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 some teR, 0

Q: 4. Evaluate fe fds where f(r, y, z) , ) and 0 <t< 1. = 12r2 and the path C is given by r(t)

A:

Q: 18/156) If the Wronskian W of f and g is te' and if (t) = t, find g(t) %3D

A: Given that gx, y=t2et with ft=tTo find gt

Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = &lt;x, 0&gt;. C is any path from (a, b) to (c, d).

Q: If A = x a, – zy a,, evaluate § A · dī around the path L defined by 0 < x < 2,0 < y < 1 as shown in…

A: Stroke's theorem:         It is also known as Generalized stroke's theorem. It states that the…

Q: If the rst line has the form Ax + By + C = 0 and the second line has the form Dx + Ey + F = 0 and…

A: Given,  The first line  Ax + By + C = 0  and  is to pass through the origin  The second line  Dx +…

Q: 1. Show that S, has the presentation (x, y|x² = y = (xy)² = 1). %3D

A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

Q: Let X1,..., X, be independent such that X;|p ~ Bernoulli(p) with pE (0,1), and p~ Uniform(0, 1).…

A: c) Given that X1,X2,.....,Xn be independent such that Xi|p~Bernoulli (p). f(Xi|p)=pxi(1-p)1-xi…

Q: 10. Suppose that a point (X, Y) is chosen at random from the circle S defined as follows: s= (x.…

A:

Q: Compute I ds over the path C : ř(t) = (t², 2t, t²) from (0,0,0) to (4, 4, 4).

A: Given path is:

Q: If the joint p.m.f. of X and Y is given byh(x , y) = k(x+ y)2, x = -1, 0, 1 and y = 0, 2, 4.where k…

A:

Q: Let XN(3, 4) and let Y~ N(2, 3). Define S = X + 3Y, where 'X and Ya atistically independent RVs.…

A: Given that: X~N3,4Y~N2,3S=X+3Y, X and Y are…

Q: 25 If A E M, is symmetric, prove that AT is symmetric.

A: Symmetric matrix

Q: Find the fowien senies fon | X), -T<X <A

A:

Q: Given u(x, t) = t-²e¯, 2 , compute:

A: We can find the required answer as below.

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: Q1. Compute xdx x'+10 using trapezoidal and Simpson's 1/3 rules with the number of points 3 and 5.

A: The given integral is, I=∫01xdxx3+10 Dividing the interval (0,1)  into 2 intervals (for 3 points)…

Q: Let C be the shortest path that connects z to ri and /= J, (sinhz + 1)dz. Then O 1 cosh r+ n(1+) O…

A:

Q: Find the length of the curve y = In(e - 1) – In(e + 1), In 2 sxsIn 3.

A:

Q: 1. Find the interval(s) wheref(x) = xe6* is concave down, if such an interval exists.

A:

Q: -2 Let f(t,y) = y + 3t?e'. Then f satisfies a Lipschitz condition in y with L = 2, given that 1<t<2.…

A:

Q: Compute I ds over the path C : ř(t) = (t², 2t, t²) from (0, 0,0) to (1, 4, 4).

A:

Q: In ΔFGH,  FH is extended through point H to point I, m∠FGH=(x−2)∘,  m∠GHI=(4x−5)∘, m∠HFG=(x+19)∘.…

A: Given that     FGH is triangle. FH is extended through point H to point I .

Q: The area bounded by the curve y = 7x and the lines z= 2 and =9 above the z-aris is O 539.00 square…

A:

Q: 2 let C be an arbiterery curve from l0,0) to D). (2,4) Evala vute: (3x² oy, x-2)- dio

A: To find the value of the given line integral.

Q: Suppose that U(x; y) = xe' + ysin z, and that x, y and z can be measured with maximum possible…

A:

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: 3. If z = f (x, y), where x = e“ – e® and y = e® – e", show that UTM UTM az az UTM X : az - ду dx…

A:

Q: 1. Prove that given a line € and a point P not on e, there exists a line through P perpendicular to…

A: Explanation of the answer is as follows

Q: 9 Give with an not example of T: W≤X →Y, to XT Ext. .R. boundedag. - keše bes

A: Please find the solution innext step this is an non-example of Tietze extension theorem in Topology.

Q: 28. (a) Prove that for x e R" and s < t, Ds(x) C D;(x).

A: The notation Dsx represent  an open ball of radius s, centered at x∈ℝn. Let x=x1, x2, …, xn,…

Q: In Let x, y ER" be given. (a) If ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

A:

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: Q1. Compute xdx x' +10 using trapezoidal and Simpson's 1/3 rules with the number of points 3 and 5.

A: we have  I=∫01x dxx3+10dx  n=3h=b-an=1-03=13…

Q: If S e-t)rLnt dt, Find the valae of (n)

A:

Q: Let X,X, X be i.i.d. with the following PMF. ...... 25 0.6 ; k = 1 Pk)= 1 0.4 ; k = -1 and let Y =…

A: Answer: For the given data,

Q: 1. If x, y, z E R, prove that (a) \x+y\ < \x| + \yl- (b) |x– y| < |x – z|+|z– y|, and (c) ||지| -…

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

Q: Q.1: lel I 1,2,3]. Ts Prove that Tz and T3 Gre topologi es for X Ta is topolagy far y Ti is nat G…

A:

Q: and further we require that u(r, z) is bounded as z - +oo. Find an expression for the steady state…

A:

Q: For p = (1, 0) and x = (x₁,x2,...,xn) ER", define ||2|p:= - ². |P Show that n jyjx|py|q for x, y…

A:

Q: Let Xi. ---- Xnd Glle e) and Tet TX = Exi , a) show tha T) is unbiased estimator for WcO) =nG: E…

A:

Q: Suppose that U(x; y) = xe' + ysin z, and that x, y and z can be measured with maximuma possible…

A:

Q: 10- If f(u, v, w) is differentiable and u = x - y, v = y - z, and w = z - x, show that fe fe fe = 0.…

A:

Q: 1. Evaluate | (x- y)ds over the path C:r(t) = 4ti + 3tj for 0 <t<2.

A: Answer

Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

A:

Q: In C[0, 27], with the inner product -2 (f, 8) = | f(x)g(x) dx, show that the set S = {1, sin x, cos…

A: Given: Inner product defined as: &lt;f,g&gt; = ∫02πf(x)g(x)dx To show that the set S = { 1, sinx,…

Q: Find f, and f and evaluate each at the given point. F(x, y) = e*y?, (In 4, 2) f(x, y) f,(x, y) =…

A:

Q: 1.) Given: T and R trisect SW XS = XW ZS = ZW Prove: XT = XR S T

A: Given that: T and R trisect SW.XS=XW∠S=∠W

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.