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If some solution (x, y, z) of 1
3 x +
y + 2z = −1 satisfies ( x − p ) 2 + y 2 + z 2 = 1 , find the range of values of p .
(5 marks) 2001ALP MATH 1−6 −5− 10. a n = a n −1 + 2bn −1
a1 = 1
and , n = 2, 3, 4, … .
Let b1 = 1 bn = a n −1 + bn−1
(a) Show that for any positive integer n ,
(i) a n , b n > 0 and a n 2 − 2bn 2 = (−1) n ; (ii) (1 + 2 ) n = a n + bn 2 .
(4 marks) (b) For n = 1, 2, 3, … , define u n = an
.
bn un + 2
.
un +1 (i) Show that u n +1 = (ii) Show that u 2 n −1 < 2 and u 2 n > 2 . (iii) Show that u n + 2 = 3u n + 4
.
2u n + 3 { u1 , u 3 , u 5 , " } is strictly
{ u 2 , u 4 , u 6 , " } is strictly Hence show that the sequence
increasing and the sequence
decreasing.
(iv) Show that the sequences { u2 , u4 , u6 , " } { u1 , u 3 , u 5 , " } and converge to the same limit. Find this limit.
(11 marks) 2001ALP MATH 1−7 −6− Go on to the next page 11. (a) Show that θ
1 + cosθ + i sin θ
= i cot .
1 − cosθ − i sin θ
2
(3 marks) (b) Let n be a positive integer. Show that all the roots of the equation
( z − 1) n + ( z + 1) n = 0
..........(*)
can be written as iα k , where α k ∈ R , k = 0, 1, ! , n − 1 .
(4 marks) (c) If iα k ( k = 0, 1, ! , n − 1 ) are the roots of (*) in (b), using the
relations between the roots and coefficients, show that ∑ n −1
k =0 α k 2 = n(n − 1) .
(5 marks) (d) Let P0 , P1 , ! , Pn −1 be the n points in an Argand plane representing
the roots of (*) in (b), and O be the origin. Q is the point representing
r (cos β + i sin β ) where r ≥ 0 and β ∈ R . If d k is the distance
between Pk and Q , show that ∑ n −1
k =0 dk 2 is independent of β .
(3 marks) 2001ALP MATH 1−8 −7− 12. The position vectors of four points A , B , C and D are a = 7i + 8 j + 3k ,
b = 2i − 7 j + 13k , c = 17i − 6 j + 3k and d = r (2i − 4 j − 5k ) respectively,
where r is a nonzero real number.
(a) Show that a , b and c are linearly independent.
(3 mark...
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This note was uploaded on 06/07/2012 for the course MATHS 2002 taught by Professor Lo during the Fall '02 term at Wisc Oshkosh.
 Fall '02
 lo
 Math

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