Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  OctNov  (P19709/12)  Q#7
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Question
The diagram shows a solid cylinder standing on a horizontal circular base with centre O and radius 4 units. Points A, B and C lie on the circumference of the base such that AB is a diameter and angle BOC = 90^{o}. Points P, Q and R lie on the upper surface of the cylinder vertically above A, B and C respectively. The height of the cylinder is 12 units. The midpoint of CR is M and N lies on BQ with BN = 4 units.
Unit vectors and are parallel to and and unit vector is vertically upwards.
Evaluate and hence find angle MPN.
Solution
We are required to evaluate the dot product first.
The scalar or dot product of two vectors


Since
Therefore, we need to find
Let us first find
A vector in the direction of
For the given case;
Therefore, we need the position vectors
First, we find the position vector
Let us find the vector
Since this is the position vector of point
· It is given that
· It is given that
· It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
Then the position vector of
Next, we find the position vector
Let us find the vector
· It is given that
· It is given that
· It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
Now we can find
Let us first find
A vector in the direction of
For the given case;
Therefore, we need the position vectors
First, we find the position vector
Let us find the vector
Since this is the position vector of point
· It is given that
· It is given that
· It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
We have already found position vector
Now we can find
Finally, we are able to find dot product
The scalar or dot product of two vectors


Since
Now we need to find the angle MPN.
We recognize that
Hence, we use scalar/dot product of
The scalar or dot product of two vectors


Since
We have from (i) that;
The scalar or dot product of two vectors
Where





For the given case;
Therefore;
Equating both scalar/dot products we get;
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