6. Consider a = 7+ 2i and B = 3 – 4i in Z[i]. Find o and p in Z[i] such that < N(B). = Bo +p with N(p) -

Q: 9. Suppose g(u) = | V1+t³dt. Find the value of (g¬1)'(0) if it exists. If it does not exist, explain…

A:

Q: Let z=3+7i and w=7−8i . Find out the following in the x+iy form. z2 + w zw2

A: Given  z=3+7i and w=7−8i

Q: If the function (x) =sin 2(2n+1)x for some positive integer number n, then the conclusion of Rolle's…

A: Given that: f(x)=sin2(2n+1)x on the interval π8, 3π8.and satisfies the Rolle's theorem at…

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

A:

Q: s expansion of 2 , valid in the re of 1/z2 is:

A:

Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: 1. To show that the transform x - Ax = (x1- 2.x2 + x3,3x1+ x2,2x2 – x3), = (r1,#2, 13) E R³ is…

A:

Q: 6. Find a²z дгу of z = In(xy)

A:

Q: Prove that Hence deduce that L{x"; p} r(n + 1) pn+1 " if n>-1, p > 0. L{x-¹/2; p} = √√(π/p),p> 0

A:

Q: Show that \{e ^ (- x), x * e ^ (- x)\} is a fundamental set of solutions of y^ prime prime +2y^…

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Q: Compute 4n — бп? lim Vlan| for a,n %3D 3n2 +7 n-00

A:

Q: Verify by mathematical induction 1. 2(2i– 1)° = n²(2n² – 1) for all n21. 33 n(n+1) 2. Prove: Vn e…

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Q: 2. (i) (ii) Prove the following results : cov (x2, x1.23)=cov (x3, x1.23)=0, cov (E 1.23, X1-23) =…

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Q: х Prove that 1 < < sin x for all x e (0,5). - ド|N

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Q: Show that F0F1 . . . Fn−1 = Fn − 2 for all n ≥ 1, where Fi is the i-th fermat number

A: To Show -

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: 4) For A = ajax + azāy + azaz , B = b,ax + b2ay + bzaz, show that A.B cos 0 = A.AVB.B

A:

Q: How does Abel's Identity allow us to conclude that if the Wronskian is 0 at a point t, then the…

A: To prove: By Abel's identity conclude that if Wronskian is 0 at a point t, then the Wronskian is…

Q: X1,, Xn ~ N(-3,2) (random variables) ... Z, = 1(X,X2 + X1X3 + X1X4 + •..+ X1X„), n > 3 %3D (a) Find…

A: From the given information, X1, .....,Xn~N-3,2 And Zn=1nX1X2+X1X3+...+X1Xn, n≥3 Consider,…

Q: 1. Show that sinz is unbounded on C and lyl <|sinz] <ebl z=x+yi € C. for

A: Given:         We have to show that sinz is unbounded in ℂ and y≤sinz≤ey   for z=x+yi ∈ℂ Function…

Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

A:

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,(+) =…

A: true

Q: Show that x2 + x + 4 is irreducible over Z11.

A:

Q: Show that x2+x+4 is irreducible over Z11.

A:

Q: Let z = 8+ 8i and w = 5 – 3i . Find out the following in the x + ijorm. 1. z2 + w 2. žw? 1 3. wz 4.…

A: Given that  Z=8+8i and w=5-3i

Q: Prove that Z(Sn) = {e} for n ≥ 3.

A:

Q: 1. Condense and expand. a) Condense as a single quantity: 1 In(3)+- In(x? +1)-3 In(1+x) 3 b) Expand:…

A: Given, (a)  To condense ln3 + 13ln(x2+1) -3ln(1+x) (b) To expand lnx2+1x+2(1+x)5

Q: Prove that Theorem 13.1 (vi) using the simple ftn, nonnegative ftn and general ftn. (vi) If X is…

A: vi) From the given information, X is measurable in N.  There exists a sequence of simple functions,…

Q: (b) f(z) = tan¯"(5z) - (In(2"))®

A: We have to find derivative

Q: 1. Prove that fraction where m is z-4

A: Given: zn-1z-1 ⇒zn-1nz-1 ----(1) Let apply the limit in equation (1) limz→1zn-1nz-1 ----(2) Let n=pq…

Q: (2) Let f(x) = cos (2x). Find the Hermite interpolation H3(x) using xo = 0 and x1 = , and use error…

A: Given: fx=14cos2x We have to find the Hermite interpolation H3x using x0=0,x1=π4…

Q: Suppose &lt;let A= 1 and you know all the cofactors in C. How can you find A?

A: Given: Determinant of a matrix A  =  1 and all the co factors are known.   To find: The matrix A.

Q: 2.2. Given: y and xo = -0 < x1 < x2 < ... < *n < xn+1 of steps to determine i such that x; < y <…

A: According to the given information it is required to find the minimum number of steps to find 'i'…

Q: 1. Prove that L(t)=. s2

A:

Q: FIND THE UNTQUE SOLUTION OF AU= 0 34/x1<4 SUCH THAT U(x) 5 on x =3, U(x)=6 on /×/ =4 on 2 in R in R…

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Q: Let z = x + yi, z2 = 1– 0.5i , and z, + z2 = 3 – 1.5i. Find In(z,) – In(z2) %3D A) In(25) ± 2niN D)…

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Q: dv Prove that 4u/2n

A: We use Gamma function to prove the given result.

Q: How does Abel's Identity allow us to conclude that if the Wronskian is 0 at a point t, then the…

A:

Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

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Q: - In the domain of Gaussian integers Z[i], the element 43 is O irreducible O reducible O unit

A:

Q: Assume C and D are chosen at random from {1,2...,n}. Let pn be the probability that C + D is a…

A: The number of pairs (c,d)  with sum m2 is m2-1 for m2≤n and 2n-m2+1 for n&lt;m2&lt;2n. Hence,…

Q: 7.1) 19) Evaluate the integral.

A:

Q: Compute for the value of tanh-(1 – i) In 5 my + 2.13i In /2 + 2.13i Inv - 4.25i 2 In /2 + 2.13i

A: Let, tanh-11-i=z. Thus, solve for z:…

Q: Let z = 3 + 9i and w = 2 – 7i . Find out the following in the x + iy form. 4. z-1 – w-1

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Q: (a) Prove that x, y = -1, y = 1 1 (b) Prove tha and n: m and

A: This is a problem of Number Theory.

Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

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: Consider x1 = (I11, F12)" and x2 = (12n,I2)". Let x = (x}, x})T and IPPP 1 P P Cov(x) = PP1p PPP1…

A: The  first pair of canonical variables, or  first canonical variate pair is the pair linear…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.