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AlgebraQuestion and Answers: Page 237

Question Number 111208 Answers: 1 Comments: 3

let p a prime number s.t p≥7 and a=333......3_(p−1 times) Show that 11∣a.

$$\:\mathrm{let}\:{p}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{s}.\mathrm{t}\:{p}\geqslant\mathrm{7}\:\mathrm{and}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}=\underset{{p}−\mathrm{1}\:{times}} {\mathrm{333}......\mathrm{3}} \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\mathrm{11}\mid{a}. \\ $$

Question Number 111175 Answers: 1 Comments: 0

Question Number 111134 Answers: 0 Comments: 0

Question Number 111062 Answers: 0 Comments: 2

Question Number 111029 Answers: 1 Comments: 0

Question Number 111008 Answers: 0 Comments: 3

((√(x+1))/(y+2)) + ((√(y+2))/(x+1)) =1 => x=?

$$\frac{\sqrt{\boldsymbol{{x}}+\mathrm{1}}}{\boldsymbol{{y}}+\mathrm{2}}\:+\:\frac{\sqrt{\boldsymbol{{y}}+\mathrm{2}}}{\boldsymbol{{x}}+\mathrm{1}}\:=\mathrm{1}\:\:\:\:\:\:=>\:\:\boldsymbol{{x}}=? \\ $$

Question Number 110897 Answers: 3 Comments: 0

(1)4x−4 ≤ ∣x^2 −3x+2 ∣ find the solution set (2) ((1+cos ((α/2))−sin ((α/2)))/(1−cos ((α/2))−sin ((α/2))))=?

$$\left(\mathrm{1}\right)\mathrm{4x}−\mathrm{4}\:\leqslant\:\mid\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{2}\:\mid\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\: \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}+\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}{\mathrm{1}−\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}=? \\ $$

Question Number 110879 Answers: 0 Comments: 1

Question Number 110843 Answers: 2 Comments: 0

(x+1)^((x+1)) =(√2) find all values of x (Please step by step)

$$\left({x}+\mathrm{1}\right)^{\left({x}+\mathrm{1}\right)} =\sqrt{\mathrm{2}}\:\:\:\:\:\:{find}\:{all}\:{values}\:{of}\:{x} \\ $$$$\left({Please}\:{step}\:{by}\:{step}\right) \\ $$

Question Number 110837 Answers: 0 Comments: 0

Question Number 110799 Answers: 0 Comments: 0

What is the he minimum value of (√(x^2 +1))+(√((y−x)^2 +25))+(√((z−y)^2 +4))+(√((9−z)^2 +16)) if (x, y and z) ∈ R

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{What}\:{is}\:{the}\:{he}\:{minimum}\:{value}\:{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}+\sqrt{\left({y}−{x}\right)^{\mathrm{2}} +\mathrm{25}}+\sqrt{\left({z}−{y}\right)^{\mathrm{2}} +\mathrm{4}}+\sqrt{\left(\mathrm{9}−{z}\right)^{\mathrm{2}} +\mathrm{16}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{if}\:\:\left({x},\:{y}\:{and}\:{z}\right)\:\in\:\mathbb{R}\: \\ $$

Question Number 110845 Answers: 1 Comments: 0

(x^2 +3x−10)^(x^3 −9x) = (x^2 +3x−10)^(3x^2 −8x)

$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3x}−\mathrm{10}\right)^{\mathrm{x}^{\mathrm{3}} −\mathrm{9x}} \:=\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3x}−\mathrm{10}\right)^{\mathrm{3x}^{\mathrm{2}} −\mathrm{8x}} \\ $$

Question Number 110728 Answers: 1 Comments: 0

a+b=9 , ab=20 a−b=?

$${a}+{b}=\mathrm{9}\:\:,\:\:{ab}=\mathrm{20}\:\:\:\:\:{a}−{b}=? \\ $$

Question Number 110727 Answers: 1 Comments: 0

a^2 +b^2 =10 , ab=13 , a^3 +b^3 =?

$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{10}\:\:\:,\:\:{ab}=\mathrm{13}\:\:\:,\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =? \\ $$

Question Number 110706 Answers: 0 Comments: 0

Question Number 110675 Answers: 2 Comments: 0

Two polynomials P and Q satisfy P(−2x+Q(x))=Q(−2x+P(x)). Given that Q(x)=x^2 −4 and P(x)=ax+b. Find 2a+b.

$$\mathrm{Two}\:\mathrm{polynomials}\:\mathrm{P}\:\mathrm{and}\:\mathrm{Q}\:\mathrm{satisfy} \\ $$$$\mathrm{P}\left(−\mathrm{2x}+\mathrm{Q}\left(\mathrm{x}\right)\right)=\mathrm{Q}\left(−\mathrm{2x}+\mathrm{P}\left(\mathrm{x}\right)\right). \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{Q}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} −\mathrm{4}\:\mathrm{and} \\ $$$$\mathrm{P}\left(\mathrm{x}\right)=\mathrm{ax}+\mathrm{b}.\:\mathrm{Find}\:\mathrm{2a}+\mathrm{b}. \\ $$

Question Number 110620 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 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 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?

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

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