If a, b, c are in A.P., then prove that the following are also in A.P. (i) a²(b + c), b²(c + a), c²(a + b) 1 1 1 (11) √b²+ √e ² √e + √a® √a + √b (ii) 1 a ( ² + ² ), o ( ² + ² ) , ( ² + 1 ) 1 - C с a a (iii) a

Q: 4. Solve. A) 3 ()² − 7 (¹) + 4 = 0

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: 4. Find the product AB, where a) A= 0 -1 -1 --1.B= -3 2 - 2 -1 03 -1 -3 -2 0 b) A=1 0 -1 B = 2 4 -1…

A: AB=1010-1-1-11001-11-10-101     =-11001-11-11 (b)AB=1-3012221-11-123-103-1-3-202…

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A: Given : A Positive Integer Is Said To Be Congruent, If It Is The Area Of A Right Triangle With Sides…

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Q: The expanded form of (3x + 2)(2x – 5) is А) 6х2 — 11х— 10 B) 6x? + 11x – 7 C) бх2 — 10 D) 6x2 – 19x…

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Q: Without simplifying the expression, use MATLAB to evaluate 5+6-3-√2 3 3.17 23

A: Given expression: Objective: Evaluate the given expression using MATLAB without simplifying it.

Q: Prove that if x, y € R, x > 0 and y > 0, then √√√x+y√√x + √y.

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Q: Use z = a + bi and w = c + di to show that z + w =7 + W.

A: Consider a complex number z=a+bi Then, the complex conjugate of above complex number is given by:…

Q: z = (a + bi)4n+ (b + ai) 4n; a ∈ ℝ ; b ∈ ℝ; n ∈ ℕ. Show that z is a real number.

A: Properties of complex number : If z is real number than z=z¯ . Suppose z=x+i y than z¯=x- iy .…

Q: 2. Let Z [√2] = { a+b√2 la,be Z}. Define addition and multiplication on Z[√2] as follows: (a+b√2)+(c…

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Q: Prove there are no natural numbers a and b such that √a+b = √a + √b.c

A: Assume such a,b exists and prove by contradiction

Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

A:

Q: Let u = (5,5,0) and v= = (5,5,5√√2). Compute |uxv|. Then sketch u, v, and uxv. |uxv= (Simplify your…

A: I have explained everything with in the solution. Please go through that.

Q: 1 Evaluate [ (x² + --- e2x +c 2 a) + In x+ b) 3 +C 3 e2x .3 c) + In x- d) + In x - 3 +c +c 2 2

A: We know  (1) integration of Xn dx =  Xn+1 / (n+1)  + C  (2) integration of emx dx = emx / m + C  (3)…

Q: Prove c (A + B) = cA + cB   (c + d)A = cA + dA   c(dA) = (cd)A  IA= A

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Q: Show that [o 2] A 3 0 has no A = L-U factorization where L is lower triangular and U is upper…

A:

Q: ply DeMorgan's Laws un - && (Ir || a))

A: Introduction: The formula for de morgan's law is given by, ~(a∧b)=~a∨~b~(a∨b)=~a∧~b

Q: 2. a| = 3, b|| V6, the angle between a and b is 45° 3. |a| = 6, |b|= 5, the angle between a and b is…

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Q: Show that is isomorphic to Z2.

A:

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: 4. Find the degrees of the following extensions. (a) Q(√3) over Q. (b) Q(√3, i) over Q(i). (c)…

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Q: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

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Q: Find the term independent of of in the expansi Ansion of √x 3 + 3 222 to

A: Given expression is x3+32x210. The aim is to find the term independent in the expansion of the above…

Q: Prove that

A:

Q: If a, b, c are in A.P., then prove that the following are also in A.P. (i) a²(b + c), b²(c + a),…

A: The given problem is to prove the given terms in A.P when the given a,b,c are in A.P.

Q: Suppose that P(A) = 0.1, P(B) = 0.3, and P(A|B) + P(B|A) = 0.8. Find P(A ∩ B).

A: From the information, given thatThe formula of the conditional probability, for any two events…

Q: True or false:(a) If A2 is defined then A is necessarily square.(b) If AB and BA are defined then A…

A: We know that a square matrix is a matrix which has equal number of rows and columns. n×n is called…

Q: 1. If x=27-3k and y = 7+57-4k, express each of the following in terms of i, j, k a) 5x-2y

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Q: Show that Q(√5, √7) = {a+b√5 +c√7 + d√√35 | a, b, c, d € Q} = Q(√5 + √7).

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Q: why do we assume a|b and prove that way instead of ac|bc and prove back to a|b

A: why do we assume a|b and prove that way instead of ac|bc and prove back to a|b .

Q: Show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

A: We have to show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

Q: Prove that {4a + 126 : a, b E Z} = {4c: c € Z}.

A:

Q: Q24 prove that b) 3i 3i √2+i √2-i = 2 ?

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Q: Which choice gives an example that supports the conjecture, and a counterexample that nows the…

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Q: A) |2x² + 5x + 1 = 1 B) 12/1 x-1 x+1 C) 5x + | ≥ 1

A:

Q: 2.2.12 Prove that if x and y positive real numbers, then Vx+y # Vx+ Vy. Argue by contradiction. are

A: Let x and y are positive real numbers. Also let x+y=x+y Now squaring on both side,we get…

Q: Prove that √2 is not an element of Q(?).

A: