Q4: 1- Prove that (1-0+(1+ () 0. 2. Put in the form a+ib ((1+i))- ((1-i)).

Q: %3D 2. Prove that -24y of 3. =2x

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Q: Prove that Σ (2n+1) P(x) no (1-2hx+h²) ¾/2₂

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Q: 9. Suppose g(u) = | V1+t³dt. Find the value of (g¬1)'(0) if it exists. If it does not exist, explain…

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

Q: 2 1-X ie 2=tny then prove that da

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Q: - Find the antiderivative, FCx), and Check your answer of the funetionçiven xZ+3x+5 by f (x)=

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Q: ¤1 + ag – r3 = 0, -2r1 – 12 + 5r3 = 0, -#1 + 4.x3 = 0 find the Rank(A), where 1 1. -1 A = -2 -1 -1…

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Q: Prove that v = In (x- 2)^2+(y-N)^2 is harmonic in every region does not include the point (2,N) and…

A: Given v=lnx-22+y-N2 where N=44. So, v=lnx-22+y-442                                    ........1 To…

Q: Suppose the conjugate z is equal to the reciprocal 1/ z. What are all possible z's?

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Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: Q3)b) Use Taylor theorem to show that 2x-2를 <2 sinz <2z-2% +2를1

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Q: (b) Prove that xe" 2 for some x in (0, 1). %3D

A: By the intermediate value property of continuous function if a function f:(a, b) --&gt; R be a…

Q: a+ib a) If x+ iy = prove that x? + y² = 1. a-ib

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Q: A homogeneous linear ODE with constant coefficientshas a characteristic equation having only the…

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Q: Evaluate the integral. 2 (3u - 5)(u + 2) du -1

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Q: 1 dæ using the residue theorem. (2² + 1)° Evaluate the improper intergal -00

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Q: Q1. Verify Euler's theorem for the homogeneous funetion, fxy.2)-axy +byz + czu

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Q: Find the infimum and supremum of the set X = {3 – Vztn ine N}. -

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Q: Use Fermat's Theorem in your proof that Vn E Z>², 2 is irrational. I

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Q: Solve the folowing equation on [0,2pi) 2sqrt2 +4cosx=0

A: In the given equation, the cosine function is the trigonometric function:   Rewrite the given…

Q: Show that F0F1 . . . Fn−1 = Fn − 2 for all n ≥ 1, where Fi is the i-th fermat number

A: To Show -

Q: Save the following Sef of Lincay equations using over relaxoation 10X, +X2 + X3 12 X, + 10X2 + X3 =…

A: Rewrite the given system of equations, to find x1, x2 and x3. Multiply each right hand side…

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: 4. Let z = -4 + 3i and zw = -7 - 2i. By showing clear steps, find w in the form of | | a+bi.

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Q: 0, 2 is defined by A natural cubic spline S on 5. So()12x - 0 if 1 (1) S(r)2b(x1)c(x - 1)2d(x - 1)…

A: 5.Note that, the natural cubic spline must satisfy the conditions

Q: Find the nverse 2 -1 2 A =|1 0 1 f it enists. 0 3 -1

A: The given matrix is A=2-1210103-1. To find the inverse matrix, augment it with the identity matrix…

Q: 1 1 b. From (i) prove that, xb + x ² + 1 + x² + x ² + 1 ² + 1 = 1 xa +x+b+1 c. From (ii) given in…

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Q: Using the 2-transform solve Untz + 4unti 734n=3" with u-0,, =1 %3D

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Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

Q: 3) Prove the following statement using the Intermediate Value Theorem (IVT): 2ª + 3* = 4* has a…

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Q: IRI- 2. <| sinz|< -e 2

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Q: Prove that the equation * + + = 1 has no solutions in positive integers x , Y, Z.

A: x/y + y/z + z/x = 1 if x , y , z are positive integers then case 1 :  If all three numbers are…

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: Find the rank and nullity of the 2. 4 9. A z -7 2. -5 4 L7 -9 2 -4 verify the that dimension…

A: The given matrix is  A=20464-7201-5247-92-4 According to dimension theorem for any matrix A of size…

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: We have to use the Euler method

Q: Jse Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: Given: dydx=2x-3y+1, y(1)=6 f(x,y)=2x-3y+1, x0=1, y0=6, h=0.1…

Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

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Q: Q2: Show that whether (Z16,.0) is field or not?

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Q: -3 [-1+v3i]

A: Explanation of the answer is as follows

Q: 6. Use the IVT to prove that x2+ In r = has a solution. %3D

A: The Intermediate value theorem says: If the continuous function attains two values then it must…

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: For f(x) = 1 –x2on [−2, 1], do the hypotheses and conclusion of Rolle’s Theorem hold?

A: Rolle's Theorem: Rolle's theorem states that if a function f is continuous and differentiable on the…

Q: Find leng th of the Cardioid Vsa6 from e -oto0 - 1 %3D

A: Use the formula of the length of the polar curve. L=∫0πr2+drdθ2dθ Here, r=aθ3 and drdθ=ddθ(aθ3)=3aθ2…

Q: Prove that JI = J [1 0 0] I = |0 1 0 Lo 0 -1 2 -31 J = 4 5 -6 %3D –7 8 -9 1.

A: Considering the matrix multiplication of J and I

Q: Q1. Prove that KT = = -.

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Q: Let z = 3 + 9i and w = 2 – 7i . Find out the following in the x + iy form. 4. z-1 – w-1

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Q: Compare the graphs of y = Vx, y = Vx, y = x, y = x², and y = x' on the interval 0 < x s 2. Write a…

A: Differentiate the expression of the slope with respect to x. For any value of x in the interval…