Questions C3 (1301 questions)

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OCR MEI C3 2011 June Q9
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\). The endpoints of the curve are P \((-\pi, 1)\) and Q \((\pi, 3)\), and \(f(x) = a + \sin bx\), where \(a\) and \(b\) are constants. \includegraphics{figure_9}
  1. Using Fig. 9, show that \(a = 2\) and \(b = \frac{1}{2}\). [3]
  2. Find the gradient of the curve \(y = f(x)\) at the point \((0, 2)\). Show that there is no point on the curve at which the gradient is greater than this. [5]
  3. Find \(f^{-1}(x)\), and state its domain and range. Write down the gradient of \(y = f^{-1}(x)\) at the point \((2, 0)\). [6]
  4. Find the area enclosed by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \pi\). [4]
OCR MEI C3 2014 June Q1
3 marks Moderate -0.8
Evaluate \(\int_0^{\frac{\pi}{4}} (1 - \sin 3x) \, dx\), giving your answer in exact form. [3]
OCR MEI C3 2014 June Q2
5 marks Standard +0.3
Find the exact gradient of the curve \(y = \ln(1 - \cos 2x)\) at the point with \(x\)-coordinate \(\frac{1}{4}\pi\). [5]
OCR MEI C3 2014 June Q3
4 marks Standard +0.3
Solve the equation \(|3 - 2x| = 4|x|\). [4]
OCR MEI C3 2014 June Q4
7 marks Standard +0.3
Fig. 4 shows the curve \(y = f(x)\), where $$f(x) = a + \cos bx, \quad 0 \leq x \leq 2\pi,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \((0, 3)\) and \((2\pi, 1)\). \includegraphics{figure_4}
  1. Find \(a\) and \(b\). [2]
  2. Find \(f^{-1}(x)\), and state its domain and range. [5]
OCR MEI C3 2014 June Q5
5 marks Standard +0.3
A spherical balloon of radius \(r\) cm has volume \(V\) cm\(^3\), where \(V = \frac{4}{3}\pi r^3\). The balloon is inflated at a constant rate of 10 cm\(^3\) s\(^{-1}\). Find the rate of increase of \(r\) when \(r = 8\). [5]
OCR MEI C3 2014 June Q6
8 marks Moderate -0.3
The value \(£V\) of a car \(t\) years after it is new is modelled by the equation \(V = Ae^{-kt}\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.
  1. Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 e^{-0.2t}.$$ Calculate how much value, to the nearest £100, this car has lost after 1 year. [2]
  2. At the same time as Brian buys his car, Kate buys a new hatchback for £15000. Her car loses £2000 of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures. [3]
  3. Find how long it is before Brian's and Kate's cars have the same value. [3]
OCR MEI C3 2014 June Q7
4 marks Standard +0.3
Either prove or disprove each of the following statements.
  1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.' [2]
  2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]
OCR MEI C3 2014 June Q8
18 marks Standard +0.3
Fig. 8 shows the curve \(y = f(x)\), where \(f(x) = \frac{x}{\sqrt{2 + x^2}}\). \includegraphics{figure_8}
  1. Show algebraically that \(f(x)\) is an odd function. Interpret this result geometrically. [3]
  2. Show that \(f'(x) = \frac{2}{(2 + x^2)^{\frac{3}{2}}}\). Hence find the exact gradient of the curve at the origin. [5]
  3. Find the exact area of the region bounded by the curve, the \(x\)-axis and the line \(x = 1\). [4]
    1. Show that if \(y = \frac{x}{\sqrt{2 + x^2}}\), then \(\frac{1}{y^2} = \frac{2}{x^2} + 1\). [2]
    2. Differentiate \(\frac{1}{y^2} = \frac{2}{x^2} + 1\) implicitly to show that \(\frac{dy}{dx} = \frac{2y^3}{x^3}\). Explain why this expression cannot be used to find the gradient of the curve at the origin. [4]
OCR MEI C3 2014 June Q9
18 marks Standard +0.8
Fig. 9 shows the curve \(y = xe^{-2x}\) together with the straight line \(y = mx\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P. The dashed line is the tangent at P. \includegraphics{figure_9}
  1. Show that the \(x\)-coordinate of P is \(-\frac{1}{2}\ln m\). [3]
  2. Find, in terms of \(m\), the gradient of the tangent to the curve at P. [4]
You are given that OP and this tangent are equally inclined to the \(x\)-axis.
