). Find 1 |z-j|=2 Z

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 {X;} be i.i.d.~ Exp(0) and a; > 0. Denote by Z = min{a,X1,..., amX,}. Suppose that 0. Prove that…

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Q: Given 21 = Z3 Prove jk 13 2.

A: Given,  ∠1≅∠3 To prove  j∥k

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Find the value of k such that A is singular. 1 0 5 3 -1 0 2 k k =

A:

Q: Prove that ||A||1 = maxi<j<n{E#1 laij|}. i=1

A:

Q: Find the value of k such that A is singular. 1 0 4 5 -1 0 3 3 k k =

A: Topic:- Matrix

Q: Let n, k, m EN with n2k21. prove that

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Q: Let x, y, z e Z Prove that if x – 2y + 3z is odd, then 4xy – 6z is even.

A:

Q: Prove that the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

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Q: 14. Given that Z calculate /Z/ and arg Z. %3D 3+ i'

A: Argument and modules of complex number

Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

A:

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: Prove that ((¬(p → r)) ≡ (r → p)) is a wff.

A: We have to solve given problem:

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

A:

Q: b) If d = 1, prove that (1 if a is odd gcd(a, 2b) %3D if a is even

A:

Q: Prove this! P (an) = W/ NM) [M-N (p-q)" ] 8 Npq %3D exp

A: your question is not clear i think some error in the question plz confirm it.....

Q: Prove that if 0 < a < y, then ỹ – VI < Vy– a.

A:

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

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Q: Show that if A is nonsingular and AB = O for an n × nmatrix B , then B = O .

A: We have given  matrix A is non singular and AB=0 for an n×n matrix  B. We have to prove that B=O

Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: Let q E Q and x E R - Q. Prove that q + XER - Q.

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Q: Let a,b E R with a # 0, b # 0. Prove that (ab)¯1 = a°1b´1. %3D

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Q: Assume that =y+In(x), and n=x, then u xX а. O a. 1 Ob. 1 1 с. O c. x² d. 1 Unn x 1 O e. x² е. 2 +Unn…

A: ANSWER :

Q: Let n E N coprime with p. Show that there exists x, y E Z coprime and such that p | x² + ny² is and…

A: To prove the given claim we have to show that :- x,y ∈ Z ; gcd(x,y)=1 ; p | x2 + ny2   implies…

Q: (a) Let r = ri+ yj + zk and let r = ||r||. Prove that - (1) a

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Q: Let x, y, A, u E Z. Prove that if 1 = Ax + µy then r, y are relatively prime.

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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: Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x,g (y) = 3y + 4 and h (z) =…

A: Given, Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x, g (y) = 3y + 4 and…

Q: Use Shapiro-Wilks and a = 0.05 for the test and show all parts. * %3D Но: (а) На: (b) F and PValue…

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Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: A := {x € (0,1)|x= ar, for some r € Q} re that sup(A) = 1.

A:

Q: (a) Let r(t) = ti+tej+fk. Find ((1),4 × (?),1) - (1)a tuj

A: First we find out first derivative of r(t) with respect to 't' then find out 2nd derivative and then…

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: Prove that ||A||1 = max1<j<n{E1 |aij|}. m

A: We will be using the definition of norm. And will verify that the formula given for ||A||11 is…

Q: Find the value(s) of k such that A is singular.

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Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

A:

Q: ,32 Let C: z-il=3 and I = -dz. Then %3D z2 1=0 None of these O I= 4nti O I= 6ni O I= 8ni

A: The solution is given as

Q: (2) Let (x)S Hg(x)S Hc(x). Prove that AN(BỤC)=(ANB)U(ANC).

A: 2. To prove the above equality we need to show that A∩B∪C⊂A∩B∪A∩C andA∩B∪A∩C⊂A∩B∪C

Q: Let z E C. Prove that if 1+z +...+z"-1 – nz" = 0, then |2| <1.

A: Given that, z∈C and, 1+z+...+zn-1-nzn=0 It is known that, 1+z+z2+z3+....+zn-1=1(rn-2-1)(r-1) So the…

Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: Solve the following Linear Homo, y" – 4y" – 5y' = 0 - %3D

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

Q: P:(A UB) cB Q :AC(A nB) Q => P I have to prove that p=>Qond that

A:

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: 2. Let A € M, (R) be given. (a) If AA = 0n, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

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Q: Find the value of k such that A is singular. 1 0 6 4 -1 0 6 2 k

A: Given. A is a 3X3 matrix. We need to find the value of k for which the matrix is singular. Singular…

Q: Find the value(s) of k such that A is singular.

A: to find the value of k

Q: Determine the ratio and interval of con (-1)"x" in=0 n6" O R.O.C=6, 1.0.C: x E [-6, 6) R.O.C=1,…

A: Here we have ∑n=0∞(-1)nxnn6n=∑n=0∞an ⇒an=(-1)nxnn6n

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.