Prove in an integral domain, a prime element is always irreducible.

Q: 1. f/² sin () cos (+) sin (²) d dx (Consider initially the first 2 terms)

A: We have to use the formulas for 2sin(A)cos(B)-sin(A+B)+sin(A-B) and 2sin(A)sin(B)=cos(A-B)-cos(A+B).…

Q: 23 Find a 4 by 4 permutation matrix (call it A) that needs 3 row exchanges to reach the end of…

A: Consider the 4×4 matrix given below, A=0001100001000010 Here we need to apply 3 row exchanges to…

Q: Find the laplace transform of cos at using the formula £{f(t)} = √e* f (t) dElencefind the laplace…

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Q: dice In the experiment of throwing two fair let (A) be the event that the is odd, (B) Be the event…

A: Two fair dice are thrown. Let A be the event that the first die is odd and B be the event that the…

Q: D²-2D+1)y = e^x logx

A: Given differential equation is D2−2D+1y=exlogx. We have to find the general solution of the given…

Q: ² - x ²² = y² dz ४x d²z Find the solution of xy' буду

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Q: a. (y + 2)²0-2x log y a²u Əy² b. uxx + xy Uxy + y²u² = 0 c.t²u=√√2+x²u əx² at² + koy - Inu = 2e-x, k…

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Q: What is the Cartesian product of A = {1, 2} and B = {a,b,c}?

A: Given A = 1, 2   and   B = a, b, c

Q: Let K be a Galois extension of F of degree pn where p is a prime. Show that there exist Galois…

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Q: Q2): A polluted lake has an initial concentration of a bacteria of (1*10' part/m³), while the…

A: Given that,The differential equation that governs the concentration (C) of the pollutantas a…

Q: Determine the eigenvalues and eigenvectors of the following matrix: 6 [39]

A: Use the given matrix.

Q: (i) Draw the graph whose adjacency matrix is given in Fig. 2.25. (ii) Draw the graph whose incidence…

A: We have to Draw the graph whose adjacency matrix is given in Fig. 2.25. and Draw the graph whose…

Q: Write the following English sentences in symbolic form-  I will dance only if you sing. Neither…

A: We can divide the statement into two premises. Let us denote P: I will dance and Q: you sing.

Q: Define an operator T: fºº → f° by T(5₁) = (¹) # and bounded. for each (₁) E. Show that T is linear…

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Q: (1 + inxy) dx + (1 +5) dy = 0

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Q: Find the tangent line approximation 7 to the graph of fat the given point. Then complete the table.…

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Q: Find two unit vectors orthogonal to a = (-2,-1, 3) and b = (5,-2, 1) Enter your answer so that the…

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Q: (sinxy +xy cos xy) dx + x² cosx y d y = о

A: The solution is given as

Q: Example 10.12. Solve the following by Euler's modified method dy = log (x + y), y (0) = 2 dx at x =…

A: Here, we need to solve this differential equation using modified Euler's method.

Q: 1 Р ven as If th ● q 2 x are real and positiv llowing relations shc

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Q: Solve the following PDE:

A: (a). Given,                 yuxy+u=0                                       ...(1)

Q: Example 9.15. Solve the simultaneous difference equations 3ux, 3u + v Ux+1-50 = subject to the…

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Q: VI. Write TRUE if the statement is correct, otherwise write FALSE. There is no need to justify your…

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Q: eeded to be solve correctly in

A: Given, the helix                r(t)=cosπt2, sinπt2, t intersects the sphere x2+y2+z2=2 in two…

Q: Find the number of all ordered 4-tuples of integers that are greater or equal to -3 whose sum is 1.…

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Q: Find the Laplace Transformation of -3x y" +3y + 2y 4e when y (0) = 2, y' (0) = 4 28² +16s+30 OY (s)…

A: Given differential equation: y''+3y'+2y=4e-3x, y0=2 and y'0=4To find: Laplace transform of the…

Q: r(t) = (7t cos t-7 sin t)j + (7t sin t + 7 cos t) k OT= (-7 sin t)j + (7 cos t)k OT= (-sin t)j +…

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Q: A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes…

A: Here , particular brand of shirt comes in 12 colors. It has female version and male version. And it…

Q: (2xte)dxt xe dyzo

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Q: Find the unit tangent vector of the given curve. r(t) = (7t cos t - 7 sin t)j + (7t sin t + 7 cos t)…

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Q: Use the second Taylor polynomial P₂(x) for f(x) = x² + ln(1 + x) about æ = 0 and evaluate ƒ ƒ(x)dx.…

A: Given function is fx=x2+ln1+x. We have to evaluate  ∫01fxdx by using the second Taylor polynomial…

Q: 13. For the following digraphs, determine ISI, the cardinality of S if: a) S is the transitive…

A: A={a,b,c,d,e} The given relation as R={(b,c), (b,e), (c,e), (e,b), (e,c), (d,a)} For transitive…

Q: Find an equation of the surface of revolution by revolving the plane curve 3y = √(9- [(x-1)] ^2)…

A: Given:3y = (9-(x-1)2)

Q: Evaluate L-1{- [6] S -} + 169) (s - 9) (5².

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Q: For the equation, 9x² +8/3xy + y²-8=0, complete the following questions. a. Rewrite the equation in…

A: The general form of a conic equation is Ax2+Bxy+Cy2+Dx+Ey+F=0.  If B2-4AC>0, then the conic is a…

Q: Find the solution of xy 흐르-x=y dydx

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Q: Determine the eigenvalue X, eigenvector v, and generalized eigenvector w of the following matrix.…

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Q: Integrate the following functions Ver R: f(x,y) = 3x-2y. the given regi the circle x² + y² registre

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Q: Recall that the differential equation for the instantaneous charge q(t) on the capacitor in an…

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Q: 6. i,How many permutations are required to be sure of predicting the outcome of tossing a six sided…

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Q: Write the following English sentences in symbolic form- 1. If I will go to Australia, then I will…

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Q: 3 42 2 -1 3 Let B = Compute det(B) by hand by reducing B to upper -1 31 triangular form using row…

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Q: How. Find the volumes for the following solids whose heights are defined by ECx,y) and bases are the…

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Q: • Evaluat the integral:- 푼 S S. seč (cosx) dx dy Sin y

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Q: 3. fe2x+Inx² sin x² dx = = 2x -+√e²x cos x² dx

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Q: Q.3: Solve the following set of four linear equations using Gauss-Seidel iteration method; Continue…

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Q: 2. Let T: R³ R³ be the linear transformation given by x1 + x2 + x3 +((:))-[ T X2 2x1 x2 + 7x3 -4x₁ +…

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Q: Give at least three (3) examples, with complete and detailed solution, onsolving Differential…

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Q: Find d(u, v), if u = (2, -4,2,-1,-6) and = (4, -6, 2,-1,-6). OA√93-√61 O B. √300 OC √73 OD V8 4 O E…

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Q: Sketch the graph of f with emphasis on concavity. Label asymptotes with their equations and…

A: To sketch the graph of the function with the given detail.   −∞,−1 -1 −1,1 1 1,2 2 2,3 3 3,∞ f'…