20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 – 2) P(2), show that all the zeroes

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: 8 If C is a circle of radius r with centre z,, in the complex z-plane and if n is a non-zero…

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Q: Q2/ fincd lachre trans form for at X- 3x+2X=lee

A: given equationx"-3x'+2x=1-e2tby assuming that initial condition is zero.x"=s2X(s)and x'=sX(s)and…

Q: 2. Consider the following quadratic form; F(x,y) = ax? +2bxy + cy. Examine under which conditions on…

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Q: Find the first piece S1 (x) of a natural cubic spline that interpolates the set of points (0,1),…

A: Formula for cubic spline interpolation is…

Q: 6. Show that the sets S = (1,5) and T = (0, 00) are equinumerous by finding a bijection from S to T.…

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Q: Let x, y e Z. Prove that if 3x+7y is odd or 5x + 6y is even, then x or y is even.

A: We have given that for x,y belongs to integers. If 3x+7y is odd or 5x+6y is even then x or y is even…

Q: of discon tinuity of the the following func Hions (a) g(x,y) = stuind %3D (b) b) f laiy)

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Q: 8 Use the Cauchy integral formulae to evaluate 22 – 1 - (9 – 22)(z + 2i))³ | where C is a circle |z…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: 1d) Prove: If z=Zo is a zero of order n of flz), then it is zero of order 2n of

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Q: 4) Use Green's Theorem to evaluate f √1+x³dx + 2xydy. C is the triangle with vertices (0,0), (1,0),…

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Q: -1/2 8. Ž Pn (x) q^ generobr x,2) = (1-2xz + 2?) of legendre polynañials ,oblin the (2n+) xPa W=h+D…

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Q: 3. Find all the fixed points and classify cach of them: f(r,u) ryev,

A: Solution:- Fx,y=xye-x2-y2 Put Fx=0 and Fy=0 Fx=e-x2-y2y-2x2=0and Fy=y-2y2 e-x2-y2=0 So,  0,0,12,12,…

Q: 8 (6562) 243 를-), then xo = If the length of the curve y = 3 x7 from the point (0, 0) to the point…

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Q: 9- Show that H= {0 E S6=(6)o = 6} is %3D subgeoup of and find its wight cosets S6 ets left Casets ·…

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Q: 6. Use a substitution to find S (x* + 1)" z³dx.

A: Here we use substitution we find∫x4+111x3dxHere we substitute x4+1=du…

Q: Show that the function f(z) = from [0, 0) to [0, ∞) has a fixed point c. Hint: Set f(r) = z and show…

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Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: 2. Find three positive integers x, y, and z if their product is 27 and the sum is a minimum.

A: Given three positive numbers are  x, y, and z Product = xyz = 27 And sum is minimum

Q: 3. Find the constants bo, co, a1, b1, C₁ so that So(x)=1+ box + cox² + x³, S(x) = { S₁(x) = a₁ - 2x…

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Q: A cubic spline S is defined by: So(x) = 3(x – 2) + (x – 2)² – 4(x – 2)ª 32 if 25x53 S(x) s,(+) =…

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Q: 13 3. Use the IVT to prove that y = r Inr has at least one zero on 2 2

A: Above function is continuous function in given interval. So we can apply IVT

Q: Prove that z'e'-z = 1 has exactly two roots inside the circle |z] = 1.

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Q: 2 1 Find a mat rix P such that P- 2 0 0 3 0 3 [11 O AP = 1 0 -1 0] BP = 1 1 1 1 OCP = 0 1 1 -1 ODP =…

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Q: A natural cubic spline S on [0,2] is defined by (So(x) = 1+ 2x – x³ S(x) = Is, (x) = a + b(x – 1) +…

A: Given: Natural cubic spline S on [0,2] is defined as: Sx=S0x=1+2x−x3 if 0≤x≤1S1x=a+bx−1+cx−12+dx−13…

Q: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

A: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

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Q: c If in is an involution of X, show that (x, y)in = (yin, xin) is an involution of X2). %3D

A: Let X be a lattice We have to show that (x, y)in=(yin, xin) is an X[2] X[2] is a 2 x 2 matrix We…

Q: 1. Do not use a theorem to find the ff. Z if yz? + z = 5xy – 1. %3D

A: In the multivariable function, if we have to find the partial derivative of one variable with…

Q: (2) If U = (1 – 2xy + y²)-³ prove that, x- = y?U® %3D

A: Since you have asked for a multiple question, we will solve the first question for you. If you want…

Q: 3. 2. If A is 2x2 matri x aud Á = A, but %3D A+ A. show that A= 0

A: The solution is :

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: Q) prove that (1) -2P xSinx 2.

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

Q: 9:Find the absolute exlemum Verlues of flx) x²-4x+24 over the interNal Cl,41

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Q: A cubic spline S is defined by: (So(x)%D 1+x- x (5,(x) = 2 - 5(x – 1) + (x – 2) if 0 SxS1 if 15xS2…

A: As we all know that Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously…

Q: 4. Sketch the proph of the following funcfions; メーメ+4 (x+4 a) f(x) = (x+4) x b) gex) = %3D

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Q: -3 [-1+v3i]

A: Explanation of the answer is as follows

Q: Find the clomain for the following funclions:- -y=2X (x+11 b-ys a - ৮-9 (3-X) C. y= Jx +4

A: We find domain of following function

Q: Find all triples of integers (x, y, z) that satisfy x² + 2y² = z2.

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Q: 3. Show that à = (6xy + z³)î + (3x² – z)ĵ + (3xz2 – y)k is irrotational. Find p such that A Vọ.

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Q: 9) suppose that horx)=1, f,(x)=x ond dz(X) Prove that Ba(f, - where Ba is a Bernyterh Polymomial.

A: Given functions are f0(x)=1, f1(x)=x and f2(x)=x2 Since Bernstein polynomial is…

Q: Let S-(xeZ vmod12-8) and T (xe2| 4x). Prove that SCT but TgS La s-(rez| 3reZz, x-12r+7) and T-(xeZ|…

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Q: 1 : Determine the manimom and Nalve of the function minimumn x'tyス condition under the *+ 2 7?== 1.

A: To find the absolute maximum and minimum of the function using lagrange multiplier

Q: 7.1) 19) Evaluate the integral.

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Q: 5. If 4=2xy į +3x°y -3pyzk is solenoidal at(1,1.1). Find p.

A: DivA=0

Q: D Reduce the Q.F 22,xz t 22,X3 2ズュズ3 weag ayf of oothogonal toans fermation by an Ans: y+92-295

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Q: I. Solve using De Moivre's Theorem. Given that z is not real number and z = 1, show that w = z-1 z+1

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.