10 probs: If f(n)=sum[r=1,n](1/r), find f(2^n - 1); If 4-digit numb. chosen at random

[gautamthapa]

\(\displaystyle \mbox{1. If }\, \displaystyle \, f(n)\, =\, \sum_{r = 1}^n\, \dfrac{1}{r},\,\)\(\displaystyle \mbox{ then }\, f\left(2^n\, -\, 1\right)\,\mbox{ lies between:}\)

. . . . .\(\displaystyle \mbox{a) }\, \dfrac{1}{2}\, \mbox{ and }\, 2\). . .\(\displaystyle \mbox{b) }\, \dfrac{1}{n}\, \mbox{ and }\, \dfrac{2}{n}\). . .\(\displaystyle \mbox{c) }\, \dfrac{n}{2}\, \mbox{ and }\, n\). . .\(\displaystyle \mbox{d) }\, \dfrac{1}{2^n}\, \mbox{ and }\, \dfrac{1}{2^{n-1}}\)

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\(\displaystyle \mbox{2. If a four-digit number is chosen at random, the probability}\)

\(\displaystyle \mbox{that is contains not more than two different digits is:}\)

. . . . .\(\displaystyle \mbox{a) }\, \dfrac{13}{200}\). . .\(\displaystyle \mbox{b) }\, \dfrac{21}{373}\). . .\(\displaystyle \mbox{c) }\, \dfrac{1}{125}\). . .\(\displaystyle \mbox{d) }\, \dfrac{23}{9000}\)

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\(\displaystyle \mbox{3. If }\, U_n\, =\, x^{2n}\, -\, x^n\, +\, 1,\, \mbox{ then the integer }\, n\, \mbox{ such that }\, U_n\, \)

\(\displaystyle \mbox{divisible by }\, U_1\, \mbox{ should be of the form:}\)

. . . . .\(\displaystyle \mbox{a) }\, 3m\, \pm\, 1,\, m\, \in\, \mathbb{N}\). . .\(\displaystyle \mbox{b) }\, 6m\, \pm\, 1,\, m\, \in\, \mathbb{N}\)

. . . . .\(\displaystyle \mbox{c) }\, m\, \pm\, 1,\, m\, in\, \mathbb{N}\). . . ..\(\displaystyle \mbox{d) }\, 4m\, \pm\, 1,\, m\, \in\, \mathbb{N}\)

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\(\displaystyle \mbox{4. If }\, \alpha\, \mbox{ and }\, \beta\, \mbox{ are two distinct roots of }\, x^2\, +\, 2\,\left(\lambda\, -\, 3\right)\, x\, +\, 9\, =\, 0,\)

\(\displaystyle \mbox{find the real values of }\, \lambda\, \mbox{ such that }\, \alpha,\, \beta\, \in\, (-6,\, 1)\, \mbox{ are:}\)

. . . . .\(\displaystyle \mbox{a) }\,\lambda\, \in\, (6,\, \infty)\). . .\(\displaystyle \mbox{b) }\, \lambda\, \in\, [-6,\, 1)\). . .\(\displaystyle \mbox{c) }\, \left(6,\, \dfrac{13}{2}\right)\). . .\(\displaystyle \mbox{d) }\, \left(6,\, \dfrac{27}{4}\right)\)

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\(\displaystyle \mbox{5. The area of a tria}\mbox{ngle whose vertices are the incenter, centroid, and }\)

\(\displaystyle \mbox{circumcenter of a tria}\mbox{ngle with sides }\, 14,\, 50,\, \mbox{ and }\, 48\, \mbox{ is:}\)

. . . . .\(\displaystyle \mbox{a) }\, 15\, \mbox{ sq. units}\). . .\(\displaystyle \mbox{b) }\, 16\, \mbox{ sq. units}\). . .\(\displaystyle \mbox{c) }\, 17\, \mbox{ sq. units}\). . .\(\displaystyle \mbox{d) }\, 18\, \mbox{ sq. units}\)

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\(\displaystyle \mbox{6. If }\, \)\(\displaystyle \large{ Z^{(i)^{(i)^{(i)}}} }\)\(\displaystyle ,\, \mbox{ where }\, i\, =\, \sqrt{\strut -1\,},\, \mbox{ then }\, \Big|\, Z\, \Big|\, =\)

. . . . .\(\displaystyle \mbox{a) }\, 1\). . .\(\displaystyle \mbox{b) }\, e^{-\pi/2}\). . .\(\displaystyle \mbox{c) }\, \dfrac{\pi^2}{4}\). . .\(\displaystyle \mbox{d) }\, (-1)^{\,e^{-\pi/2}}\)

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Paragraph (For Problems 7 to 10)

Let ABC be a triangle with inradius r and circumradius R. Its incircle touches the sides BC, CA, and AB and A₁, B₁, and C₁, respectively. The incircle of triangle A₁B₁C₁ touches its sides at A₂, B₂, and C₂, respectively, and so on.

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\(\displaystyle \mbox{7. In triangle }\, A_4 B_4 C_4,\, \mbox{ the value of }\, A_4\, =\)

. . . . .\(\displaystyle \mbox{a) }\, \dfrac{3\pi\, +\, A}{8}\). . .\(\displaystyle \mbox{b) }\, \dfrac{5\pi \, -\, A}{8}\). . .\(\displaystyle \mbox{c) }\, \dfrac{5 \pi \, -\, A}{16}\). . .\(\displaystyle \mbox{d) }\, \dfrac{5 \pi \, +\, A}{16}\)

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\(\displaystyle \mbox{8. }\, \)\(\displaystyle \displaystyle \lim_{n\, \rightarrow \, \infty}\, A_n\, =\)

. . . . .\(\displaystyle \mbox{a) }\, 0\). . .\(\displaystyle \mbox{b) }\, \dfrac{\pi}{6}\). . .\(\displaystyle \mbox{c) }\, \dfrac{\pi}{4}\). . .\(\displaystyle \mbox{d) }\, \dfrac{\pi}{3}\)

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\(\displaystyle \mbox{9. }\, \dfrac{B_1 C_1}{BC}\, =\)

. . . . .\(\displaystyle \mbox{a) }\, \sin\left(\dfrac{B}{2}\right)\, \sin\left(\dfrac{C}{2}\right)\). . . . . . . . . ..\(\displaystyle \mbox{b) }\,\cos\left(\dfrac{B}{2}\right)\, \cos\left(\dfrac{C}{2}\right)\)

. . . . .\(\displaystyle \mbox{c) }\, \cos\left(\dfrac{B\, -\, C}{2}\right)\, -\, \sin\left(\dfrac{A}{2}\right)\). . . . .\(\displaystyle \mbox{d) }\, \sin\left(\dfrac{A}{2}\right)\, +\, \cos\left(\dfrac{B\, -\, C}{2}\right)\)

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\(\displaystyle \mbox{10. }\, \dfrac{\Delta A_1 B_1 C_1}{\Delta ABC}\, =\, \lambda,\, \dfrac{r}{R},\, \mbox{ where }\, \lambda\, =\)

. . . . .\(\displaystyle \mbox{a) }\, 1\). . .\(\displaystyle \mbox{b) }\, \dfrac{1}{2}\). . .\(\displaystyle \mbox{c) }\, \dfrac{1}{3}\). . .\(\displaystyle \mbox{d) }\, \dfrac{1}{4}\)

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[stapel]

\(\displaystyle \mbox{1. If }\, \displaystyle \, f(n)\, =\, \sum_{r = 1}^n\, \dfrac{1}{r},\,\)\(\displaystyle \mbox{ then }\, f\left(2^n\, -\, 1\right)\,\mbox{ lies between:}\)

. . . . .\(\displaystyle \mbox{a) }\, \dfrac{1}{2}\, \mbox{ and }\, 2\). . .\(\displaystyle \mbox{b) }\, \dfrac{1}{n}\, \mbox{ and }\, \dfrac{2}{n}\). . .\(\displaystyle \mbox{c) }\, \dfrac{n}{2}\, \mbox{ and }\, n\). . .\(\displaystyle \mbox{d) }\, \dfrac{1}{2^n}\, \mbox{ and }\, \dfrac{1}{2^{n-1}}\)

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\(\displaystyle \mbox{2. If a four-digit number is chosen at random, the probability}\)

\(\displaystyle \mbox{that is contains not more than two different digits is:}\)

. . . . .\(\displaystyle \mbox{a) }\, \dfrac{13}{200}\). . .\(\displaystyle \mbox{b) }\, \dfrac{21}{373}\). . .\(\displaystyle \mbox{c) }\, \dfrac{1}{125}\). . .\(\displaystyle \mbox{d) }\, \dfrac{23}{9000}\)

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\(\displaystyle \mbox{3. If }\, U_n\, =\, x^{2n}\, -\, x^n\, +\, 1,\, \mbox{ then the integer }\, n\, \mbox{ such that }\, U_n\, \)

\(\displaystyle \mbox{divisible by }\, U_1\, \mbox{ should be of the form:}\)

. . . . .\(\displaystyle \mbox{a) }\, 3m\, \pm\, 1,\, m\, \in\, \mathbb{N}\). . .\(\displaystyle \mbox{b) }\, 6m\, \pm\, 1,\, m\, \in\, \mathbb{N}\)

. . . . .\(\displaystyle \mbox{c) }\, m\, \pm\, 1,\, m\, in\, \mathbb{N}\). . . ..\(\displaystyle \mbox{d) }\, 4m\, \pm\, 1,\, m\, \in\, \mathbb{N}\)

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\(\displaystyle \mbox{4. If }\, \alpha\, \mbox{ and }\, \beta\, \mbox{ are two distinct roots of }\, x^2\, +\, 2\,\left(\lambda\, -\, 3\right)\, x\, +\, 9\, =\, 0,\)

\(\displaystyle \mbox{find the real values of }\, \lambda\, \mbox{ such that }\, \alpha,\, \beta\, \in\, (-6,\, 1)\, \mbox{ are:}\)

. . . . .\(\displaystyle \mbox{a) }\,\lambda\, \in\, (6,\, \infty)\). . .\(\displaystyle \mbox{b) }\, \lambda\, \in\, [-6,\, 1)\). . .\(\displaystyle \mbox{c) }\, \left(6,\, \dfrac{13}{2}\right)\). . .\(\displaystyle \mbox{d) }\, \left(6,\, \dfrac{27}{4}\right)\)

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\(\displaystyle \mbox{5. The area of a tria}\mbox{ngle whose vertices are the incenter, centroid, and }\)

\(\displaystyle \mbox{circumcenter of a tria}\mbox{ngle with sides }\, 14,\, 50,\, \mbox{ and }\, 48\, \mbox{ is:}\)

. . . . .\(\displaystyle \mbox{a) }\, 15\, \mbox{ sq. units}\). . .\(\displaystyle \mbox{b) }\, 16\, \mbox{ sq. units}\). . .\(\displaystyle \mbox{c) }\, 17\, \mbox{ sq. units}\). . .\(\displaystyle \mbox{d) }\, 18\, \mbox{ sq. units}\)

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\(\displaystyle \mbox{6. If }\, \)\(\displaystyle \large{ Z^{(i)^{(i)^{(i)}}} }\)\(\displaystyle ,\, \mbox{ where }\, i\, =\, \sqrt{\strut -1\,},\, \mbox{ then }\, \Big|\, Z\, \Big|\, =\)

. . . . .\(\displaystyle \mbox{a) }\, 1\). . .\(\displaystyle \mbox{b) }\, e^{-\pi/2}\). . .\(\displaystyle \mbox{c) }\, \dfrac{\pi^2}{4}\). . .\(\displaystyle \mbox{d) }\, (-1)^{\,e^{-\pi/2}}\)

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Paragraph (For Problems 7 to 10)

Let ABC be a triangle with inradius r and circumradius R. Its incircle touches the sides BC, CA, and AB and A₁, B₁, and C₁, respectively. The incircle of triangle A₁B₁C₁ touches its sides at A₂, B₂, and C₂, respectively, and so on.

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\(\displaystyle \mbox{7. In triangle }\, A_4 B_4 C_4,\, \mbox{ the value of }\, A_4\, =\)

. . . . .\(\displaystyle \mbox{a) }\, \dfrac{3\pi\, +\, A}{8}\). . .\(\displaystyle \mbox{b) }\, \dfrac{5\pi \, -\, A}{8}\). . .\(\displaystyle \mbox{c) }\, \dfrac{5 \pi \, -\, A}{16}\). . .\(\displaystyle \mbox{d) }\, \dfrac{5 \pi \, +\, A}{16}\)

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\(\displaystyle \mbox{8. }\, \)\(\displaystyle \displaystyle \lim_{n\, \rightarrow \, \infty}\, A_n\, =\)

. . . . .\(\displaystyle \mbox{a) }\, 0\). . .\(\displaystyle \mbox{b) }\, \dfrac{\pi}{6}\). . .\(\displaystyle \mbox{c) }\, \dfrac{\pi}{4}\). . .\(\displaystyle \mbox{d) }\, \dfrac{\pi}{3}\)

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\(\displaystyle \mbox{9. }\, \dfrac{B_1 C_1}{BC}\, =\)

. . . . .\(\displaystyle \mbox{a) }\, \sin\left(\dfrac{B}{2}\right)\, \sin\left(\dfrac{C}{2}\right)\). . . . . . . . . ..\(\displaystyle \mbox{b) }\,\cos\left(\dfrac{B}{2}\right)\, \cos\left(\dfrac{C}{2}\right)\)

. . . . .\(\displaystyle \mbox{c) }\, \cos\left(\dfrac{B\, -\, C}{2}\right)\, -\, \sin\left(\dfrac{A}{2}\right)\). . . . .\(\displaystyle \mbox{d) }\, \sin\left(\dfrac{A}{2}\right)\, +\, \cos\left(\dfrac{B\, -\, C}{2}\right)\)

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\(\displaystyle \mbox{10. }\, \dfrac{\Delta A_1 B_1 C_1}{\Delta ABC}\, =\, \lambda,\, \dfrac{r}{R},\, \mbox{ where }\, \lambda\, =\)

. . . . .\(\displaystyle \mbox{a) }\, 1\). . .\(\displaystyle \mbox{b) }\, \dfrac{1}{2}\). . .\(\displaystyle \mbox{c) }\, \dfrac{1}{3}\). . .\(\displaystyle \mbox{d) }\, \dfrac{1}{4}\)

What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

 

[gautamthapa]

What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

actually i have been given a lot more to do ...thats i did not got somuch time tosolve these all...ihope u cansolve them....

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[gautamthapa]

What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

plzz i want solution to these problems...

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[gautamthapa]

1. A root of unity is a complex number that is a solution to the equation z ⁿ = 1 for some positive integer n. The number of roots of unity that are also the roots of the equation z ² + az + b = 0, for some integers a and b, is:

. . . . .(A) 6. . .(B) 8. . .(C) 9. . .(D) 10

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2. If z is a complex number such that z + (1/z) = 2 cos(3 degrees), then the value of z ²⁰⁰⁰ + (1/z ²⁰⁰⁰) + 1 is equal to:

. . . . .(A) 0. . .(B) -1. . .(C) sqrt(3) + 1. . .(D) 1 - sqrt(3)

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can u guys solve the first two questions for me

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[stapel]

i did not got so much time to solve these all.

You don't have enough time to do any work on any of the entire assignment, but you have enough time to wait days for other people to do them for you...? That doesn't make much sense, now, does it?

Please re-read the "Read Before Posting" announcement, and then reply with a complete listing of your thoughts and efforts on each exercise with which you are still experiencing difficulty. Thank you!

 

[gautamthapa]

You don't have enough time to do any work on any of the entire assignment, but you have enough time to wait days for other people to do them for you...? That doesn't make much sense, now, does it?

Please re-read the "Read Before Posting" announcement, and then reply with a complete listing of your thoughts and efforts on each exercise with which you are still experiencing difficulty. Thank you!

i was new to these forums but seems like u guys dont want to help me...thats not good...2 problems i posted but still no one answered...forget about the attachments but what about the text which i posted?


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[Harry_the_cat]

plzz i want solution to these problems...

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What...and then you'll hand in the assignment as your own work! How can you think it's ok to do that????

By the way, if you guess all 10 answers you'll have a 0.079 chance of getting more than 5 right! Odds aren't in your favour. Time to do some work buddy!

 

[gautamthapa]

What...and then you'll hand in the assignment as your own work! How can you think it's ok to do that????

By the way, if you guess all 10 answers you'll have a 0.079 chance of getting more than 5 right! Odds aren't in your favour. Time to do some work buddy!

ok now there i knw that no one here is to help you...
i will not post my homework...here...
actually that was not a homework just some difficult questions that i was not able to do...

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[Harry_the_cat]

ok now there i knw that no one here is to help you...
i will not post my homework...here...
actually that was not a homework just some difficult questions that i was not able to do...

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We're more than willing to help you understand but just need to see some effort on your part!

 

[gautamthapa]

We're more than willing to help you understand but just need to see some effort on your part!

i have done what i can...

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[Deleted member 4993]

i have done what i can...

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But you have shown nothing!!

It seems that you cannot even start the problem. So we will need to make sure you know the definitions:

1. A root of unity is a complex number that is a solution to the equation z ⁿ = 1 for some positive integer n. The number of roots of unity that are also the roots of the equation z ² + az + b = 0, for some integers a and b, is:

. . . . .(A) 6. . .(B) 8. . .(C) 9. . .(D) 10

What is the definition of "root of unity"?

What the definition of "solution to the equation z ⁿ = 1 "?

Please answer those - so that we know where should we begin to help you.

 

[gautamthapa]

i am currently studying in class 11 and with knowledge that i have i can say that...
when x^n=1 = cos0 +isin0 =cos2kpi + i sin 2k pi

so the eqn like x^n =1 has nth roots of unity...correct me if i am wrong...
sum of roots of unity is 0...
product of all roots of unity is (-1)^n-1
and i knw little bit about geometry too but not too deep...
i am a beginner...



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[Deleted member 4993]

i am currently studying in class 11 and with knowledge that i have i can say that...
when x^n=1 = cos0 +isin0 =cos2kpi + i sin 2k pi

so the eqn like x^n =1 has nth roots of unity...correct me if i am wrong...
sum of roots of unity is 0...
product of all roots of unity is (-1)^n-1
and i knw little bit about geometry too but not too deep...
i am a beginner...

1. A root of unity is a complex number that is a solution to the equation z ⁿ = 1 for some positive integer n. The number of roots of unity that are also the roots of the equation z ² + az + b = 0, for some integers a and b, is:

. . . . .(A) 6. . .(B) 8. . .(C) 9. . .(D) 10

Now tell us:
what are the roots of z ² + az + b = 0