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AlgebraQuestion and Answers: Page 231

Question Number 120943    Answers: 3   Comments: 1

∫(x^3 +3)^4 dx=?

$$\int\left({x}^{\mathrm{3}} +\mathrm{3}\right)^{\mathrm{4}} {dx}=? \\ $$

Question Number 120929    Answers: 5   Comments: 2

Question Number 120904    Answers: 1   Comments: 1

Question Number 120882    Answers: 1   Comments: 0

Question Number 120855    Answers: 2   Comments: 0

... elementary calculus... :: α,β are roots of equation of : x^2 −6x−2=0 define :: t_n =α^n −β^n (n≥1) then evaluate : A=((t_(10) −2t_8 )/(2t_9 )) =??? ...m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:...\:{elementary}\:\:{calculus}... \\ $$$$\:\:::\:\alpha,\beta\:{are}\:{roots}\:{of}\:\:{equation} \\ $$$$\:\:\:\:\:{of}\::\:{x}^{\mathrm{2}} −\mathrm{6}{x}−\mathrm{2}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:{define}\:::\:{t}_{{n}} =\alpha^{{n}} −\beta^{{n}} \:\left({n}\geqslant\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{then}\:\:{evaluate}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{A}=\frac{{t}_{\mathrm{10}} −\mathrm{2}{t}_{\mathrm{8}} }{\mathrm{2}{t}_{\mathrm{9}} }\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.\mathrm{1970}... \\ $$

Question Number 120839    Answers: 3   Comments: 0

Question Number 120809    Answers: 0   Comments: 0

Find the largest number of positive integers that can be found in such a way that any two of them a and b ( a≠b) satisfy the next inequality ∣a−b∣≥((ab)/(100))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{found}\:\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way} \\ $$$$\mathrm{that}\:\mathrm{any}\:\mathrm{two}\:\mathrm{of}\:\mathrm{them}\:{a}\:\mathrm{and}\:{b}\:\left(\:{a}\neq{b}\right)\: \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{next}\:\mathrm{inequality}\:\mid{a}−{b}\mid\geqslant\frac{{ab}}{\mathrm{100}} \\ $$

Question Number 120780    Answers: 2   Comments: 2

solve x^2^x =−(1/2)

$${solve}\:{x}^{\mathrm{2}^{{x}} } =−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 120711    Answers: 3   Comments: 0

show that 1+2+3+4...=((−1)/8)

$${show}\:{that}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}...=\frac{−\mathrm{1}}{\mathrm{8}} \\ $$

Question Number 120636    Answers: 0   Comments: 23

selective Binomial

$${selective}\:{Binomial} \\ $$

Question Number 120601    Answers: 1   Comments: 0

Find the range of values of x for which the expansion of the binomial (2 − 3x)^(−4) is valid. I need help with explanation please

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{for}\: \\ $$$$\:\mathrm{which}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\:\left(\mathrm{2}\:−\:\mathrm{3}{x}\right)^{−\mathrm{4}} \:\mathrm{is}\:\mathrm{valid}.\: \\ $$$$\:{I}\:{need}\:{help}\:{with}\:{explanation}\:{please} \\ $$

Question Number 120583    Answers: 0   Comments: 0

Question Number 120558    Answers: 3   Comments: 0

Question Number 120556    Answers: 0   Comments: 2

Question Number 120545    Answers: 1   Comments: 2

((√x))^(x/( (√x))) = (1/( (√2))) x=?

$$\left(\sqrt{\boldsymbol{{x}}}\right)^{\frac{\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{x}}}}} \:=\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\boldsymbol{{x}}=? \\ $$

Question Number 120529    Answers: 1   Comments: 1

Question Number 120480    Answers: 4   Comments: 0

show that ∀ n ∈N^∗ Σ_(k=1) ^n k(n−k)=(((n−1)(n+1))/6)

$${show}\:{that}\:\forall\:{n}\:\in\mathbb{N}^{\ast} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left({n}−{k}\right)=\frac{\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$

