Analysis and Approaches for IBDP Maths Ebook 2 | Page 104

Your Practice Set – Analysis and Approaches for IBDP Mathematics
D , E and F are the points such that ABCFED is a prism, where the triangle ABC is
congruent to the triangle DEF. It is given that the total surface area of the prism
ABCFED is 2(30 � 3) .
(e) Find the exact volume of the prism ABCFED .
[4]
3. The vector equations of the lines L
1
and L
2
are
y �14
x� 6� � z� 6 respectively.
2
�14� � �5�
� � � �
r � 18 � t
�6
and
� 8 � � �2�
� � � �
(a) Find the coordinates of P , the point of intersection of L
1
and L
2
.
(b) R( a , 4, 3) is a point lying on L
2
. Find the value of a .
[5]
[2]
�
(c) Q is a point on L
1
such that RQ � � 4 i � 4 j�k
. Find the coordinates of Q .
[4]
Let T and S be the points on L
1
and L
2
with parameters t � 8 and s �� 4 respectively,
where s is the parameter of the vector equation of L
2
.
(d) Find the exact area of the quadrilateral QRST .
(e)
U is a point such that UQ is perpendicular to QRST , and the volume of the
pyramid QRSTU is 162 29 . Find the shortest distance between U and QRST .
4. The coordinates of the points O , A , B , C and D are (0, 0, 0) , (3, 0, 0) , (3, 0, 3) ,
(0, 0, 3) and (0, � 3, 0) respectively. E is a point on the line segment BD such that
AE
� BD .
(a) Show that the coordinates of E are (2, � 1, 2) .
(b)
Find the angle between the plane ABD and the plane BCD.
(c) F is a point on the y -axis such that the volume of the pyramid OABCF is 15.
Find the possible values of DF .
[8]
[2]
[6]
[9]
[6]
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