Questions C4 (1219 questions)

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OCR MEI C4 Q2
18 marks Standard +0.3
2 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b46db958-aa88-47fb-8db3-786472791577-2_916_1255_464_397} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\). Hence calculate the angle BAC .
  2. Verify that the equation of the plane ABC is \(x + y - 2 z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane.
  3. Given that \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB .
OCR MEI C4 Q3
18 marks Standard +0.3
3 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, Oy due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.
  1. Verify that \(\left. \overrightarrow { \mathrm { AB } } = \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
    [0pt] [3]
  2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical.
    [0pt] [6]
    A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
  3. Find the coordinates of the point where the pipeline meets the layer of rock.
  4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.
OCR MEI C4 Q4
17 marks Standard +0.3
4 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point \(\mathrm { A } ( 1,2,2 )\), and enters a glass object at point \(\mathrm { B } ( 0,0,2 )\). The surface of the glass object is a plane with normal vector \(\mathbf { n }\). Fig. 7 shows a cross-section of the glass object in the plane of the light ray and \(\mathbf { n }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b46db958-aa88-47fb-8db3-786472791577-4_689_812_341_662} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the vector \(\overrightarrow { \mathrm { AB } }\) and a vector equation of the line AB . The surface of the glass object is a plane with equation \(x + z = 2\). AB makes an acute angle \(\theta\) with the normal to this plane.
  2. Write down the normal vector \(\mathbf { n }\), and hence calculate \(\theta\), giving your answer in degrees. The line BC has vector equation \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 0 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } - 2 \\ - 2 \\ - 1 \end{array} \right)\). This line makes an acute angle \(\phi\) with the
    normal to the plane. normal to the plane.
  3. Show that \(\phi = 45 ^ { \circ }\).
  4. Snell's Law states that \(\sin \theta = k \sin \phi\), where \(k\) is a constant called the refractive index. Find \(k\). The light ray leaves the glass object through a plane with equation \(x + z = - 1\). Units are centimetres.
  5. Find the point of intersection of the line BC with the plane \(x + z = - 1\). Hence find the distance the light ray travels through the glass object.
OCR MEI C4 Q1
8 marks Standard +0.3
1
  1. Find the point of intersection of the line \(\left. \left. \mathbf { r } = \begin{array} { r } - 8 \\ - 2 \\ 6 \end{array} \right) + \lambda \begin{array} { r } - 3 \\ 0 \\ 1 \end{array} \right)\) and the plane \(2 x - 3 y + z = 11\).
  2. Find the acute angle between the line and the normal to the plane.
OCR MEI C4 Q2
7 marks Standard +0.3
2 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(( 1,3 , - 2 ) , ( - 1,2 , - 3 )\) and \(( 0 , - 8,1 )\) respectively.
  1. Find the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\).
  2. Show that the vector \(2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the equation of the plane ABC .
  3. Write down normal vectors to the planes \(2 x - y + z = 2\) and \(x - z = 1\). Hence find the acute angle between the planes.
  4. Write down a vector equation of the line through \(( 2,0,1 )\) perpendicular to the plane \(2 x - y + z = 2\). Find the point of intersection of this line with the plane.
  5. Find the cartesian equation of the plane through the point \(( 2 , - 1,4 )\) with normal vector $$\mathbf { n } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right) .$$
  6. Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 12 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)$$
OCR MEI C4 Q12
Standard +0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff20b83a-5e38-437e-8115-5b0a6a54fa9d-2_745_1300_256_399} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
  1. Find the length AE .
  2. Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
  3. Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30 .$$ Write down a vector normal to this plane.
  4. Show that the vector \(\left( \begin{array} { l } 4 \\ 3 \\ 5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE .
  5. Find the angle between the planes ABC and ABDE .
OCR MEI C4 Q1
6 marks Moderate -0.5
1 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )\) and \(\mathrm { C } ( 10,5 , - 5 )\), relative to the origin O . Show that \(\overrightarrow { \mathrm { OC } }\) can be written in the form \(\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and C from the fact that \(\overrightarrow { \mathrm { OC } }\) can be expressed as a combination of \(\overrightarrow { \mathrm { OA } }\) and \(\overrightarrow { \mathrm { OB } }\) ?
OCR MEI C4 Q2
5 marks Moderate -0.8
2 Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).
Find constants \(\lambda\) and \(\mu\) such that \(\lambda \mathbf { a } + \mu \mathbf { b } = 4 \mathbf { j } - 3 \mathbf { k }\).
