Theorem 4.10. Let p be an odd prime. Then among 1, 2, ,p-1, exactly (p-1)/2 many are quadratic residues mod p and (p-1)/2 many are non-residues. The quadratic residues are 1², 2², ... (₁=¹) ² 2 mod p.

Q: space. connected subsets of X such that C; is non-empty, then C = C; is also

A:

Q: A study showed the following migration matrix, which describes the movement of the population of the…

A:

Q: define A(x) by A(x) = Σ a(n), n≤x a: Let a(n) be an arithmetical function. We y<n<x where A(x) = 0…

A: As per the question we have to prove the given Abel's Summation Theorem, which is an applied form of…

Q: If fel²() and gelices. prove that fgeles.

A:

Q: A scale of 1 in 10,000 is adopted for a photogrammetric survey for the region of 200 sq. km.…

A:

Q: a.V odv

A:

Q: 1. Determine the convergence of the following sequences. C. 3n {(²1+5) ³7}

A:

Q: Let f(x) = sin(sin(sin.x)) for for all x in R (f(x) x-0 g (x), and g(x) = (sinx), VxER, then lim is…

A:

Q: If y, and y₂ are linearly independent solutions of ty" + 4y + tey = 0 and if W(y₁.₂)(1) = 4, find…

A:

Q: Theorem Let f: G H be a group homomorphism. Then, Im f≤ H.

A: Let us consider the mapping f:G→H . Then f is group homomorphism if  f(x·y)=f(x)·f(y) where, x,y∈G…

Q: prove that 12 div A; = √√√x² {√8 8¹k Ax} = dxr div A¹, where A; and Ai are the covariant and…

A:

Q: 3. A (-3, 2,-5) and B (6, 7, 2) are two points in R³. Determine: [3K, 2A] a) ABI b) A unit vector in…

A:

Q: Use the method of variation of parameters to solve the initial value problem x' = Ax + f(t), x(a) =…

A:

Q: Solve y'-y cosx = cos x

A: We now solve the differential equation, y' -ycosx=cosx . This is a first order homogeneous equation.

Q: 7 3х +4 3. их н dx X

A:

Q: г 3х +4 *3их н dx X

A: The given problem is to evaluate the given indefinite integral of given rational function.

Q: SHOW that the continuous image Path - connected space is path Connected.

A:

Q: 5. Solve the equation y-y'y"=1, given that y=√2 and y'- and y'= when r=0.

A:

Q: Find a least squares solution of Ax=b by (a) constructing the normal equations for and (b) solving…

A:

Q: A tank has pure water flowing into it at 10 liters/min. The contents of the tank are kept thoroughly…

A:

Q: What will the flow rate be in milliliters per hour for vancomycin 1g/500 mL IV, if it is to be…

A: Solution: Given,           Total volume of vancomycin = 500 ml          Total time = 90 minutes =…

Q: Find the Wronskian of two solutions of the differential equation t²y" — t(t — 5)y' + (t− 6)y = 0…

A:

Q: Let f: R" x R → R define as follows f(x,y) = then f is i=1

A:

Q: Determine the longest interval in which the initial value problem is certain to have a unique twice…

A:

Q: Find the value of lim ((n + 1) cos (n + 1) cos¯¹ (1) − n) n

A: Given: limn→∞2πn+1cos-11n-n To find: The value of given limit.

Q: Find & simplify (a) equation of the tangent plane to a surface 2 = = f(x, y) = = ln (√√9x² + 4y² –…

A: The given problem is to find the equation of tangent plane to the given surface at given point…

Q: noon, a large pool contains only 26 gallons of water. A hose is set up so that the pool is filled at…

A: Introduction: We use the term "Rate" to describe the comparison and ratio of two separate…

Q: A) Compute F.dà using the Divergence Theorem. SHOW WORK. S B) Compute the flux of F out S, which is…

A: Given that S is the outward oriented closed cylinder  x2+y2=9, 0≤z≤2 and  F→=xi^+yj^+2zk^ Solution…

Q: 2. The differential equation (x² +1)y"+ay": = 0 and the following data are assumed; X y 0 0…

A:

Q: Apply the method of undetermined coefficients to find a particular solution to the following system.…

A: Given:x'=5x-7y+10, y'=x-3y-2e-2tApply the method of undetermined coefficients to find a particular…

Q: Siven: dy + хо з aht hxE X2 +xy 12 = 0

A: To solve the differential equation of the form dydx=f(x,y) where f(x,y) is a homogenous function…

Q: The current I(t) in an LC series circuit is governed by the initial value problem below. Determine…

A:

Q: Let L₁ be the line passing through the point P₁=(5, 2, 3) with direction vector d=[1, 3, 1]', and…

A:

Q: Given the following initial value problem. dy/dt yt²-1.1y where y(0) = 1 The solution at t 0.75; Le…

A: Given dydt=yt2-1.1y where y(0)=1, h=0.25

Q: (a) Find the general solution of the given system of equations. x = ( 8³ )x x' 3 -4 5 -5

A: Please find the attached solution.

Q: Twelve teachers are to be assigned to three schools. In how many ways can this be done if: i. a…

A:

Q: The formula A(t) = 19.6e0.0138t (4) models the population, A(t), of a US state in millions, t years…

A:

Q: 3. Express the negation of these propositions using quantifiers, and then express the negation in…

A: Negation of the quantifies  Quantifiers  Negation of quantifiers ∀ ∃ ∃ ∀ Some  All

Q: Consider a system of two toy railway cars (i.e., frictionless masses) connected to each other by two…

A:

Q: Use the Maclaurin series expansion for the sin x to estimate sin(pi/3) x5 x3 sinx=x−= + 3! 5! 7!…

A:

Q: Use the table to the right to answer the following question. How many kilograms of isotope would be…

A:

Q: 14. Solve the differential equation dy t dt y y(0)=1 by the Euler method with h=0.1 to get y(0.2).…

A:

Q: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

Q: √√√x√1-x-¹ dx

A:

Q: Use the method of variation of parameters to solve the initial value problem x' = Ax + f(t), x(a) =…

A:

Q: Determine the values of a, if any, for which all solutions of the differential equation y" (2a −…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 3. PDE r + a²t = 0, has the solution :

A:

Q: Theorem Let f: G H be a group homomorphism. Then, Im fs H.

A: Let us consider the mapping f:G→H. Then f is group homomorphism if  f(x·y)=f(x)·f(y) , x,y∈G Image…

Q: 3. Given these data 1 2 3 5 7 8 3 6 1999 289 442 f(x) (b) Use Lagrange polynomials of order 1…

A:

Q: During the Christmas holidays, a group of friends went out for dinner at a restaurant in Montego…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…