2) x+y+kz= 2 3x+4y+2z=k 2x+3y-z =1 a) for have only one solution ks b) for be unsolvable k? c) forhave than one soton k more

Q: then 7(1 - а)М? + 6(1 - а) Мт %3D ВM?, 7(1 - а)т? + 6(1 — а)Мт %3D Вт?, Subtracting we obtain Y(1 –…

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Q: 2. Use the inclusion-exclusion principle to find the number of solutions to the equation r+y+: = 19,…

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Q: how many intergers solutions to x1 + x2 + x3 + x4 = 29 are there for which x1 ≥ 5, x2 ≥ 2, x3 ≥ 6…

A: The given equation is -  x1 + x2 + x3 + x4 = 29 where x1 ≥ 5, x2 ≥ 2, x3 ≥ 6 and x4 ≥ 5 We have to…

Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

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Q: In 26-36, write a proof. 3 12 1. a Ex. 26 Ex. 27 26. Given: mz1 + mz2% 180°, mz3 + m2 180°. Prove:…

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Q: 3. prove that: det (AB) = det (A) det (B) If a - (;). » - ( :) 2 -1 7 A = 3 2 4

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Q: 5. Prove that (Z, × Z6)/((2,3)) S Z12.

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Q: Prove that S3 = Inn(S3) = Aut(S3).

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Q: 17. Find a, b, c in Q such that (1+ V4)\(2 – V2) = a + bV2 +cV4 Note that such a, b, c exist, since…

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Q: 15, If neZ, then nº +3n +4 is even. (Try cases.) ho bn 6. If two integono

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Q: Prove both sides equal πU (PN ₂) = (π Up) N (π UV).

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Q: 2. Suppose A is an m x n matrir, x ɛ Rª and Ax = b. Then b is in 1) R" | 2) R™ 3) RmfnT 4) R™n

A: Given :  A is an mxn matrix , x∈Rn and Ax=b   To find :  b is in which space of the given spaces.

Q: In rank - sign test, the null hypothesis u = µo can be rejectedin favor of %3D the alternative u < u…

A: Signed rank test HypothesisHo:μ=μ¯oHo:μ&lt;μ¯oWe reject the null hypothesis if W+≤Wαwe reject  the…

Q: Prove using the IVT. 2x = bx has a solution if b &gt; 2.

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Q: 16. Write a two column proof. Given: OSTUV U V. Prove: AVXS = AUXT T S 8.

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Q: Then, we assume that Wn+1 - H Wn - l kn+1 kin 1 2 E Wn-p Wn-p p=0 p=1 (6) n+1 - € Zn - € Vn+1 = 1 Vn…

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Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: 27. Prove in detail that Q(/3+ 7) = Q(/3, v7).

A: We have to prove given equation.

Q: 6. Given: 21 and 22 are complementary Prove: SX 1 WX S. 2,

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Q: 3. prove that: det (AB) = det (A) det (B) a - (; ), » - ( ) 2 -1 7 2 B = 2 в 3 %3D

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Q: 5. Prove that if x, y ER, then x + y ≤ x + y. Use the following cases: (i.) x, y ≥ 0, (ii.) x, y <…

A: We now prove that if x,y ∈ ℝ , then x+ y≤ x+y with the help of given three cases.

Q: Give a natural deduction proof of V x B(x) from hypotheses 1. Vx (A(x) V B(x)) and 2. V y (- A(y))

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Q: Prove that Vtan a de – 7//2.

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Q: Use an inclusion-exclusion principle argument to determine the number non-negative integer solutions…

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Q: 7) Assume dim V = n, dim U = k, dim W = n - k, and U + W = V. Prove that Un W = {0} and V = U OW.

A: Given, dim V=n, dim U=k. dim W=n-k, and U+W=V.To prove that  U∩W=0 and  V=U⊕W.

Q: A library subscribes to two different weekly news magazines, each of which is supposed to arrive in…

A: Given, Let the events be: The corresponding probabilities are:

Q: 9. Write a 2-column proof: Given: p // q , Zg = (59 + 5x)° , %3D Za = (11x – 7)° Prove: Za = 114°…

A: Explanation of the answer is as follows

Q: 4. Demonstrate that Re(iz) = -Im(z);

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Q: 4. Prove that, VR, -11-

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Q: 8. How many integer solutions are there to the inequality Y1 + y2 + Y3+ y4 0, 0 < y2 ≤ 10, 0≤ Y3 ≤…

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Q: Prove that. 2/nJn-(n+2)Jne2 + (n+yjJney --- 3. end hence deduce that. 1xJn - (nti)Jnas -…

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Q: (2 Let xinN(0.1), n = 2 and T= Xx +X2 Show that Tis a Sufficiency Stakistic.

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Q: (4). Show that of X~UCO,!) as Y=-n-X)~ txponenkalla),

A: i) Uniform distribution : If X follows Uniform distribution over interval (a, b ) then cumulative…

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A: given premises 1) x~Ax⊐Dx 2) Ey~By∨~Dy⊐ ~~Dy∨~Ay 3) xAx∨Dx⊐~Ax∨~Dx /ExAx∨ Bx

Q: Prove that Z(D,) # {1} if and only if 2|n (hint: if 2|n look at ri).

A: Let us suppose, Dn=e,r,r2,..,n-1,f,rf,..,rn-1f and nis divided  by 2. Then, n2 is a integer. We see…

Q: 3. Prove that: na” u[n] = az (a-z)2

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Q: 341 Show that the Neumann problem v'u = 1 in r+ y' < 1 du - 2 an on x² + y' = 1 does not have a…

A: We will use greens theorem

Q: Suppose ỹ = cje (:)- + czetx () is the general solution to = Aj. What is A O We don't have enough…

A: Use given differential equation y' = A y to find it.

Q: 5. Demonstrate that Im(iz) = Re(z).

A: It is required to demonstrate the problem 5. Problem 5 is solved below in detail.

Q: Q-5 Consider NLP problem, max 8() --2+ 2ィ-ズメューメュ+1 ダ」ン0 ズェン」 19 (2). Satsty fnst1 2nd order. For ist…

A: Given: Let's consider the NLP problem max sx=-x12+2x1-x1x2-x2+1 subject to x1≥0x2≥1,xa=12,1  satisfy…

Q: 3) Ll a, = 5, Qz = 13, end Anta 5 anti - 6 an (n71). Prove by inbuetem that an =2* +3" (n. %3D =

A: The solution is given as

Q: 3. Suppose at a small college, 30% of the students are freshmen, and 60% of the students live on…

A: Given, The probability that the students are freshmen is pf=0.30 The probability that the students…

Q: Find the unique fixed prbability vecter for the regular stochostic matrx A= ) 2 5 2 0 3 3

A: The following information has been given: The regular stochastic matrix is : A=311252033 We need to…

Q: Find the first devdivative 2. 2. u = x2+ 2 ut1 %1 ).

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Q: If p,q,r are prime number but p is not equal to q, then what is gcd(p,q)? What is the smallest…

A: It is given that p and q are prime numbers. Assume that gcdp, q=d. Then, d is the greatest common…

Q: -. Show that both R and R2 may be covered by countably many open balls.

A: Soution.

Q: Show for Re(z) > 1 -dr = r(2) ((2). %3D et - 1

A: