4. Let R be the set of all real numbers, xER and n E N such that n 2 2. If (1+x) > 0, prove that P, : (1+ x)" >1+ nx is true vnE N.

Q: 1. Let y be a positive real number. Prove that for every n e N there exists a unique positive real…

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Q: Prove that Σ (2n+1) P(x) no (1-2hx+h²) ¾/2₂

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Q: .2 is strictly monotone Prove by definition that the function b(x) = increasing on [0, ∞) and is…

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Q: Let X = {x E Z|x = Y = {y € Z]y = 3n for some integer n}. 6m for some integer m} and let %D a. Prove…

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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: 6) The Ying of integers(2) is ideal of veal number (R) an DTrue D Fulse

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Q: Find a counterexample to Lie's Theorem when k is not algebraically closed and chark = 0. %3D

A: According to Lie's theorem, for any finite-dimensional linear representation of a finite-dimensional…

Q: The set of integers is not dense in cofinite topo log y of R True False O

A: The set of integers is not dense in co finite topology on ℝ.

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

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Q: Let fca) satis fies Cauchy Riemann Eguation , then (f(z)" is also satis fies cauchy Riemann Eguation…

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

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Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r>d/2 where x, y, z belongs to Rn So locus z…

Q: Find the infimum and supremum of the set X = {3 – Vztn ine N}. -

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Q: 2. Let f : (0, –→R such that f(x) = x². Show that f(x) < r. • Show that f has no fixed point on (0,…

A: The given function is f:(0,13]→R and defined as f(x)=x2 . we have to show that f has no fixed point…

Q: 6. Suppose that f(x) = x" + an-1r*-1+ ...+ ao E Z[x]. Ifris rational and x –r divides f(x), prove…

A: We will use division algorithm to prove the fact.

Q: 63. Given n nonzero numbers o1,...,on, show that the minimum value of S(x1....,xn) = xfo7 +..+ x…

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Q: (2) Find all the numbers z such that elz = 10i.

A: Consider the given. eiz=10i

Q: Find a number t such that In(4t + 3) = 5 %3D

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Q: Show that if x ~ Nm (u, 2), where N is positive definite, then (x – u)'N-I (x – µ) ~ Xm: 2

A: Given  x~Nmμ,Ω, where Ω is positive definite.

Q: 13 3. Use the IVT to prove that y = r Inr has at least one zero on 2 2

A: Above function is continuous function in given interval. So we can apply IVT

Q: Find all the values of x in the interval [0,2pi], that satisfy the question.  10cos(x)+5sin(2x)=0

A: Given equation is Solve the given equation for x

Q: there exists a real number s>0 such that |2x-5|<.01 wherenever |x-2.5|<s

A: We can prove that there exists S such that given condition holds. Actually we find a particular…

Q: nd the supremum and infimum of the set D defined by D:= :neN

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Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

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Q: Let x be a real number. Prove that there is a positive integer n such that the distance between nx…

A: Let x be a real number  To prove there is a positive integer n such that the distanc between nx and…

Q: If u, v are given unequal numbers anu (un-1+ Pn-1), vn = V (u –1 Vn-1), where n2 2 , %3D Un %3D 2…

A: Given

Q: Q) prove that (1) -2P xSinx 2.

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

Q: 5. Find all pairs (m, n) of positive integers such that y(y(nm)) = n, where p is Euler's totient…

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Q: (It v5)" (1 Vsy (b) Prove that f. for all integers n> 1.

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Q: Find the absolute maximum and the absolute minimum of U(q) 20002 (15 5q)4 on the nterval 04.

A: First find the derivative of the given function and compute the critical points.

Q: Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for

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Q: 2 (x) 2. prove that s

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Q: Suppose that thế fühctlons are defined for all real numbers x as follows. r(x)=3x s (x) = 2x² Write…

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Q: (c) Prove that 10 is not separable.

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Q: (a)Show that,if x is small enough for its cube and higher powers to be 1 neglected, = 1-x +. by…

A: It is given that x is small enough for its cube and higher powers to be neglected. It is required to…

Q: 3 We can de fine the fallaing tunction on real mumbers ( called "absohite value") ; abs (x) = x if…

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

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Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: Q1. Prove that KT = = -.

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Q: The Lucas numbers L; are defined as follows: (1) Lo = 2, L = 1; and %3D (2) L, = L,-1+ L-, for n E…

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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 of k such that A is singular. 0 4 6 -1 0 4 2 k k =

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Q: 2) Let t be the nth triangular number. Prove that t +3 n + 6 is n triangular.

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Q: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

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Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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Q: Suppose that fo sf,:X-gy and g,-3 YoZ Prove that (Hint : First use ce) to show that gfe g,, and then…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.