Questions — OCR (4907 questions)

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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]
OCR C3 2010 June Q4
7 marks Standard +0.3
\includegraphics{figure_4} The diagram shows part of the curve \(y = \frac{k}{x}\), where \(k\) is a positive constant. The points A and B on the curve have \(x\)-coordinates 2 and 6 respectively. Lines through A and B parallel to the axes as shown meet at the point C. The region R is bounded by the curve and the lines \(x = 2\), \(x = 6\) and \(y = 0\). The region S is bounded by the curve and the lines AC and BC. It is given that the area of the region R is \(\ln 81\).
  1. Show that \(k = 4\). [3]
  2. Find the exact volume of the solid produced when the region S is rotated completely about the \(x\)-axis. [4]
OCR C3 2010 June Q5
7 marks Standard +0.8
  1. Solve the inequality \(|2x + 1| \leqslant |x - 3|\). [5]
  2. Given that \(x\) satisfies the inequality \(|2x + 1| \leqslant |x - 3|\), find the greatest possible value of \(|x + 2|\). [2]
OCR C3 2010 June Q6
10 marks Standard +0.3
  1. Show by calculation that the equation $$\tan^2 x - x - 2 = 0,$$ where \(x\) is measured in radians, has a root between 1.0 and 1.1. [3]
  2. Use the iteration formula \(x_{n+1} = \tan^{-1}\sqrt{2 + x_n}\) with a suitable starting value to find this root correct to 5 decimal places. You should show the outcome of each step of the process. [4]
  3. Deduce a root of the equation $$\sec^2 2x - 2x - 3 = 0.$$ [3]
OCR C3 2010 June Q7
10 marks Standard +0.8
\includegraphics{figure_7} The diagram shows the curve with equation \(y = (3x - 1)^4\). The point P on the curve has coordinates \((1, 16)\) and the tangent to the curve at P meets the \(x\)-axis at the point Q. The shaded region is bounded by PQ, the \(x\)-axis and that part of the curve for which \(\frac{1}{3} \leqslant x \leqslant 1\). Find the exact area of this shaded region. [10]
OCR C3 2010 June Q8
9 marks Standard +0.3
  1. Express \(3 \cos x + 3 \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [3]
  2. The expression T\((x)\) is defined by T\((x) = \frac{8}{3 \cos x + 3 \sin x}\).
    1. Determine a value of \(x\) for which T\((x)\) is not defined. [2]
    2. Find the smallest positive value of \(x\) satisfying T\((3x) = \frac{8}{3}\sqrt{6}\), giving your answer in an exact form. [4]