  1. Show that \(m = e^{-2}\), and find the exact coordinates of P. [4]
  2. Find the exact area of the shaded region between the line OP and the curve. [7]
OCR MEI C3 2016 June Q1
3 marks Moderate -0.8
Find the exact value of \(\int_0^{\frac{1}{4}\pi} (1 + \cos \frac{1}{2}x) dx\). [3]
OCR MEI C3 2016 June Q2
5 marks Standard +0.8
The functions \(f(x)\) and \(g(x)\) are defined by \(f(x) = \ln x\) and \(g(x) = 2 + e^x\), for \(x > 0\). Find the exact value of \(x\), given that \(fg(x) = 2x\). [5]
OCR MEI C3 2016 June Q3
5 marks Challenging +1.8
Find \(\int_{-1}^4 x^{-\frac{1}{2}} \ln x dx\), giving your answer in an exact form. [5]
OCR MEI C3 2016 June Q4
4 marks Moderate -0.8
By sketching the graphs of \(y = |2x + 1|\) and \(y = -x\) on the same axes, show that the equation \(|2x + 1| = -x\) has two roots. Find these roots. [4]
OCR MEI C3 2016 June Q5
7 marks Standard +0.3
The volume \(V\) m³ of a pile of grain of height \(h\) metres is modelled by the equation $$V = 4\sqrt{h^3 + 1} - 4.$$
  1. Find \(\frac{dV}{dh}\) when \(h = 2\). [4]
At a certain time, the height of the pile is 2 metres, and grain is being added so that the volume is increasing at a rate of 0.4 m³ per minute.
  1. Find the rate at which the height is increasing at this time. [3]
OCR MEI C3 2016 June Q6
8 marks Standard +0.3
Fig. 6 shows part of the curve \(\sin 2y = x - 1\). P is the point with coordinates \((1.5, \frac{1}{12}\pi)\) on the curve. \includegraphics{figure_6}
  1. Find \(\frac{dy}{dx}\) in terms of \(y\). Hence find the exact gradient of the curve \(\sin 2y = x - 1\) at the point P. [4]
The part of the curve shown is the image of the curve \(y = \arcsin x\) under a sequence of two geometrical transformations.
  1. Find \(y\) in terms of \(x\) for the curve \(\sin 2y = x - 1\). Hence describe fully the sequence of transformations. [4]
OCR MEI C3 2016 June Q7
4 marks Standard +0.8
You are given that \(n\) is a positive integer. By expressing \(x^{2n} - 1\) as a product of two factors, prove that \(2^{2n} - 1\) is divisible by 3. [4]
OCR MEI C3 2016 June Q8
18 marks Standard +0.8
Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x+4}}\) and the line \(x = 5\). The curve has an asymptote \(l\). The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q. \includegraphics{figure_8}
  1. Show that for this curve \(\frac{dy}{dx} = \frac{x + 8}{2(x + 4)^{\frac{3}{2}}}\). [5]
  2. Find the coordinates of the point P. [4]
  3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\). [9]
OCR MEI C3 2016 June Q9
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = e^{2x} + k e^{-2x}\) and \(k\) is a constant greater than 1. The curve crosses the \(y\)-axis at P and has a turning point Q. \includegraphics{figure_9}
  1. Find the \(y\)-coordinate of P in terms of \(k\). [1]
  2. Show that the \(x\)-coordinate of Q is \(\frac{1}{4}\ln k\), and find the \(y\)-coordinate in its simplest form. [5]
  3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\ln k\). Give your answer in the form \(ak + b\). [4]
The function \(g(x)\) is defined by \(g(x) = f(x + \frac{1}{4}\ln k)\).
    1. Show that \(g(x) = \sqrt{k}(e^{2x} + e^{-2x})\). [3]
    2. Hence show that \(g(x)\) is an even function. [2]
    3. Deduce, with reasons, a geometrical property of the curve \(y = f(x)\). [3]
OCR MEI C3 Q1
5 marks Moderate -0.3
You are given that \(y^2 = 4x + 7\).
  1. Use implicit differentiation to find \(\frac{dy}{dx}\) in terms of \(y\). [2]
  2. Make \(x\) the subject of the equation. Find \(\frac{dx}{dy}\) and hence show that in this case \(\frac{dx}{dy} = \frac{1}{\frac{dx}{dy}}\). [3]
OCR MEI C3 Q2
4 marks Moderate -0.8
  1. Expand \((e^x + e^{-x})^2\). [1]
  2. Hence find \(\int (e^x + e^{-x})^2 dx\). [3]
OCR MEI C3 Q3
6 marks Moderate -0.8
  1. Sketch the graph of \(y = |3x - 6|\). [2]
  2. Solve the equation \(|3x - 6| = x + 4\) and illustrate your answer on your graph. [4]
OCR MEI C3 Q4
4 marks Standard +0.3
Find \(\int x \sin 3x dx\). [4]
OCR MEI C3 Q5
4 marks Moderate -0.3
Make \(x\) the subject of \(t = \ln \sqrt{\frac{5}{(x-3)}}\). [4]
OCR MEI C3 Q6
7 marks Standard +0.3
The function f(x) is defined as \(f(x) = \frac{\ln x}{x}\). The graph of the function is shown in Fig. 6. \includegraphics{figure_6}
  1. Give the coordinates of the point, P, where the curve crosses the \(x\)-axis. [1]
  2. Use calculus to find the coordinates of the stationary point, Q, and show that it is a maximum. [6]