let us assume that /3 is a rational number .show that V3 = where a,b are co – primes b

Q: deduce

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Q: Evaluate the definite integral.

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Q: Show that log, q is irrational for any distinct prime numbers p and q. (Distinct here p# q.) means

A: To Determine :-  logp q is irrational for any distinct prime numbers p and q.

Q: Consider the following number. 0.59595959.. Show that the number is rational by writing it as a…

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Q: Use induction onn to prove that, for all integers n > 1, we have cos(nº) sin(n0) - sin(n®) cos(n®).…

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Q: 32 n2と

A: We are going to solve

Q: 5. Determine whether the foll (a) sin (x²) x7/2 dax

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Q: is: The number of zeros of sin(-) is: `z-1

A: If y=f(z), then the variable z is called the input of the function, and y is called the output of…

Q: Prove that L (sin² x 2/3 ; p} = 1/ 108 р x (1²,5 + 4)

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Q: Find x such that 15777 x (mod 11). 0sx<11. You must show all the detailed steps.

A: (3) We have to find the value of x such that 15777≡xmod11, 0≤x≤11. Since, 15777=14×11+377 Hence,…

Q: Use the substitution formula to evaluate the integral. O 10 క్వాక్ - 5

A: The given integration ∫01x+4dx

Q: In the given question, verify the identity:-   cosh2 x =- 1 + cosh 2x/ 2

A: To verify: cosh2x=-1+cosh 2x2

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

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Q: Suppose n is an integer. So Show that for n+0, ſ x cos(nx)dx = { п n is even (a) 2 п is odd n2

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Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

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Q: Evaluate the definite integral 8 | (2x + 4)dr 2

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Q: *70. Verify the following statements. a. Iff (x) = sin x and n is a nonnegative integer, then %3D…

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Q: B- If x+ 2 cos0 prove that x" +2 cosro by used complex number.

A: we have to show that the following

Q: Evaluate the definite integral 7 | (4x + 2)dr

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Q: Use residue to evaluate the integral 3+ 2 cos 0 do 5 + 4 cos 0 0.

A: Given integral is, Let C denote the unit circle |z|=1. So,

Q: EVALUATE ap- x2 + 64 O x – 8tan + c 8 O none O x – tan -1 +c 8 1 uis2 – a 8 x - -sin +

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Q: Evaluate the integral : sin 60 do 2+ cos 60 using the substitution u = 2+ cos 60. sin 60 do = +C 2+…

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Q: Prove by induction that (cos 0+i sin 0)" cos n0 +i sin ne, for all negative integers n. Question 1.…

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Q: Find the number of pairs (m, n) of positive integers with 1 < m <n< 30 such that there exists a real…

A: Hope this will help you

Q: Let h(x) = tan(3x + 7). Then n'(5) is and h'"(5) is

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Q: Evaluate the definite integral. 10 -4 dr V4r + 9

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Q: use method of false position. to Selution acccurate to- within lo -3x°=0 for osxsI and 3€x<

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Q: Which is larger  2^100 or F100 with F representing a Fibonacci number? Prove which is larger.

A: Increment in each term in Fibonacci series = FnFn-1  Increment in each term in 2n series = 2

Q: (7 - t)/

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Q: Evaluate the integral. 9-sin 4x 4.

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Q: Evaluate the integral / sin 50 do 2+ cos 50 using the substitution u = 2+ cos 50. sin 50 do 2+ cos…

A: Recall: ∫1xdx=ln(x)+C

Q: xsinnx I= dx = 2 sinx

A: …

Q: © Find the singular value deomposition of the matix I- A =

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Q: Prove, that (nezond n73) If x"=5,+hen x is not a rational number.

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Q: Show that the following number is rational by writing it as a ratio of two integers. 2 +

A: Given                   29+25 So we know that rational number is in form of pqand where q is not…

Q: NIIVISIDIE D 4. Prove that v3 is not rational. 5 Provo

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Q: the pth term of AP is a and the qth term is p,prove that it's nth term is (p+q-n)

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Q: result by induction. 88. Prove that d2n d2n (cos x) (1)" sin x dr2n (sin x)= and (-1)" cos x. dr2n

A: a) By induction method, prove for n =1.

Q: 9. Show that 1729 is a Carmichael number. 10. Show that 2465 is a Carmichael number.

A: See the attachment

Q: So I need to keep the 4 and the cubed in the integral??

A: Yes, you have missed the "4" and the cube.

Q: Q4. Expand tan 0 in powers of [0 – ().

A: we have to expand tanθ in powers of θ-π4

Q: Which of these is a Subring of Rational numbers Q? Assume, m, n ∈ Z such that n ̸= 0 and m/n is in…

A: Both set forms subring of Rational number.

Q: Find the root (7 – 7i)/2 O +7/2[cos(x/4) – isin(x/4)] O +V98[cos(r/8) – isin(x/8)] O +7/2[cos(x/8) +…

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Q: Denote by fn the nth Fibonacci number. Show that when n is a positive integer, fofi + fif2 + +…

A: Given: fnis nth Fibonacci number. To show: f0f1 + f1f2+···+f2n-1f2n=f2n2

Q: Solve and prove fne Sum and produ cts OF 1he Roots OF X" -3x 12=O

A: