1. Let p be a prime and let g be a primitive root modulo p. Let a be an integer which is relatively prime to p. Prove that a has a square root modulo p (i.e., there exists an integer b such that b² = a (mod p)) if and only if the discrete logarithm log, (a) is even.

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Q: 2. If H and K are normal subgroups of a group G and H^K = {e}, prove that G is isomorphic to a…

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Q: (a) State True or False and proved justification for: If S is an ellipsoid and F is a constant…

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Q: 10. Laplace transform and inverse transform a) Take the Laplace transform of the following ODE and…

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Q: Evaluate If yds where S is given by z = x + y², 0≤x≤ 1,0 ≤ y ≤ 2.

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Q: What do the vertical line test and horizontal line test tell us about a graph?

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Q: 9. Second order ODE a) Consider the following nonhomogeneous ODEs, find their homogeneous solution,…

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Q: Let y(t): = 4 cost - 7(t-4)³ Is the solution of an IVP on the domain -2 ≤t ≤3. Determine an upper…

A: yt=4cost-7t-43 The domain is -2≤t≤3 Step length h=0.001

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A: The above is the Case of Glide Reflection. There are 7 cases of Frieze Patterns: 1. Translation 2.…

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Q: What is the mini order that there ex probability zero o

A: This is a question of finite math.

Q: Solve the IVP: y(4) +2y"+y=0, y(0)=1, y'(0)=-1, y"(0)=0, y" (0) = 2

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Q: Find a function f such that F(x, y, z) = evaluate F. dr along the curve C given by r(t) = (t² +1,2…

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Q: (B) Evaluate: [7zdv 72dV, where E= = {(x, y, z) |0 ≤ y ≤ 2,0 ≤x≤ (4-y²),0 ≤ z≤y}.

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Q: Use the fact that matrices A and B are row-equivalent. 2 10 0 0 2 5 1 1 3 7 22-2 15 34 11 2 4 10…

A: We are given two matrices A and B. These two are equalent matrices.

Q: a. Approximate the given quantity using Taylor polynomials with n=3. b. Compute the absolute error…

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Q: Find the area of region enclosed by x = t-t², y = t-t¹, 0≤ t ≤ 1 using Green's Theorem. Question…

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Q: 4) Solve Poisson problem for a=lı b=l ↑u (x₁b) = f₂(x) ty b (0₁4)=9₁(y) 0 Ou=f(x,y) u(a.y) = 9₂ (9)…

A: Use separation of variables for laplace problem then from there go to poisson problem

Q: On an 18-hole golf course, there are par-3 holds, par-4 holes, and par-5 holes. A golfer who shoots…

A: Let x=number of par-3 holes       y=number of par-4 holes       z=number of par-5 holesGiven, Total…

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Q: Find a vector a with representation given by the directed line segment AB. A(4, 0, -1), B(4, 3, 4)…

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Q: X Question 7 Find the matrix for T(w) = projí where v = [−1 −4 −8]ª. DNE X

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Q: Which of the following letters has a nontrivial (that is, the angle of rotation is not equal to 0°…

A: The correct answer is determined as shown in the next step.

Q: a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the…

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Q: 11) Jen works as a Nurse. Her annual salary is $62,250. As of January 1, Jen will receive a 4.28%…

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Q: 2.let and B be sets.  Prove A ∪ B = A∩B if a = b

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