Questions - OCR MEI C4 - A-Level Maths

Find the length of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
(A) Verify that $4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }$ is normal to the plane ABC .
(B) Hence find the equation of this plane.
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 . The volume of a tetrahedron is $\frac { 1 } { 3 } \times$ area of base × height.
Find the volume of the tetrahedron ABCD .
Find a vector equation of the line $l$ joining the points $( 0,1,3 )$ and $( - 2,2,5 )$.
Find the point of intersection of the line $l$ with the plane $x + 3 y + 2 z = 4$.
Find the acute angle between the line $l$ and the normal to the plane.

OCR MEI C4 Q1

1 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point $\mathrm { A } ( 1,2,4 )$, and is reflected at point B off the plane with equation $x + 2 y - 3 z = 0$, as shown in Fig. 8. $\mathrm { A } ^ { \prime }$ is the point $( 2,4,1 )$, and $M$ is the midpoint of $\mathrm { AA } ^ { \prime }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b46db958-aa88-47fb-8db3-786472791577-1_562_716_464_650} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Show that $\mathrm { AA } ^ { \prime }$ is perpendicular to the plane $x + 2 y - 3 z = 0$, and that M lies in the plane. The vector equation of the line AB is $\mathbf { r } = \left( \begin{array} { l } 1
2
4 \end{array} \right) + \lambda \left( \begin{array} { r } 1
- 1
2 \end{array} \right)$.
Find the coordinates of B , and a vector equation of the line $\mathrm { A } ^ { \prime } \mathrm { B }$.
Given that $\mathrm { A } ^ { \prime } \mathrm { BC }$ is a straight line, find the angle $\theta$.
Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the $x$ - and $z$-axes)

OCR MEI C4 Q2

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}

Write down the vectors $\overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { AC } }$. Hence calculate the angle BAC .
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.
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

9 marks

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.

Verify that $\left. \overrightarrow { \mathrm { AB } } = \begin{array} { l } 300
100
100 \end{array} \right)$ and find the length of the pipeline.
[0pt] [3]
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 $x + 2 y + 3 z = 320$.
Find the coordinates of the point where the pipeline meets the layer of rock.
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

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}

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.
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.
Show that $\phi = 45 ^ { \circ }$.
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.
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

1 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-1_1014_1407_424_406} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Relative to axes $\mathrm { O } x$ (due east), $\mathrm { O } y$ (due north) and $\mathrm { O } z$ (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: ( $0,100 , - 25$ )
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

Verify that the cartesian equation of the plane ABC is $3 x + 2 y + 20 z + 300 = 0$.
Find the vectors $\overrightarrow { \mathrm { DE } }$ and $\overrightarrow { \mathrm { DF } }$. Show that the vector $2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }$ is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point $\mathrm { R } ( 15,34,0 )$ in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
Write down a vector equation of the line RS.

OCR MEI C4 Q3

3 Verify that the point $( - 1,6,5 )$ lies on both the lines $$\mathbf { r } = \left( \begin{array} { l } 1
2
1 \end{array} \right) + \lambda \left( \begin{array} { l } 1
2
3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 0
6
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
0
2 \end{array} \right)$$ Find the acute angle between the lines.

OCR MEI C4 Q4

4 A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill boles by entering the equation of the line of the hole. A $20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}$ cuboid is to be cut and drilled. The cuboid is positioned relative to $x - y ^ { 2 }$ and $z$-axes as shown in Fig. 8.1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-3_414_740_460_302} \captionsetup{labelformat=empty} \caption{Fig. 8.1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-3_449_737_425_1062} \captionsetup{labelformat=empty} \caption{Fig. 8.2}

\end{figure} First, a plane cut is made to remove the comer at $E$. The cut goes through the points $P , Q$ and $R$, which are the midpoints of the sides $\mathrm { ED } , \mathrm { EA }$ and EF respectively.

Write down the coordinates $\boldsymbol { 0 } \mathbf { F } \mathrm { Q }$ and $\mathrm { R } \left( \begin{array} { l } F
1 \end{array} \right]$
Hence show that $\mathrm { PQ } = { } _ { - }$: and $\mathrm { PR } =$
(U) Show th,i tho $, 0010,11$ is pc,pondio,la, to the pl'ute through $P , Q$ rudd $R$ Hence find the cartesian equation of this plane. A hole is then drilled perpendicular to triangle PQR , as shown in Fig. 82. The hole passes through the triangle at the point T which divides the line PS in the ratio 2 : I , where S is the midpoint of QR .
Write down the coordinates of S , and show that the point T has coordinates $( - 5.16 \mathrm { i } , 25 )$.
Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C .

OCR MEI C4 Q6

6
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
0
2 \end{array} \right)$$ Find the acute angle between the lines. 4 A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill boles by entering the equation of the line of the hole. A $20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}$ cuboid is to be cut and drilled. The cuboid is positioned relative to $x - y ^ { 2 }$ and $z$-axes as shown in Fig. 8.1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-3_414_740_460_302} \captionsetup{labelformat=empty} \caption{Fig. 8.1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-3_449_737_425_1062} \captionsetup{labelformat=empty} \caption{Fig. 8.2}

Write down the coordinates $\boldsymbol { 0 } \mathbf { F } \mathrm { Q }$ and $\mathrm { R } \left( \begin{array} { l } F
1 \end{array} \right]$
Hence show that $\mathrm { PQ } = { } _ { - }$: and $\mathrm { PR } =$
(U) Show th,i tho $, 0010,11$ is pc,pondio,la, to the pl'ute through $P , Q$ rudd $R$ Hence find the cartesian equation of this plane. A hole is then drilled perpendicular to triangle PQR , as shown in Fig. 82. The hole passes through the triangle at the point T which divides the line PS in the ratio 2 : I , where S is the midpoint of QR .
Write down the coordinates of S , and show that the point T has coordinates $( - 5.16 \mathrm { i } , 25 )$.
Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C . 5 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The Oxy plane is horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-4_555_1004_486_565} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
Show that the vector $\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }$ is normal to the plane through $\mathrm { A } , \mathrm { D }$ and E . Hence find the equation of this plane. Given that B lies in this plane, find $a$.
Verify that the equation of the plane BCD is $x + z = 8$. Hence find the acute angle between the planes ABDE and BCD .

