Questions C4 (1219 questions)

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OCR MEI C4 2012 June Q5
6 marks Moderate -0.3
Given the equation \(\sin(x + 45°) = 2\cos x\), show that \(\sin x + \cos x = 2\sqrt{2}\cos x\). Hence solve, correct to 2 decimal places, the equation for \(0° < x < 360°\). [6]
OCR MEI C4 2012 June Q6
8 marks Standard +0.3
Solve the differential equation \(\frac{dy}{dx} = \frac{y}{x(x+1)}\), given that when \(x = 1\), \(y = 1\). Your answer should express \(y\) explicitly in terms of \(x\). [8]
OCR MEI C4 2012 June Q7
19 marks Standard +0.3
Fig. 7a shows the curve with the parametric equations $$x = 2\cos\theta, \quad y = \sin 2\theta, \quad -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}.$$ The curve meets the \(x\)-axis at O and P. Q and R are turning points on the curve. The scales on the axes are the same. \includegraphics{figure_7a}
  1. State, with their coordinates, the points on the curve for which \(\theta = -\frac{\pi}{2}\), \(\theta = 0\) and \(\theta = \frac{\pi}{2}\). [3]
  2. Find \(\frac{dy}{dx}\) in terms of \(\theta\). Hence find the gradient of the curve when \(\theta = \frac{\pi}{2}\), and verify that the two tangents to the curve at the origin meet at right angles. [5]
  3. Find the exact coordinates of the turning point Q. [3]
When the curve is rotated about the \(x\)-axis, it forms a paperweight shape, as shown in Fig. 7b. \includegraphics{figure_7b}
  1. Express \(\sin^2\theta\) in terms of \(x\). Hence show that the cartesian equation of the curve is \(y^2 = x^2(1 - \frac{1}{4}x^2)\). [4]
  2. Find the volume of the paperweight shape. [4]
OCR MEI C4 2012 June Q8
17 marks Standard +0.3
With respect to cartesian coordinates \(Oxyz\), a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2y - 3z = 0\), as shown in Fig. 8. A' is the point (2, 4, 1), and M is the midpoint of AA'. \includegraphics{figure_8}
  1. Show that AA' is perpendicular to the plane \(x + 2y - 3z = 0\), and that M lies in the plane. [4]
The vector equation of the line AB is \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).
  1. Find the coordinates of B, and a vector equation of the line A'B. [6]
  2. Given that A'BC is a straight line, find the angle \(\theta\). [4]
  3. Find the coordinates of the point where BC crosses the \(Oxz\) plane (the plane containing the \(x\)- and \(z\)-axes). [3]
OCR MEI C4 2013 June Q1
8 marks Moderate -0.3
  1. Express \(\frac{x}{(1 + x)(1 - 2x)}\) in partial fractions. [3]
  2. Hence use binomial expansions to show that \(\frac{x}{(1 + x)(1 - 2x)} = ax + bx^2 + ...\), where \(a\) and \(b\) are constants to be determined. State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2013 June Q2
7 marks Standard +0.3
Show that the equation \(\cos ec x + 5 \cot x = 3 \sin x\) may be rearranged as $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ Hence solve the equation for \(0° \leq x \leq 360°\), giving your answers to 1 decimal place. [7]
OCR MEI C4 2013 June Q3
7 marks Moderate -0.8
Using appropriate right-angled triangles, show that \(\tan 45° = 1\) and \(\tan 30° = \frac{1}{\sqrt{3}}\). Hence show that \(\tan 75° = 2 + \sqrt{3}\). [7]
OCR MEI C4 2013 June Q4
8 marks Moderate -0.3
  1. Find a vector equation of the line \(l\) joining the points \((0, 1, 3)\) and \((-2, 2, 5)\). [2]
  2. Find the point of intersection of the line \(l\) with the plane \(x + 3y + 2z = 4\). [3]
  3. Find the acute angle between the line \(l\) and the normal to the plane. [3]
OCR MEI C4 2013 June Q5
6 marks Standard +0.3
The points A, B and C have coordinates \(A(3, 2, -1)\), \(B(-1, 1, 2)\) and \(C(10, 5, -5)\), relative to the origin O. Show that \(\overrightarrow{OC}\) can be written in the form \(\lambda\overrightarrow{OA} + \mu\overrightarrow{OB}\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points O, A, B and C from the fact that \(\overrightarrow{OC}\) can be expressed as a combination of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\)? [6]
OCR MEI C4 2013 June Q6
18 marks Standard +0.3
The motion of a particle is modelled by the differential equation $$v \frac{dv}{dt} + 4x = 0,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity. Initially \(x = 1\) and \(v = 4\).
  1. Solve the differential equation to show that \(v^2 = 20 - 4x^2\). [4]
Now consider motion for which \(x = \cos 2t + 2 \sin 2t\), where \(x\) is the displacement from a fixed point at time \(t\).
  1. Verify that, when \(t = 0\), \(x = 1\). Use the fact that \(v = \frac{dx}{dt}\) to verify that when \(t = 0\), \(v = 4\). [4]
  2. Express \(x\) in the form \(R \cos(2t - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v^2 = 20 - 4x^2\). [7]
  3. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value. [3]
OCR MEI C4 2013 June Q7
18 marks Standard +0.3
Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \includegraphics{figure_7}
  1. Find the lengths OA, OB and AC. [5]
  2. Find \(\frac{dy}{dx}\) in terms of \(u\). Hence find the angle \(\theta\). [6]
  3. Show that the cartesian equation of the curve is \(y = e^{x/5} + e^{-x/5}\). [2]
An object is formed by rotating the region OACB through \(360°\) about Ox.
