Questions — OCR C3 (339 questions)

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OCR C3 Q7
11 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with equation \(y = \cos^{-1} x\).
  1. Sketch the curve with equation \(y = 3 \cos^{-1}(x - 1)\), showing the coordinates of the points where the curve meets the axes. [3]
  2. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos^{-1}(x - 1) = x\) has exactly one root. [1]
  3. Show by calculation that the root of the equation \(3 \cos^{-1}(x - 1) = x\) lies between 1.8 and 1.9. [2]
  4. The sequence defined by $$x_1 = 2, \quad x_{n+1} = 1 + \cos(\frac{1}{3}x_n)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos^{-1}(x - 1) = x\). [5]
OCR C3 Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of the curve \(y = \ln(5 - x^2)\) which meets the \(x\)-axis at the point \(P\) with coordinates \((2, 0)\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(PQ\) and \(x = 0\).
  1. Find the equation of the tangent to the curve at \(P\). [5]
  2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures. [4]
  3. Deduce an approximation to the area of the region \(B\). [2]
OCR C3 Q9
13 marks Challenging +1.2
  1. By first writing \(\sin 3\theta\) as \(\sin(2\theta + \theta)\), show that $$\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta.$$ [4]
  2. Determine the greatest possible value of $$9 \sin(\frac{10}{3}\alpha) - 12 \sin^3(\frac{10}{3}\alpha),$$ and find the smallest positive value of \(\alpha\) (in degrees) for which that greatest value occurs. [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(3 \sin 6\beta \cos 2\beta = 4\). [6]
OCR C3 Q1
5 marks Moderate -0.3
Find the equation of the tangent to the curve \(y = \sqrt{4x + 1}\) at the point \((2, 3)\). [5]
OCR C3 Q2
5 marks Standard +0.8
Solve the inequality \(|2x - 3| < |x + 1|\). [5]
OCR C3 Q3
9 marks Moderate -0.3
The equation \(2x^3 + 4x - 35 = 0\) has one real root.
  1. Show by calculation that this real root lies between 2 and 3. [3]
  2. Use the iterative formula $$x_{n+1} = \sqrt[3]{17.5 - 2x_n},$$ with a suitable starting value, to find the real root of the equation \(2x^3 + 4x - 35 = 0\) correct to 2 decimal places. You should show the result of each iteration. [3]
OCR C3 Q4
6 marks Moderate -0.3
It is given that \(y = 5^{x-1}\).
  1. Show that \(x = 1 + \frac{\ln y}{\ln 5}\). [2]
  2. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
  3. Hence find the exact value of the gradient of the curve \(y = 5^{x-1}\) at the point \((3, 25)\). [2]
OCR C3 Q5
7 marks Moderate -0.3
  1. Write down the identity expressing \(\sin 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\). [1]
  2. Given that \(\sin \alpha = \frac{1}{4}\) and \(\alpha\) is acute, show that \(\sin 2\alpha = \frac{1}{8}\sqrt{15}\). [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(5 \sin 2\beta \sec \beta = 3\). [3]
OCR C3 Q6
9 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the graph of \(y = f(x)\), where $$f(x) = 2 - x^2, \quad x \leq 0.$$
  1. Evaluate ff(-3). [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]
OCR C3 Q7
10 marks Moderate -0.3
  1. Find the exact value of \(\int_1^2 \frac{2}{(4x - 1)^2} \, dx\). [4]
  2. \includegraphics{figure_7b} The diagram shows part of the curve \(y = \frac{1}{x}\). The point \(P\) has coordinates \((a, \frac{1}{a})\) and the point \(Q\) has coordinates \((2a, \frac{1}{2a})\), where \(a\) is a positive constant. The point \(R\) is such that \(PR\) is parallel to the \(x\)-axis and \(QR\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(PR\) and \(QR\). Show that the area of this shaded region is \(\ln(\frac{4}{e})\). [6]
OCR C3 Q8
11 marks Standard +0.3
  1. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\). [3]
  3. Solve, for \(0° < x < 360°\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1°\). [5]
OCR C3 Q9
13 marks Challenging +1.2
\includegraphics{figure_9} The diagram shows the curve with equation \(y = 2 \ln(x - 1)\). The point \(P\) has coordinates \((0, p)\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(y\)-axis to form a solid.
  1. Show that the volume, \(V \text{ cm}^3\), of the solid is given by $$V = \pi(e^p + 4e^{\frac{p}{2}} + p - 5).$$ [8]
  2. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \text{ cm min}^{-1}\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures. [5]
OCR C3 Q1
5 marks Moderate -0.3
Find the equation of the tangent to the curve \(y = \frac{2x + 1}{3x - 1}\) at the point \((1, \frac{3}{2})\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C3 Q2
5 marks Moderate -0.3
It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac{12}{13}\). Find the exact value of
  1. \(\cot \theta\), [2]
  2. \(\cos 2\theta\). [3]
OCR C3 Q3
12 marks Moderate -0.3
  1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x^5 \quad \text{and} \quad y = a - bx$$ on the same diagram, show that the equation $$x^5 + bx - a = 0$$ has exactly one real root. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[5]{53 - 2x_n}\), with a suitable starting value, to find the real root of the equation \(x^5 + 2x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places. [4]
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]