1.1 Let z, w e C. Show that |z + w? + |2 - w? = 2(|:|2 + |w?).

Q: 1: Prove that (c a) * H = (c b) * H implies that a H = b H

A: Given an equation. We need to prove that : a*H = b*H where H is a subgroup of a group ( say G). Also…

Q: Let S = {:n e N}. a) Show that inf S exists Question 16 %3D b) Show that inf S = 0

A: PART A The given set is : S=1n:n∈ℕ As we know that : ℕ=1,2,3,4,......... The set is given by :…

Q: 2 20 Let A = Which of the following is a consequence of the Cayley -Hamilton Theorem? 2 10 L-10 0]…

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Q: If u (E) =0, then E is measurable.

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Q: =[12] and U- = -[] Then A2017U is equal to -1 Let A =

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Q: 3. Show that if A-1 exists, then f eAT dr = A-(eAt – I) = (e4t – I)A-1.

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Q: Show theit the fenias +- (AtI)(28+1) (P+1)(2p41)(38+1) (uhese a>0, B>0) conveages if B>a amd…

A: Ratio test If we have series ∑ann=1∞ Then we will find the limit l = limn→∞an+1an If l<1 , then…

Q: If A= then A" + nl is equal to 1.

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Q: 2.16 If S = {r |0<< 12}, M = {x | 1 < z < 9}, and N = {r | 0<r<5}, find (a) MUN; (b) MON; (c) M'nN'.…

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Q: 7. If [G: H] = 2, prove that gH = Hg (18 from 6.5)

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Q: - Let X1, X2, ...,Xn~N(0,0) and T = Fi (T),Var(T)

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Q: 7. Solve the following first-order linear DE's. y' + (cotx)y = xcscx, ) = 1

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

Q: 1) Suppose A e M„(R) is. show: A) A is singular if and only if adj (A) is singular. n-1 B) If n > 1,…

A: The solution is given as

Q: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: Let (W, t > 0) be standard BM. Show that P(n|w,1 – Wa sc) → as n → 0.

A: From the given information,  (Wt, t≥0) be standard Brownian Motion. Note: In the given situation,…

Q: (3. Suppose T is a linear tronstormation from V to w and that {š X is a LI. söt of ve ctors in V.…

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Q: 3. Given: GH DE Prove: AFGH ~ AFDE F G 工

A: Given data: Line GH|| DE In triangle FGH and triangle FDE m∠F=m∠F  [common angle] We know that line…

Q: 2 Delermire if fW=x s is even, odd neither.

A: As we know that a function f(x) will be  1. even function if f(x)=f(-x)  2. Odd function if…

Q: If E, and E2 are measurable then (E,UE,) is not measurable .

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Q: 25. Which of the following is the answer for secx = V2 A. E + 2nk|k E Z}U {+ 2rnk|k € Z} 3 B. G+…

A: We will solve the problem

Q: 3.19 Suppose if r = 0, f(x) = {-1)" if r E n+1' n where n E N. Prove: f € R[0, 1], even though f has…

A: Sol

Q: Let m eN and a, b e Z be such that (a, m) = 1 and (b, m) Let h = 0m(a), k = 0m(b), and let n = 0m…

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Q: 4.51 Verify that BIBD with the parameter a = 8, r = 8, k = 4, and b = 16 does not exist.

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Q: 10. Let X , j= 1,.., n, be i.i.d. r.v.'s such that E(X,)= µ, o'(X)=o', both finite. j' Show that E(X…

A: The normal distribution: A continuous random variable X is said to follow normal distribution if the…

Q: 11. Let S= {| -2 }. Show that u= E span S. 1 2

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Q: 11. Suppose A\BC C. Prove that if x E A\C then x E B.

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Q: /et A 3 8 カー (-45 Show That LA+B S ZA+B) 3-5 = At B

A: Transpose of a Matrix is defined as “A Matrix which is formed by turning all the rows of a given…

Q: 4 16 Define T: R? - R° by T(x) = Ax, where A = 1 3 -

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Q: eb Let A E M₁ (TR) be given If AAt = On HOME JATOT prove that A = On - Does this hold if A EMn (C)?

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Q: 4. Let A %3D {a, b, c}, В %3D {x, у}, and C %3D {0, 1}. Find a) АхВх С b) СхВхА с) СхАх В d) ВxВxВ.

A: Given that: 4. A={a, b, c}, B={x, y}, C={0, 1}   NOTE: As per our answering guidelines, we can only…

Q: 42. Show that S 4xys + %3D 1- x2 1- y? 1– 22 (1 – a²) (1 – 3²) (1 – 2²) if xy + xx yz = 1

A: Given that  xy+yz+xx=1 Prove that the expression

Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: Which of the following can you not use to prove A CAT S A DOG? 1. CA DO II. ZC ZD III. ZA E ZOo IV.…

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Q: Let w = z. In(xy² z®), then which of the following is- O a. O b. O d. O e. 0

A: While finding the partial derivative of a multivariable function, if we have to find the partial…

Q: 1. Let P, X,Q denote distinct points of E. If X is between Pand Q, show that d(P,X) d(X,Q) 1-t, for…

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

Q: 1. Let A = {x E Z | 6 divides x} and B = {x € Z | 3 divides x}. Prove that A C B, and then prove…

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Q: 1. Find fr fy and fry if f(x,y) = y* + x?y + In xy.

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Q: 29 7 dispnove "7z: x?ex" means Provide an a st. x Pow Kat fr がそ.x メ **メ Prove that fn t : x<x!

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Q: 10. Show that S t2 if 0 1, 2(1 - e-) then Ldf %3D

A: Laplace of differential equation.

Q: 8.Prove the following relationships: a) I(X,Y)=H(X)-HOX/Y)

A: Solution

Q: Suppose A · B = 0. Show that this implies that C(B) is contained in Null(A).

A: Suppose A.B = 0 then we have to show that  Column space of B contained in the null space of A  i.e…

Q: (c) If A E M,(R) is non singular and n 2 2, prove that det(adjA) = (det A)"- (d) If A E M,(R)…

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Q: 1. Let x (x,) E l. Find x if 11-- a. x = 2 %3D

A: This is a question of functional analysis in lp space.

Q: 13. If w(x,y) = In(1+xy²), show that 2wxx + y³Wyx = 0. %3|

A: Given: w(x, y)=ln(1+xy2)  To show: 2wxx+y3wyx=0 Chain rule : ddx(f(g(x))=f'(g(x))g'(x) Quotient…

Q: D. Is there guaranteed to exist a value of x = c on -2 < c < 2, such that h'(c) = 0? Give a reason…

A: Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and…

Q: 7) Show that K± = det [ T 4 J.

A: Sol

Q: 4) Let x, y, z E Zg. Prove or disprove the following: Ifxy = xz, then y = z.

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