Analysis and Approaches for IBDP Maths Ebook 2 | Page 110

Your Practice Set – Analysis and Approaches for IBDP Mathematics
3. The Cartesian equations of the planes �
1
and �
2
are 3x � y � 2z
� 12 and
3x � y � 2z
� � 12 respectively.
(a) Show that the vector equation of the line of intersection of the planes �
1
and �
2
is
� 0 � �0�
� � � �
r � �12 � t
0
.
� 0 � �1�
� � � �
(b) The plane �
1
meets the x , y and z axes at A( a , 0, 0) , B(0, b , 0) and C(0, 0, c )
respectively. The plane �
2
meets the x , y and z axes at A � ( �, 0, 0) , B and C
respectively, where A� is the reflection point of A about the y - z plane.
[4]
(i) Write down the values of a , b , c and � .
(ii)
Hence, find the volume of the pyramid A� ABC .
(c) (i) Find the angle between AC and the x -axis.
[7]
(d)
(ii) Hence, find the size of ˆ ACA� .
The line L passing the origin O is perpendicular to the plane ABC . It is given
that the line L meets the plane A� BC at Q . Find the coordinates of Q .
[5]
[4]
4. The plane �
1
meets the x , y and z axes at A , B and C respectively. The plane �
2
meets the x , y and z axes at A , B and C� respectively, where C� is the reflection
point of C about the x - y plane. The Cartesian equation of the plane �
1
is
2x �3y � 4z
� 12 .
(a) Show that the Cartesian equation of the plane �
2
is 2x �3y � 4z
� 12 .
(b) Find the volume of the pyramid ABCC� .
(c) Find the obtuse angle between �
1
and �
2
.
[5]
(d) (i) Write down the coordinates of M , the mid-point of the line segment BC .
[6]
[4]
(ii)
Find the vector equation of the line that passes the point M and is
perpendicular to both �
1
and �
2
, giving the answer in Cartesian form.
[5]
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