Questions — OCR MEI (4456 questions)

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OCR MEI C2 Q9
3 marks Moderate -0.5
A geometric progression has 6 as its first term. Its sum to infinity is 5. Calculate its common ratio. [3]
OCR MEI C2 Q1
12 marks Moderate -0.8
Fig. 11.1 shows a village green which is bordered by 3 straight roads AB, BC and CA. The road AC runs due North and the measurements shown are in metres. \includegraphics{figure_1}
  1. Calculate the bearing of B from C, giving your answer to the nearest 0.1°. [4]
  2. Calculate the area of the village green. [2]
The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \includegraphics{figure_2} The new road is an arc of a circle with centre O and radius 130 m.
  1. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures. [2] (B) Show that the area of land added to the village green is 5300 m² correct to 2 significant figures. [4]
OCR MEI C2 Q2
5 marks Moderate -0.8
\includegraphics{figure_3} For triangle ABC shown in Fig. 4, calculate
  1. the length of BC, [3]
  2. the area of triangle ABC. [2]
OCR MEI C2 Q3
13 marks Moderate -0.3
  1. A boat travels from P to Q and then to R. As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of 045°. R is 9.2 km from P on a bearing of 113°, so that angle QPR is 68°. \includegraphics{figure_4} Calculate the distance and bearing of R from Q. [5]
  2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \includegraphics{figure_5} BC is an arc of a circle with centre A and radius 80 cm. Angle CAB = \(\frac{2\pi}{3}\) radians. EC is an arc of a circle with centre D and radius \(r\) cm. Angle CDE is a right angle.
    1. Calculate the area of sector ABC. [2]
    2. Show that \(r = 40\sqrt{3}\) and calculate the area of triangle CDA. [3]
    3. Hence calculate the area of cross-section of the rudder. [3]
OCR MEI C2 Q4
12 marks Standard +0.3
\emph{Arrowline Enterprises} is considering two possible logos: \includegraphics{figure_6}
  1. Fig. 10.1 shows the first logo ABCD. It is symmetrical about AC. Find the length of AB and hence find the area of this logo. [4]
  2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm. ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that ST = 16.2 cm to 3 significant figures. Find the area and perimeter of this logo. [8]
OCR MEI C2 Q5
12 marks Moderate -0.3
  1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A, then travel to B, then to C and finally back to A. \includegraphics{figure_7}
    1. Calculate the total length of the course for this race. [4]
    2. Given that the bearing of the first stage, AB, is 175°, calculate the bearing of the second stage, BC. [4]
  2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to Q, then a straight line back to P. The circle has radius 120 m and centre O; angle POQ = 136°. \includegraphics{figure_8} Calculate the total length of the course for this race. [4]
OCR MEI C2 Q1
5 marks Moderate -0.8
  1. Starting with an equilateral triangle, prove that \(\cos 30° = \frac{\sqrt{3}}{2}\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leqslant \theta \leqslant 2\pi\), giving your answers in terms of \(\pi\). [3]
OCR MEI C2 Q2
2 marks Easy -1.2
Use an isosceles right-angled triangle to show that \(\cos 45° = \frac{1}{\sqrt{2}}\). [2]
OCR MEI C2 Q3
5 marks Easy -1.3
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2x\) for values of \(x\) from \(0\) to \(2\pi\). [3]
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\). [2]
OCR MEI C2 Q4
3 marks Moderate -0.8
\(\theta\) is an acute angle and \(\sin \theta = \frac{1}{4}\). Find the exact value of \(\tan \theta\). [3]
OCR MEI C2 Q5
5 marks Easy -1.2
  1. Sketch the graph of \(y = \cos x\) for \(0° \leqslant x \leqslant 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leqslant x \leqslant 360°\). Label each graph clearly. [3]
  2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leqslant x \leqslant 360°\). [2]
OCR MEI C2 Q6
5 marks Moderate -0.8
  1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2\pi\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leqslant \theta \leqslant 2\pi\). Give your answers in the form \(k\pi\). [3]
OCR MEI C2 Q7
4 marks Moderate -0.8
Sketch the curve \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.68\) for \(0° \leqslant x \leqslant 360°\). [4]
OCR MEI C2 Q8
5 marks Moderate -0.8
  1. Sketch the graph of \(y = \tan x\) for \(0° \leqslant x \leqslant 360°\). [2]
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0° \leqslant x \leqslant 360°\). [3]
OCR MEI C2 Q9
4 marks Moderate -0.8
Sketch the graph of \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.2\) for \(0° \leqslant x \leqslant 360°\). [4]
OCR MEI C3 Q1
Easy -1.2
Solve the equation \(|3x + 2| = 1\).
