2. Give the condition which ensure that |ez| < 1 where z in C.

Q: 5. (a) Let a, b, ce R3. If det(a, b, c) # 0, prove that a, b and c are non-zero. (b) Prove that x x…

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Q: State TRUE or FALSE giving proper justification for the following statement. There exists a…

A: The given statement is : "There exists a continuous function f:ℝ→ℝ2 such that f(cosn)=(n,1n) for all…

Q: Give four different ways to determine when A = M₁ (F) is singular.

A: To determine the different ways when A∈Mn(F) is singular .

Q: Q.No.1.|A f:X→Y where X,Y E Z*, and 02X<23 and, 02 Y<59, X is hour of clock and Y is minute of…

A: Let f:X→Y,  where X,Y ∈ℤ+ and 0≥X≤23,0≥Y≤59. Here, X denote the hour of clock and Y denote the…

Q: S[y] given by y = cosh æ + B sinhx, 0 0,

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Q: 1. Find the T.F of the following sys.: 園 = -5 3 =][*] + € y = [12] u

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Q: 6. Find the values of c such that g(x) is continuous everywhere. fx² + c², 15c – x, x< 2 g(x) = x22…

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Q: B. Let H « G and K - G, and assume that HNK = {e}. Show that if x E H and y E K, then xy = yx

A: We can prove given result by definition of normal subgroup:

Q: Determine the values of a and b such that the function (x -a ,x 1 Is continuous of all values of x.

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Q: 1.Find the interval and radius of conv. of ln(1 – x²) ?

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Q: If f(x)= x - 4 and g(x)= Vx-2; which of the following is equal to х — 2 Option 1 – 4(SF 2) X - х — 2…

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Q: Solve: 2 0 -1 0 2 -6 1 -2 -3 -2 0 1 -5

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Q: 4. Use the mean value theorem to show that the following inequality holds for every x,y E R: |sin x…

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Q: .

A: Introduction: The Cauchy-Schwarz inequality is a fundamental inequality in mathematics that relates…

Q: 3.5 Suppose f E R[a, b] and g(x) = f(x) except at finitely many points in [a,b]. Prove: g E R[a, b]…

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Q: 4.1 Suppose that fig and g'are continuous on the interval 1= {x :asxs b). Show that a ) exists, and…

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Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Queston #o 4 Show that X, Ixe, where x, and Xe E IR X= (1,1,1, -1) Xz = (4,2,2, -1) %3D

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Q: Liouville function X(n) is given by formula

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Q: 1. Determine whether f is a function from Z to R if а. f (n) %3D tn b. f(n) = Vn² + a %3| 1 с. f (n)…

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Q: a.s. Let X, X, E|X,| < x, E|X| < x, and EX, → EX. Prove that E|X, - X| → 0. (Hint: dominate the…

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Q: If X is a RV with p.d.f f(x) = in of Y=2X-3. 2 1 in 1<x<5 and =0, otherwise. Find the p.d.f

A: Probability value for the random variable for distinct random variable is denoted using the…

Q: 2. Prove the following: i. Let c > 0. Then |x| <c -c <x<c. |x| 2 x, Vx E R. |x + yl < ]x[ + lyl vx,…

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Q: 3. Express the following in terms of z] and Zz: (a) z* (b) 1/z (c) Re{z} (d) Im{z}

A: In this question, we have to express the following in terms of z and ∠z :  (a) z* (b) 1z (c) Rez (d)…

Q: 5. Find the x-coordinates for all points of discontinuity for f(x)- 「X316, xśl x〉 1 2 c. 1, 32d.…

A: First we check the limits as x approaches to 1 from left side and right side .

Q: 5. Suppose f E L'(R²) is real-valued. Show that there exists a measurable set EC R? such that fdm2.…

A: Given that f∈L1(ℝ2). And E⊆ℝ2. The objective is to show that ∫Efdm2=∫ℝ2/Efdm2.

Q: iii. Let e > 0 be given. Briefly explain why there exists >0 such that |z − zo| < 8 ⇒ |ƒ(2) − ƒ…

A: Given that: ε&gt;0 We need to explain why there exists δ&gt;0, such that |z-z0|&lt;δ ⇒|f(z)-f(z0)|…

Q: 4) Find the value of the constant k so that the function (1 – k, if z<3, f(z) = 1– kr, if 123, is…

A: This question is related to limit and continuity of a function, we will use condition for a function…

Q: #3) For any x € R", show that ||x|| = inf{r: xerB}, where B:= {|ly|| ||y|| ≤ 1}.

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Q: Suppose S = {1, 2, 3} T = { 1, 3, 5, 7} 28. S = {x|x & S and x & T} = 29. S = {x|x€ S or x ε T} =…

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Q: If |E| <∞ and fn →f a.e. in E, then fn → f in measure. Show that this is false if LEL -m

A: Given a complete measure μ, if f0,f1,f2,…f0,f1,f2,… are measurable, and fn→f almost everywhere, then…

Q: - X1,·… , Xn i.i.d. Uniform(0, 0). .. Find X(1)+X(2) ] E

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Q: give 3. Let A = {x Rx # 0} and define f: A → R by f(x) = . Is f one-to-one? Is fonto? Prove or for…

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