OCR MEI C4 Q3
6 marks Standard +0.3
3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle. [6]
OCR C4 2009 January Q1
3 marks Easy -1.2
1 Simplify \(\frac { 20 - 5 x } { 6 x ^ { 2 } - 24 x }\).
OCR C4 2009 January Q2
4 marks Standard +0.3
2 Find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
OCR C4 2009 January Q3
9 marks Standard +0.3
3
  1. Expand \(( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. Hence find the expansion of \(\frac { ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } } { ( 1 + x ) ^ { 3 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  3. State the set of values of \(x\) for which the expansion in part (ii) is valid.
OCR C4 2009 January Q4
6 marks Moderate -0.3
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 1 + \sin x ) ^ { 2 } \mathrm {~d} x\).
OCR C4 2009 January Q5
8 marks Standard +0.3
5
  1. Show that the substitution \(u = \sqrt { x }\) transforms \(\int \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x\) to \(\int \frac { 2 } { u ( 1 + u ) } \mathrm { d } u\).
  2. Hence find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x\).
OCR C4 2009 January Q6
9 marks Moderate -0.3
6 A curve has parametric equations $$x = t ^ { 2 } - 6 t + 4 , \quad y = t - 3 .$$ Find
  1. the coordinates of the point where the curve meets the \(x\)-axis,
  2. the equation of the curve in cartesian form, giving your answer in a simple form without brackets,
  3. the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C4 2009 January Q7
10 marks Standard +0.3
7
  1. Show that the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 4 \\ - 2 \end{array} \right)\) meets the line passing through ( \(9,7,5\) ) and ( \(7,8,2\) ), and find the point of intersection of these lines.
  2. Find the acute angle between these lines.
OCR C4 2009 January Q8
12 marks Standard +0.3
8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } = 6 x y\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Show that the point \(\left( 2 ^ { \frac { 4 } { 3 } } , 2 ^ { \frac { 5 } { 3 } } \right)\) lies on the curve and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at this point.
  3. The point \(( a , a )\), where \(a > 0\), lies on the curve. Find the value of \(a\) and the gradient of the curve at this point.
OCR C4 2009 January Q9
11 marks Standard +0.3
9 A liquid is being heated in an oven maintained at a constant temperature of \(160 ^ { \circ } \mathrm { C }\). It may be assumed that the rate of increase of the temperature of the liquid at any particular time \(t\) minutes is proportional to \(160 - \theta\), where \(\theta ^ { \circ } \mathrm { C }\) is the temperature of the liquid at that time.
  1. Write down a differential equation connecting \(\theta\) and \(t\). When the liquid was placed in the oven, its temperature was \(20 ^ { \circ } \mathrm { C }\) and 5 minutes later its temperature had risen to \(65 ^ { \circ } \mathrm { C }\).
  2. Find the temperature of the liquid, correct to the nearest degree, after another 5 minutes. 4
OCR C4 2010 January Q1
4 marks Moderate -0.3
1 Find the quotient and the remainder when \(x ^ { 4 } + 11 x ^ { 3 } + 28 x ^ { 2 } + 3 x + 1\) is divided by \(x ^ { 2 } + 5 x + 2\).
OCR C4 2010 January Q2
6 marks Standard +0.3
2 Points \(A , B\) and \(C\) have position vectors \(- 5 \mathbf { i } - 10 \mathbf { j } + 12 \mathbf { k } , \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } + 6 \mathbf { j } + p \mathbf { k }\) respectively, where \(p\) is a constant.
  1. Given that angle \(A B C = 90 ^ { \circ }\), find the value of \(p\).
  2. Given instead that \(A B C\) is a straight line, find the value of \(p\).
OCR C4 2010 January Q3
5 marks Moderate -0.3
3 By expressing \(\cos 2 x\) in terms of \(\cos x\), find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x\).
OCR C4 2010 January Q4
6 marks Moderate -0.3
4 Use the substitution \(u = 2 + \ln t\) to find the exact value of $$\int _ { 1 } ^ { \mathrm { e } } \frac { 1 } { t ( 2 + \ln t ) ^ { 2 } } \mathrm {~d} t$$
OCR C4 2010 January Q5
7 marks Moderate -0.3
5
  1. Expand \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. (a) Hence, or otherwise, expand \(( 8 + 16 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
    (b) State the set of values of \(x\) for which the expansion in part (ii) (a) is valid.
OCR C4 2010 January Q6
6 marks Standard +0.3
6 A curve has parametric equations $$x = 9 t - \ln ( 9 t ) , \quad y = t ^ { 3 } - \ln \left( t ^ { 3 } \right)$$ Show that there is only one value of \(t\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and state that value.
OCR C4 2010 January Q7
8 marks Standard +0.3
7 Find the equation of the normal to the curve \(x ^ { 3 } + 2 x ^ { 2 } y = y ^ { 3 } + 15\) at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.