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Question Number 110620 Answers: 1 Comments: 0

Question Number 110619 Answers: 0 Comments: 2

someone posted this problem and my solution followed solve ∫_0 ^1 ((ln(x+(√(1+x^2 ))))/(√(1+x^2 )))dx solution y=ln(x+(√(1+x^2 ))) and dy=(1/(√(1+x^2 )))dx at x=0,y=0 and at x=1,y=ln(1+(√2)) hence I=∫_0 ^(ln(1+(√2))) (y/(√(1+x^2 )))×(√(1+x^2 ))dy=∫_0 ^(ln(1+(√2))) ydy I=[(y^2 /2)]_0 ^(ln(1+(√2))) =(1/2)ln^2 (1+(√2)) ∵∫_0 ^1 ((ln(x+(√(1+x^2 ))))/(√(1+x^2 )))dx=(1/2)ln^2 (1+(√2))

Question Number 110616 Answers: 1 Comments: 0

lim_(n→∞) ((e^(1/n) +e^(2/n) +......+e^(n/n) )/n) ho is can you help me step by step

$${lim}_{{n}\rightarrow\infty} \frac{{e}^{\frac{\mathrm{1}}{{n}}} +{e}^{\frac{\mathrm{2}}{{n}}} +......+{e}^{\frac{{n}}{{n}}} }{{n}}\: \\ $$$${ho}\:{is}\:{can}\:{you}\:{help}\:{me}\:{step}\:{by}\:{step} \\ $$

Question Number 110614 Answers: 1 Comments: 0

Question Number 110608 Answers: 0 Comments: 1

(√x) +1=0 Solution (√(x )) = −1 recalled that ϱ^(iΠ) = −1 (√x) =ϱ^(iΠ) x=(ϱ^(iΠ) )^2 ⇒ x=ϱ^(2iΠ) or x= 2cosΠ+isinΠ x has no real value!

$$\sqrt{{x}}\:+\mathrm{1}=\mathrm{0} \\ $$$${Solution} \\ $$$$\sqrt{{x}\:}\:=\:−\mathrm{1} \\ $$$${recalled}\:{that}\:\varrho^{{i}\Pi} =\:−\mathrm{1} \\ $$$$\sqrt{{x}}\:=\varrho^{{i}\Pi} \\ $$$${x}=\left(\varrho^{{i}\Pi} \right)^{\mathrm{2}} \\ $$$$\Rightarrow\:{x}=\varrho^{\mathrm{2}{i}\Pi} \\ $$$${or}\:{x}=\:\mathrm{2}{cos}\Pi+{isin}\Pi \\ $$$${x}\:{has}\:{no}\:{real}\:{value}! \\ $$

Question Number 110704 Answers: 3 Comments: 0

Find the minimum value of ((n^2 +1)/n)+(n/(n^2 +1)),n>0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}{\mathrm{n}}+\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}},\mathrm{n}>\mathrm{0} \\ $$

Question Number 110596 Answers: 1 Comments: 2

Three real numbers a,b,c satisfying ab+c=10,bc+a=11,ca+b=14. Find (a−b)(b−c)(c−a)(a−1)(b−1)(c−1)

$$\mathrm{Three}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{satisfying} \\ $$$$\mathrm{ab}+\mathrm{c}=\mathrm{10},\mathrm{bc}+\mathrm{a}=\mathrm{11},\mathrm{ca}+\mathrm{b}=\mathrm{14}.\:\mathrm{Find} \\ $$$$\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{b}−\mathrm{c}\right)\left(\mathrm{c}−\mathrm{a}\right)\left(\mathrm{a}−\mathrm{1}\right)\left(\mathrm{b}−\mathrm{1}\right)\left(\mathrm{c}−\mathrm{1}\right) \\ $$

Question Number 110595 Answers: 0 Comments: 5

Evaluate 5!•6!(mod 7!)

$$\mathrm{Evaluate}\:\mathrm{5}!\bullet\mathrm{6}!\left(\mathrm{mod}\:\mathrm{7}!\right) \\ $$

Question Number 110594 Answers: 1 Comments: 0

A polynomial satisfies f(x^2 −2)=f(x)f(−x). Assuming that f(x)≠0 for −2≤x≤2, what is the value of f(−2)+f(1)?

$$\mathrm{A}\:\mathrm{polynomial}\:\mathrm{satisfies} \\ $$$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2}\right)=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(−\mathrm{x}\right).\:\mathrm{Assuming}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\neq\mathrm{0}\:\mathrm{for}\:−\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{2},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{f}\left(−\mathrm{2}\right)+\mathrm{f}\left(\mathrm{1}\right)? \\ $$

Question Number 110593 Answers: 1 Comments: 0

Find the maximum possible integer n such that (((n−1)(n^2 +n−3))/(n^2 +4)) is also an integer

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{integer}\:\mathrm{n} \\ $$$$\mathrm{such}\:\mathrm{that}\:\frac{\left(\mathrm{n}−\mathrm{1}\right)\left(\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{3}\right)}{\mathrm{n}^{\mathrm{2}} +\mathrm{4}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{an} \\ $$$$\mathrm{integer} \\ $$

Question Number 110592 Answers: 3 Comments: 0

How many pairs of integers x and y satisfy the equation (1/x)+(1/y)=(1/(32))

$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{32}} \\ $$

Question Number 110591 Answers: 3 Comments: 0

Find the sum of all positive two−digit integers that are divisible by each of their digits.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive} \\ $$$$\mathrm{two}−\mathrm{digit}\:\mathrm{integers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{each}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}. \\ $$

Question Number 110673 Answers: 2 Comments: 0

Let k be a real number such that the inequality (√(x−3))+(√(6−x))≥k has a solution. Find the maximum value of k.

$$ \\ $$$$\mathrm{Let}\:\mathrm{k}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{inequality}\:\sqrt{\mathrm{x}−\mathrm{3}}+\sqrt{\mathrm{6}−\mathrm{x}}\geqslant\mathrm{k}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{solution}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{k}. \\ $$

Question Number 110587 Answers: 1 Comments: 0

A palindrome is a number that remains the same when its numbers are reversed. The number n and n+192 are three−digit and four−digit palindromes respectively. What is the sum of the digits of m? Can this be solved mathematically?

$$\mathrm{A}\:\mathrm{palindrome}\:\mathrm{is}\:\mathrm{a}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{remains}\:\mathrm{the}\:\mathrm{same}\:\mathrm{when}\:\mathrm{its}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{reversed}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{n}\:\mathrm{and} \\ $$$$\mathrm{n}+\mathrm{192}\:\mathrm{are}\:\mathrm{three}−\mathrm{digit}\:\mathrm{and} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{palindromes}\:\mathrm{respectively}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{m}?\: \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{mathematically}? \\ $$

Question Number 110586 Answers: 1 Comments: 0

The length of the interval of solution a≤2x+5≤b is 15. What is b−a?

$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{solution} \\ $$$$\mathrm{a}\leqslant\mathrm{2x}+\mathrm{5}\leqslant\mathrm{b}\:\mathrm{is}\:\mathrm{15}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{b}−\mathrm{a}? \\ $$

Question Number 110725 Answers: 2 Comments: 0

x^x^x^x^(...) =2 x=?

$${x}^{{x}^{{x}^{{x}^{...} } } } =\mathrm{2}\:\:\:\:\:\:\:{x}=? \\ $$

Question Number 110562 Answers: 2 Comments: 0

Let x and y be integers such that xy≠1, x^2 ≠y and y^2 ≠x. (i) Show that p∣xy−1 and p∣x^2 −y then p∣y^2 −x where p is a prime. (ii) Let p be a prime. Suppose that p∣x^2 −y and p∣y^2 −x, must p∣xy−1? [If yes, then prove it. If no, then give a counter example]

$$\mathrm{Let}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{be}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{xy}\neq\mathrm{1},\:\mathrm{x}^{\mathrm{2}} \neq\mathrm{y}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} \neq\mathrm{x}. \\ $$$$ \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{p}\mid\mathrm{xy}−\mathrm{1}\:\mathrm{and}\:\mathrm{p}\mid\mathrm{x}^{\mathrm{2}} −\mathrm{y} \\ $$$$\mathrm{then}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x}\:\mathrm{where}\:\mathrm{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{a}\:\mathrm{prime}.\:\mathrm{Suppose}\:\mathrm{that} \\ $$$$\mathrm{p}\mid\mathrm{x}^{\mathrm{2}} −\mathrm{y}\:\mathrm{and}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x},\:\mathrm{must}\:\mathrm{p}\mid\mathrm{xy}−\mathrm{1}? \\ $$$$ \\ $$$$\left[\mathrm{If}\:\mathrm{yes},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{it}.\:\mathrm{If}\:\mathrm{no},\:\mathrm{then}\:\mathrm{give}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{counter}\:\mathrm{example}\right] \\ $$

Question Number 110565 Answers: 0 Comments: 15

Let n∈N. Using the formula lcm(a,b) = ((ab)/(gcd(a,b))) and lcm(a,b,c) =lcm(lcm(a,b),c), find all the possible value of ((6•lcm(n,n+1,n+2,n+3))/(n(n+1)(n+2)(n+3)))

$$\mathrm{Let}\:\mathrm{n}\in\mathbb{N}.\:\mathrm{Using}\:\mathrm{the}\:\mathrm{formula}\:\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right) \\ $$$$=\:\frac{\mathrm{ab}}{\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right)}\:\mathrm{and}\:\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ $$$$=\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right),\:\mathrm{find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{6}\bullet\mathrm{lcm}\left(\mathrm{n},\mathrm{n}+\mathrm{1},\mathrm{n}+\mathrm{2},\mathrm{n}+\mathrm{3}\right)}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)\left(\mathrm{n}+\mathrm{3}\right)} \\ $$

Question Number 110688 Answers: 0 Comments: 0

Let a,b and c be positive integers such that ab+1∣bc+1 and bc+1∣ca+1. Show that ab+1 is the sum of two squares.

$$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$

Question Number 110551 Answers: 2 Comments: 0