OCR MEI C4 (Core Mathematics 4)

Question 1
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1 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes \(\mathrm { O } x y z\), the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{27c27c79-9aea-45a4-a000-41aac70ff866-1_798_1296_354_418} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
  2. (A) Show that the vector \(\left( \begin{array} { r } 1
    - 6
    0 \end{array} \right)\) is normal to the plane DHC.
    (B) Hence find the cartesian equation of this plane.
    (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
  3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and \(\mathrm { F } . \mathrm { Q }\) is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
  4. Find the height of Q above the ground.
Question 2
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2 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes Oxyz , are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{27c27c79-9aea-45a4-a000-41aac70ff866-2_805_854_385_615} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the length of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
  2. (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.
  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 . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
  4. Find the volume of the tetrahedron ABCD .
  5. Find a vector equation of the line \(l\) joining the points \(( 0,1,3 )\) and \(( - 2,2,5 )\).
  6. Find the point of intersection of the line \(l\) with the plane \(x + 3 y + 2 z = 4\).
  7. Find the acute angle between the line \(l\) and the normal to the plane.