1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a
1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![1. (10) An m x n rectangular array of distinct real numbers has the
property that the numbers in each row increase from left to right.
The entries in each column, individually, are rearranged so that the
numbers in each column increase from top to bottom. Prove that
in the final array, the numbers in each row will increase from left to
right.
2. (11) Determine distinct positive integers a, b, c, d, e such that the
five numbers a, b², c³, d4, e5 constitute an arithmetic progression.
(The difference between adjacent pairs is the same.)
3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub-
sets with the same sum if and only if k divides 1½n(n + 1) and
n≥ 2k - 1.
4. (6) Suppose that f(x) is a continuous real-valued function defined
on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies
(i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1.
(a) Prove that there exists a number a for which 0 < a < 1 and
f'(a) < f(1);
(b) Prove that there exists a unique number b for which a <b<1
and f'(a) = f(b)/b.
=
5. (6) For x 1 and x 0, let
-
f(x)=
−8[1 − (1 − x) 1/2]3
-
=
x2
(a) Prove that limo f(x) exists. Take this as the value of f(0).
(b) Determine the smallest closed interval that contains all values
assumed by f(x) on its domain.
(c) Prove that f(f(f(x))) = f(x) for all x ≤ 1.
6. (4) Let h(n) denote the number of finite sequences {a1, a2,.
of positive integers exceeding 1 for which k ≥ 1, a₁ ≥ a2 ≥ … … · ≥ ak
ak}
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Transcribed Image Text:1. (10) An m x n rectangular array of distinct real numbers has the
property that the numbers in each row increase from left to right.
The entries in each column, individually, are rearranged so that the
numbers in each column increase from top to bottom. Prove that
in the final array, the numbers in each row will increase from left to
right.
2. (11) Determine distinct positive integers a, b, c, d, e such that the
five numbers a, b², c³, d4, e5 constitute an arithmetic progression.
(The difference between adjacent pairs is the same.)
3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub-
sets with the same sum if and only if k divides 1½n(n + 1) and
n≥ 2k - 1.
4. (6) Suppose that f(x) is a continuous real-valued function defined
on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies
(i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1.
(a) Prove that there exists a number a for which 0 < a < 1 and
f'(a) < f(1);
(b) Prove that there exists a unique number b for which a <b<1
and f'(a) = f(b)/b.
=
5. (6) For x 1 and x 0, let
-
f(x)=
−8[1 − (1 − x) 1/2]3
-
=
x2
(a) Prove that limo f(x) exists. Take this as the value of f(0).
(b) Determine the smallest closed interval that contains all values
assumed by f(x) on its domain.
(c) Prove that f(f(f(x))) = f(x) for all x ≤ 1.
6. (4) Let h(n) denote the number of finite sequences {a1, a2,.
of positive integers exceeding 1 for which k ≥ 1, a₁ ≥ a2 ≥ … … · ≥ ak
ak}
>
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