OCR MEI C4 Q1

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 $2 x - 3 y + z = 11$.
Find the acute angle between the line and the normal to the plane.

OCR MEI C4 Q2

2 The points $\mathrm { A } , \mathrm { B }$ and C have coordinates $( 1,3 , - 2 ) , ( - 1,2 , - 3 )$ and $( 0 , - 8,1 )$ respectively.

Find the vectors $\overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { AC } }$.
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 .
Write down normal vectors to the planes $2 x - y + z = 2$ and $x - z = 1$. Hence find the acute angle between the planes.
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.
Find the cartesian equation of the plane through the point $( 2 , - 1,4 )$ with normal vector $$\mathbf { n } = \left( \begin{array} { l } 1
1
2 \end{array} \right) .$$
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

OCR MEI C4 Q12

12
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
3
2 \end{array} \right)$$ \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 .

Find the length AE .
Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30 .$$ Write down a vector normal to this plane.
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 .
Find the angle between the planes ABC and ABDE .

OCR MEI C4 Q1

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

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

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 MEI C4 2009 January Q1

1 Express $\frac { 3 x + 2 } { x \left( x ^ { 2 } + 1 \right) }$ in partial fractions.

OCR MEI C4 2009 January Q2

2 Show that $( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots$, and find the next term in the expansion.
State the set of values of $x$ for which the expansion is valid.

OCR MEI C4 2009 January Q3

3 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 2009 January Q4

4 Prove that $\cot \beta - \cot \alpha = \frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta }$.

OCR MEI C4 2009 January Q5

Write down normal vectors to the planes $2 x - y + z = 2$ and $x - z = 1$.
Hence find the acute angle between the planes.
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.

OCR MEI C4 2009 January Q6

Express $\cos \theta + \sqrt { 3 } \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $\alpha$ is acute, expressing $\alpha$ in terms of $\pi$.
Write down the derivative of $\tan \theta$. Hence show that $\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { ( \cos \theta + \sqrt { 3 } \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { \sqrt { 3 } } { 4 }$.

OCR MEI C4 2009 January Q7

7 Scientists can estimate the time elapsed since an animal died by measuring its body temperature.

Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour, estimate how long it will take for the temperature to drop
(A) from $98 ^ { \circ } \mathrm { F }$ to $89 ^ { \circ } \mathrm { F }$,
(B) from $98 ^ { \circ } \mathrm { F }$ to $80 ^ { \circ } \mathrm { F }$. In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton's law of cooling, which states that the temperature $\theta$ in degrees Fahrenheit $t$ hours after death is given by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k \left( \theta - \theta _ { 0 } \right)$$ where $\theta _ { 0 } { } ^ { \circ } \mathrm { F }$ is the air temperature and $k$ is a constant.
Show by integration that the solution of this equation is $\theta = \theta _ { 0 } + A \mathrm { e } ^ { - k t }$, where $A$ is a constant. The value of $\theta _ { 0 }$ is 50 , and the initial value of $\theta$ is 98 . The initial rate of temperature loss is $1.5 ^ { \circ } \mathrm { F }$ per hour.
Find $A$, and show that $k = 0.03125$.
Use this model to calculate how long it will take for the temperature to drop
(A) from $98 ^ { \circ } \mathrm { F }$ to $89 ^ { \circ } \mathrm { F }$,
(B) from $98 ^ { \circ } \mathrm { F }$ to $80 ^ { \circ } \mathrm { F }$.
Comment on the results obtained in parts (i) and (iv).

OCR MEI C4 2009 January Q8

8 Fig. 8 illustrates a hot air balloon on its side. The balloon is modelled by the volume of revolution about the $x$-axis of the curve with parametric equations $$x = 2 + 2 \sin \theta , \quad y = 2 \cos \theta + \sin 2 \theta , \quad ( 0 \leqslant \theta \leqslant 2 \pi ) .$$ The curve crosses the $x$-axis at the point $\mathrm { A } ( 4,0 )$. B and C are maximum and minimum points on the curve. Units on the axes are metres. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f61b7d80-8e21-4720-8e8c-259531c1b305-4_821_809_575_667} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$.
Verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $\theta = \frac { 1 } { 6 } \pi$, and find the exact coordinates of B . Hence find the maximum width BC of the balloon.
(A) Show that $y = x \cos \theta$.
(B) Find $\sin \theta$ in terms of $x$ and show that $\cos ^ { 2 } \theta = x - \frac { 1 } { 4 } x ^ { 2 }$.
(C) Hence show that the cartesian equation of the curve is $y ^ { 2 } = x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 }$.
Find the volume of the balloon.

OCR MEI C4 2010 January Q1

1 Find the first three terms in the binomial expansion of $\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }$ in ascending powers of $x$. State the set of values of $x$ for which the expansion is valid.

OCR MEI C4 2010 January Q2

2 Show that $\cot 2 \theta = \frac { 1 - \tan ^ { 2 } \theta } { 2 \tan \theta }$.
Hence solve the equation $$\cot 2 \theta = 1 + \tan \theta \quad \text { for } 0 ^ { \circ } < \theta < 360 ^ { \circ }$$

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