), the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\). [3]