  1. Find the volume of the object. [5]
OCR MEI C4 2013 June Q1
2 marks Easy -2.0
The diagram is a copy of Fig. 4. R is a place with latitude \(45°\) north and longitude \(60°\) west. Show the position of R on the diagram. M is the sub-solar point. It is on the Greenwich meridian and the declination of the sun is \(+20°\). Show the position of M on the diagram. [2] \includegraphics{figure_4}
OCR MEI C4 2013 June Q2
3 marks Easy -1.2
Use Fig. 8 to estimate the difference in the length of daylight between places with latitudes of \(30°\) south and \(60°\) south on the day for which the graph applies. [3]
OCR MEI C4 2013 June Q3
2 marks Easy -1.2
The graph is a copy of Fig. 6. The article says that it shows the terminator in the cases where the sun has declination \(10°\) north, \(1°\) north, \(5°\) south and \(15°\) south. Identify which curve (A, B, C or D) relates to which declination. [2] \includegraphics{figure_6}
\(10°\) north:
\(1°\) north:
\(5°\) south:
\(15°\) south:
OCR MEI C4 2013 June Q4
4 marks Moderate -0.5
In lines 94 and 95 the article says "Fig. 8 shows you that at latitude \(60°\) north the terminator passes approximately through time \(+9\) hours and \(-9\) hours so that there are about 18 hours of daylight." Use Equation (4) to check the accuracy of the figure of 18 hours. [4]
OCR MEI C4 2013 June Q5
7 marks Standard +0.3
  1. Use Equation (3) to calculate the declination of the sun on February 2nd. [3]
  2. The town of Boston, in Lincolnshire, has latitude \(53°\) north and longitude \(0°\). Calculate the time of sunset in Boston on February 2nd. Give your answer in hours and minutes using the 24-hour clock. [4]
OCR MEI C4 2014 June Q1
5 marks Moderate -0.3
Express \(\frac{3x}{(2-x)(4+x^2)}\) in partial fractions. [5]
OCR MEI C4 2014 June Q2
5 marks Moderate -0.3
Find the first three terms in the binomial expansion of \((4+x)^{\frac{1}{2}}\). State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2014 June Q3
5 marks Moderate -0.3
Fig. 3 shows the curve \(y = x^3 + \sqrt{(\sin x)}\) for \(0 \leqslant x \leqslant \frac{\pi}{4}\). \includegraphics{figure_3}
  1. Use the trapezium rule with 4 strips to estimate the area of the region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{4}\), giving your answer to 3 decimal places. [4]
  2. Suppose the number of strips in the trapezium rule is increased. Without doing further calculations, state, with a reason, whether the area estimate increases, decreases, or it is not possible to say. [1]
OCR MEI C4 2014 June Q4
8 marks Moderate -0.3
  1. Show that \(\cos(\alpha + \beta) = \frac{1 - \tan \alpha \tan \beta}{\sec \alpha \sec \beta}\). [3]
  2. Hence show that \(\cos 2\alpha = \frac{1 - \tan^2 \alpha}{1 + \tan^2 \alpha}\). [2]
  3. Hence or otherwise solve the equation \(\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \frac{1}{2}\) for \(0° \leqslant \theta \leqslant 180°\). [3]
OCR MEI C4 2014 June Q5
7 marks Standard +0.3
A curve has parametric equations \(x = e^{2t}, y = te^{2t}\).
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\). [4]
  2. Find the cartesian equation of the curve in the form \(y = ax^b \ln x\), where \(a\) and \(b\) are constants to be determined. [3]
OCR MEI C4 2014 June Q6
6 marks Standard +0.8
Fig. 6 shows the region enclosed by the curve \(y = (1 + 2x^2)^{\frac{1}{2}}\) and the line \(y = 2\). \includegraphics{figure_6} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\). [6]
OCR MEI C4 2014 June Q7
18 marks Standard +0.3
Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}
  1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
    1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
    2. Hence find the equation of this plane. [2]
  3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]
The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).
  1. Find the volume of the tetrahedron ABCD. [2]
OCR MEI C4 2014 June Q8
18 marks Standard +0.8
Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel. \includegraphics{figure_8.1} The height of the water surface above the hole \(t\) seconds after opening the hole is \(h\) metres, where $$\frac{dh}{dt} = -A\sqrt{h}$$ and where \(A\) is a positive constant. Initially the water surface is 1 metre above the hole.
  1. Verify that the solution to this differential equation is $$h = \left(1 - \frac{1}{2}At\right)^2.$$ [3]
The water stops leaking when \(h = 0\). This occurs after 20 seconds.
  1. Find the value of \(A\), and the time when the height of the water surface above the hole is 0.5 m. [4]
Fig. 8.2 shows a similar situation with a different barrel; \(h\) is in metres. \includegraphics{figure_8.2} For this barrel, $$\frac{dh}{dt} = -B\frac{\sqrt{h}}{(1+h)^2},$$ where \(B\) is a positive constant. When \(t = 0\), \(h = 1\).
  1. Solve this differential equation, and hence show that $$h^{\frac{1}{2}}(30 + 20h + 6h^2) = 56 - 15Bt.$$ [7]
  2. Given that \(h = 0\) when \(t = 20\), find \(B\). Find also the time when the height of the water surface above the hole is 0.5 m. [4]
Edexcel C4 Q1
6 marks Moderate -0.3
  1. Find the binomial expansion of \((2 - 3x)^{-3}\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
  2. State the set of values of \(x\) for which your expansion is valid. [1]