page 09 MARKS 16. Planes 1π , 2 π and 3 π have equations: : : : 1 2 3 2 4 3 5 2 1 7 11 11 π x y z π x y z π x y az − + = − − − = − + + = − where a∈\. (a) Use Gaussian elimination to find the value of a such that the intersection of the planes 1π , 2 π and 3 π is a line. (b) Find the equation of the line of intersection of the planes when a takes this value. The plane 4 π has equation x y z − + + = 9 15 6 20. (c) Find the acute angle between 1π and 4 π . (d) Describe the geometrical relationship between 2 π and 4 π . Justify your answer. 17. (a) Given ( ) x f x e = 2 , obtain the Maclaurin expansion for ( ) f x up to, and including, the term in x3. (b) On a suitable domain, let ( ) tan g x x = . (i) Show that the third derivative of ( ) g x is given by ( ) sec tan sec g x x x x = + ′′′ 4 2 2 2 4 . (ii) Hence obtain the Maclaurin expansion for ( ) g x up to and including the term in x3. (c) Hence, or otherwise, obtain the Maclaurin expansion for tan x e x 2 up to, and including, the term in x3. (d) Write down the first three non-zero terms in the Maclaurin expansion for tan sec x x e x e x + 2 2 2 2 . [END OF QUESTION PAPER] 4 2 3 1 2 3 2 2 1 page 10 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 11 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 12 [BLANK PAGE] DO NOT WRITE ON THIS PAGE