(Proof by cases: (p₁ V P2 V (Pn →q)) Prove or disprove. V Pn) → q = (P₁ → q)^(p2 →q)^^...^ "The 4th power of any integer never has remainder 2 when divided by 3."

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

A:

Q: b) 4(x – 4)(x + 2)² (2x – 1)³ < 0 - -

A:

Q: x - x3 + x² – x + 1 is reducible over Q True False

A:

Q: In(x²) = 2 ln(x) for all x True O False

A: ln(x^2) =2ln(x) for all x

Q: Show that P(n, 1) + P(n, 2) = (22) – 2 (2), for integers n ≥ 2.

A:

Q: Show that 1+2+3+...+n= ("""). (";"').

A:

Q: (1+ 2x + 4x + 8x° + · · · )(3x + 3x* + 3x° + · · · )(x° + 2x° + 3x° + •..)

A:

Q: Evaluate: L(e* -3t * sinh 7t).

A: Given: Le-3tsinh7t

Q: (a) (b) Use the quotient-remainder theorem to find the gef{77, 42}. Find ALL integer solutions to…

A: Using quotient remainder theorem we can find the gcf is 7 and all integer solution are q=6n+5 and…

Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

A:

Q: 5. (20) Prove by induction that for positive integers n, 4|(5"+1 + 32n-1).

A:

Q: If C₁ * x² = c(1) ²¹ + a₂ (1) ²³ t² is the complementary solution of the nonhomogeneous system 3…

A: If the nonhomogeneous system is x'=3-22-2x+-2tt4-1 If the complimentary solution is…

Q: Find 1 (9) Lp(p – 1). p(p²-1)3 (10) L-1 (p2+1)2,

A:

Q: Evaluate In x x (4 - In² x) - In 14- In² x | +C - 2 In 14-In² x | +C 12 -dx In | x (4- In²x) | +C In…

A:

Q: If P(A) = P(B) = P(AB), show that P(AB + AB) = 0 [AB = An B]. %3D

A:

Q: Prove that product notation d/n = nτ(n)/2.

A:

Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: Prove that n E (-1)*s = 1 n1, N2, N3 ni+n2+n3=n (Hint: think about the trinomial theorem.)

A:

Q: Find L[cosh 7t] s2-49 7 s2+49 s2+49 7 2 O O O C

A:

Q: 1- x**! 1- x n+1 6 (x # 1) r=0

A:

Q: Prove that 1¹x 2² x 3³ x...xn" < n(n+1) 2 (2n+1) 3

A:

Q: (a) Prove that 1^2+2^2+3^2+………..+n2= n(n+1) (2n+1) divided by 6 For all positive integers n.  (b)…

A:

Q: if f(x) = 2+2x - 4x sqaured find f prime of 5

A: The objective of this question is to find the derivative of the function f(x) = 2 + 2x - 4x^2 at the…

Q: If a, b, c are non-zero real numbers, then bc ca ab ca ab bc vanishes, when ab bc ca ca 1.1 -+- 0…

A:

Q: While proving (a*2(2a+1 is an integer for the case a = 6q +1 which of the following will be obtained…

A: We are given with a(a+1)(2a+1)6 We need to substitute a=6q+1

Q: n Prove that & (31²-2₁) = 2n³ tn ²-n 2=1 2 Goal: 2 (k+1) ³) + (1+₁)² = (K+1) 2

A:

Q: Suppose X₁ ~ N(6, 1) and X₂ ~ N(7,2), and X₁ is independent of X₂. Find P(X₁ < X₂) and P(X₁ < X₂ +…

A: We know linear combination of normal distribution is also a normal distribution. Therefore any…

Q: beta

A: According to the question,        First,  define some of the terms for…

Q: For any positive integers m, n (

A:

Q: There exists two odd integers whose average is odd. 1 1 1 There exists distinct positive integers a,…

A: (a) Consider two odd integers 2m+1, 2n+1 such that m+n is even. Now, Consider the average of the…

Q: (b) Prove that 32n²+1, for n ≥ 1.

A: The given inequality is 3n≥2n2+1, for n≥1. We have to prove this inequality.

Q: Q2/ prove [A] [A] = [I] if [2 4 -5 [A]= 2 7 1 3 4

A:

Q: 1 Use convolution theorem to evaluate L-1 (s²+9)²)

A:

Q: 4[(b+f)-(c+r)][a (d+g)-(e+s)][(c+r)(d+g)+a(e+s)(b+f)] [((e+s) – (d+g))(a+1)] K ae ([(b+ f) – c] +…

A: Given, x1-Q=a(d+g)-(e+s)b+f-c+rc+rd+g+ae+sb+fa(e+s)…

Q: Use substitution method to prove step-by-step that (P^Q)^(P = ¬(Q^R))^(S = R) = –¬S.

A: We are asked to prove the following using the substitution method, P∧Q∧P⇒~Q∧R∧S⇒R⇒~S We will show…

Q: 2. (a) Show that (z − 1)(26 + 25 + 24 + z³ + z²+z+1) = z7 - 1. (b) Show that if ¢ is a 7th root if…

A: