Name: 1. Show the following propositions using a direct proof. (a) Proposition. If a is an odd integer, then a² + 3a + 5 is odd. (b) Proposition. Suppose a, b E Z. If alb then a²|b². (c) Proposition. f(x) = X x² Section: 1 I work is an odd function. (d) Use the identity sin² (x) + cos² (x) = 1, to show the identity tan² (x) + 1 = sec² (x).
Name: 1. Show the following propositions using a direct proof. (a) Proposition. If a is an odd integer, then a² + 3a + 5 is odd. (b) Proposition. Suppose a, b E Z. If alb then a²|b². (c) Proposition. f(x) = X x² Section: 1 I work is an odd function. (d) Use the identity sin² (x) + cos² (x) = 1, to show the identity tan² (x) + 1 = sec² (x).
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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How can we write the following as a direct proof?
![# MATH 140 - Lecture 9 Homework
**Name:** [______________]
**Section:** [______________]
### 1. Show the following propositions using a direct proof.
**(a) Proposition.** If \( a \) is an odd integer, then \( a^2 + 3a + 5 \) is odd.
**(b) Proposition.** Suppose \( a, b \in \mathbb{Z} \). If \( a|b \) then \( a^2|b^2 \).
**(c) Proposition.** \( f(x) = \frac{x^2}{x - \frac{1}{x}} \) is an odd function.
**(d)** Use the identity \( \sin^2(x) + \cos^2(x) = 1 \), to show the identity \( \tan^2(x) + 1 = \sec^2(x) \).
*Note: Some exercises are from Richard Hammock's "Book of Proof".*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38e21668-df36-4535-9b97-aa4a56ba16f0%2Fd18883e1-07a0-4b49-9d02-f568afed0295%2Fqhvh349_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# MATH 140 - Lecture 9 Homework
**Name:** [______________]
**Section:** [______________]
### 1. Show the following propositions using a direct proof.
**(a) Proposition.** If \( a \) is an odd integer, then \( a^2 + 3a + 5 \) is odd.
**(b) Proposition.** Suppose \( a, b \in \mathbb{Z} \). If \( a|b \) then \( a^2|b^2 \).
**(c) Proposition.** \( f(x) = \frac{x^2}{x - \frac{1}{x}} \) is an odd function.
**(d)** Use the identity \( \sin^2(x) + \cos^2(x) = 1 \), to show the identity \( \tan^2(x) + 1 = \sec^2(x) \).
*Note: Some exercises are from Richard Hammock's "Book of Proof".*
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