Complete the missing components of the know-show table to prove the statement be- low. Alternatively, you may construct your own table to prove the statement using the strategy that comes to your mind. Statement: For all integers n, if n is odd, then n³ + 4n+5 is even. Step Know P P1 n³ is odd P2 P3 5 is odd 0 Step Reason Hypothesis Product of even and odd is even 5 = 2(2)+1 Show Reason

Consider the following false statement: For all integers a and b, if ab = 1 (mod 8), then a = 1 (mod 8) or b = 1 (mod 8). (a) Which of the following could be used as a counterexample. Select all that apply. a = -7 and b = −7 a = 1 and b = 23 ☐ a = 3 and b: = −5 ☐ a = 4 and b = 6 □ a = −1 and b = −9

1. Given X' = X 3 e2t (a) Verify that X₁(t) = (e) and X2(t) = (et) - are solutions to the given system. (b) Verify that X₁(t) and X2(t) form a fundamental set on the interval (-∞, ∞). (c) Write the general solution to the given system. (d) Find the solution that satisfies the initial condition X(0) = ( 2 ).

Prove that a relation X defined on a set A that is reflexive, symmetric and antisymmetric is an equivalence relation and determine the equivalence classes.

Let X be the relation defined on the power set of the set integers P(Z) by AXB whenever A U B is a finite set of integers. Prove whether or not X is reflexive, symmetric, antisymmetirc or transitive

Page < 1 of 2 - ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). −1 0 01 i) If A = 0 0 2 0, then its eigenvalues are ₁ = 1,λ₂ = 2, and 13 0 0 = : 0. (TRUE FALSE) ii) A linear transformation is operation preserving because the same result occurs whether you perform the operations of addition and scalar multiplication before or after applying the linear transformation. ( TRUE FALSE) iii) A linear transformation that is one-to-one and onto is called an isomorphism. (TRUE FALSE) iv) If the standard matrix A for the linear transformation T: R³ → R³ is -1 0 01 A = 2 00, then T is invertible. (TRUE FALSE) 0 1 1. v) Let A, B, and C be square matrices of order n. If A is similar to B and B is similar to C, then A is similar to C. ( TRUE FALSE) 2) a) i) Find the matrix that produces the counterclockwise rotation of 30° about the z-axis. ii) Find the image of the vector (1,1,1) for the rotation described in i). b) Give a geometric description…

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1. Except for the door and floor, a shed is built entirely out of plywood. How many square meters of plywood are needed to build the shed? (1 foot 0.3048 m) 10 ft. 7 ft. 3 ft. 18 ft. 17 ft. 15 ft.

1. Leigh plans to invest $2500 for 4 years. She is considering two options: • A compound interest Guaranteed Investment Certificate (GIC) earning 5.6% per year for 4 years • A simple interest government bond earning 6.1% per year for 4 years a) Determine the total interest earned for each option at the end of each year. GIC Year Principal Interest earned (5.6%) Total interest 1 2 3 4 Government Bond [2 Marks] Interest earned (6.1%) Year Principal 1 2 3 4 b) Which is the better option? Explain. Total interest c) If Leigh can only invest for 3 years, which is the better option? Explain.