Test 6

Pure Mathematics – Inequalities

p.1

King’s College 2007 – 2008 F.6 Pure Mathematics Revision Test 6 Time Allowed: 50 minutes Total Mark: 30 1.

(1991) (a) Let ak, bk (k = 1, 2, …, n) be non-zero numbers. (i) Prove the Schwarz’s inequality 2

⎧ n 2 ⎫⎧ n 2 ⎫ ⎧ n ⎫ ⎨∑ a k ⎬⎨∑ bk ⎬ ≥ ⎨∑ a k bk ⎬ . ⎩ k =1 ⎭⎩ k =1 ⎭ ⎩ k =1 ⎭

(ii) If p ≤

bk ≤ q for k = 1, 2, …, n, prove that ak

pqa k2 − ( p + q )a k bk + bk2 ≤ 0 for k = 1, 2, …, n. Deduce that

n

n

n

k =1

k =1

k =1

( p + q )∑ ak bk ≥ ∑ bk2 + pq∑ a k2 .

(iii) If 0 < m ≤ a k ≤ M and 0 < m ≤ bk ≤ M for k = 1, 2, …, n, prove by using (ii) or otherwise, that 2

2

⎫ ⎧ n 2 ⎫⎧ n 2 ⎫ 1 ⎛ M m ⎞ ⎧ n ⎨∑ a k ⎬⎨∑ bk ⎬ ≤ ⎜ + ⎟ ⎨∑ a k bk ⎬ . ⎭ ⎩ k =1 ⎭⎩ k =1 ⎭ 4 ⎝ m M ⎠ ⎩ k =1

(10 marks) (b) Using (a) or otherwise, show that 2 2 2 2 ⎧⎪ n ⎛ 1 ⎞ ⎫⎪ 169 ⎛ 1⎞ 1⎞ 1 ⎞ ⎫⎪⎧⎪ n ⎛ ⎛ ⎜ n + ⎟ < ⎨∑ ⎜1 + k ⎟ ⎬⎨∑ ⎜1 − k +1 ⎟ ⎬ < ⎜n + ⎟ . 3⎠ 9⎠ ⎪⎩ k =1 ⎝ 3 ⎠ ⎪⎭⎪⎩ k =1 ⎝ 3 ⎠ ⎪⎭ 144 ⎝ ⎝

(5 marks)

Pure Mathematics – Inequalities

2.

p.2

(1997) Let x1, x2, y1, y2, z1 and z2 be positive numbers such that x1y1 – z12 > 0 and x2y2 – z22 > 0. (a) Let D1 = x1y1 – z12 and D2 = x2y2 – z22. Using A.M. ≥ G.M. , show that y2 y D1 + 1 D2 ≥ 2 D1 D2 , (i) y1 y2 (ii)

y2 y D1 + 1 D2 ≤ x1 y 2 + x 2 y1 − 2 z1 z 2 , y1 y2

(iii) (x1 + x 2 )( y1 + y 2 ) − ( z1 + z 2 ) ≥ 4 D1 D2 . 2

(9 marks) (b) Show that

8

(x1 + x2 )( y1 + y 2 ) − (z1 + z 2 )

2

≤

1 x1 y1 − z1

2

+

1 x2 y2 − z 2

2

,

and if the equality holds, then x1 = x2, y1 = y2 and z1 = z2. (6 marks)

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