Question Number 120475    Answers: 1   Comments: 0

Question Number 120472    Answers: 0   Comments: 0

A=5x23^(−) ^6 show that A≡x−4[7] deduct the value of x for which A is divisible by 7

$${A}=\overline {\mathrm{5}{x}\mathrm{23}}\:^{\mathrm{6}} \:{show}\:{that}\:{A}\equiv{x}−\mathrm{4}\left[\mathrm{7}\right] \\ $$$${deduct}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{A}\:{is}\: \\ $$$${divisible}\:{by}\:\mathrm{7} \\ $$

Question Number 120471    Answers: 1   Comments: 0

solve in Z x^3 +2x+1≡1[4]

$${solve}\:{in}\:\mathbb{Z}\:{x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{1}\equiv\mathrm{1}\left[\mathrm{4}\right] \\ $$

Question Number 120470    Answers: 0   Comments: 0

solve in function of n: 2^n ≡x−4[3] n∈N

$${solve}\:{in}\:{function}\:{of}\:{n}: \\ $$$$\mathrm{2}^{{n}} \equiv{x}−\mathrm{4}\left[\mathrm{3}\right] \\ $$$${n}\in\mathbb{N} \\ $$

Question Number 120469    Answers: 1   Comments: 0

c alculate the rest of the division of 2^n by 3 ; n ∈ N

$${c}\:{alculate}\:{the}\:{rest}\:{of}\:{the}\:{division}\:{of} \\ $$$$\mathrm{2}^{{n}} \:{by}\:\mathrm{3}\:;\:{n}\:\in\:\mathbb{N} \\ $$

Question Number 120464    Answers: 1   Comments: 0

Question Number 120455    Answers: 0   Comments: 0

Question Number 120355    Answers: 3   Comments: 2

Question Number 120325    Answers: 0   Comments: 6

Let f:R→R be a function satisfying the functional relation (f(x))^y +(f(y))^x =2f(xy) for all x, y ∈R and it is given that f(1)=1/2. Answer the following questions. (i) f(x+y)= (A) f(x)+f(y) (B) f(x)f(y) (C) f(x^y y^x ) (D) ((f(x))/(f(y))) (ii) f(xy)= (A) f(x)f(y) (B) f(x)+f(y) (C) (f(x))^y (D) (f(xy))^(xy) (iii) Σ_(k=0) ^∞ f(k)= (A) 5/2 (B) 3/2 (C) 3 (D) 2

$$\mathrm{Let}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{functional}\:\mathrm{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({f}\left(\mathrm{x}\right)\right)^{\mathrm{y}} +\left({f}\left(\mathrm{y}\right)\right)^{\mathrm{x}} =\mathrm{2}{f}\left(\mathrm{xy}\right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{x},\:\mathrm{y}\:\in\mathbb{R}\:\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{given}\:\mathrm{that}\:{f}\left(\mathrm{1}\right)=\mathrm{1}/\mathrm{2}.\:\mathrm{Answer} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{questions}. \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\:\:\:\:{f}\left(\mathrm{x}+\mathrm{y}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:{f}\left(\mathrm{x}\right){f}\left(\mathrm{y}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:{f}\left(\mathrm{x}^{\mathrm{y}} \mathrm{y}^{\mathrm{x}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\:\frac{{f}\left(\mathrm{x}\right)}{{f}\left(\mathrm{y}\right)} \\ $$$$\left(\boldsymbol{\mathrm{ii}}\right)\:\:\:\:{f}\left(\mathrm{xy}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{f}\left(\mathrm{x}\right){f}\left(\mathrm{y}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:{f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\left({f}\left(\mathrm{x}\right)\right)^{\mathrm{y}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\left({f}\left(\mathrm{xy}\right)\right)^{\mathrm{xy}} \\ $$$$\left(\boldsymbol{\mathrm{iii}}\right)\:\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({k}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:\mathrm{5}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{3}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{2} \\ $$

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