OCR MEI C3 Q2
Moderate -0.8
Given that \(\arcsin x = \frac{1}{6}\pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).
OCR MEI C3 Q3
Moderate -0.8
The functions \(f(x)\) and \(g(x)\) are defined for the domain \(x > 0\) as follows: $$f(x) = \ln x, \quad g(x) = x^3.$$ Express the composite function \(fg(x)\) in terms of \(\ln x\). State the transformation which maps the curve \(y = f(x)\) onto the curve \(y = fg(x)\).
OCR MEI C3 Q4
Moderate -0.8
The temperature \(T°C\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20e^{-0.05t}, \quad \text{for } t \geq 0.$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature. Find the time at which the temperature is \(40°C\).
OCR MEI C3 Q5
Moderate -0.3
Using the substitution \(u = 2x + 1\), show that \(\int_0^1 \frac{x}{2x + 1} dx = \frac{1}{4}(2 - \ln 3)\).
OCR MEI C3 Q6
Standard +0.8
A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve.
OCR MEI C3 Q7
Standard +0.3
Fig. 7 shows the curve defined implicitly by the equation $$y^2 + y = x^3 + 2x,$$ together with the line \(x = 2\). \includegraphics{figure_7} Find the coordinates of the points of intersection of the line and the curve. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.
OCR MEI C3 Q8
17 marks Standard +0.3
Fig. 8 shows part of the curve \(y = x \sin 3x\). It crosses the \(x\)-axis at P. The point on the curve with \(x\)-coordinate \(\frac{1}{6}\pi\) is Q. \includegraphics{figure_8}
  1. Find the \(x\)-coordinate of P. [3]
  2. Show that Q lies on the line \(y = x\). [1]
  3. Differentiate \(x \sin 3x\). Hence prove that the line \(y = x\) touches the curve at Q. [6]
  4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac{1}{72}(\pi^2 - 8)\). [7]
OCR MEI C3 Q9
19 marks Standard +0.2
The function \(f(x) = \ln(1 + x^2)\) has domain \(-3 \leq x \leq 3\). Fig. 9 shows the graph of \(y = f(x)\). \includegraphics{figure_9}
  1. Show algebraically that the function is even. State how this property relates to the shape of the curve. [3]
  2. Find the gradient of the curve at the point P\((2, \ln 5)\). [4]
  3. Explain why the function does not have an inverse for the domain \(-3 \leq x \leq 3\). [1]
The domain of \(f(x)\) is now restricted to \(0 \leq x \leq 3\). The inverse of \(f(x)\) is the function \(g(x)\).
  1. Sketch the curves \(y = f(x)\) and \(y = g(x)\) on the same axes. State the domain of the function \(g(x)\). Show that \(g(x) = \sqrt{e^x - 1}\). [6]
  2. Differentiate \(g(x)\). Hence verify that \(g'(\ln 5) = \frac{1}{4}\). Explain the connection between this result and your answer to part (ii). [5]
OCR MEI C3 2011 January Q1
7 marks Moderate -0.8
Given that \(y = \sqrt[3]{1 + x^2}\), find \(\frac{dy}{dx}\). [4]