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GeometryQuestion and Answers: Page 107

Question Number 8359 Answers: 1 Comments: 1

prove that 10 divides 11^(10) −1

$${prove}\:{that}\: \\ $$$$\mathrm{10}\:{divides}\:\mathrm{11}^{\mathrm{10}} −\mathrm{1} \\ $$

Question Number 8357 Answers: 0 Comments: 1

who is bigger ? 1000^(1000) or 101^(999 ) please answer in details.

$${who}\:{is}\:{bigger}\:? \\ $$$$\mathrm{1000}^{\mathrm{1000}} \:{or}\:\mathrm{101}^{\mathrm{999}\:} \\ $$$${please}\:{answer}\:{in}\:{details}. \\ $$$$ \\ $$

Question Number 8356 Answers: 0 Comments: 0

find factor: a^(10) +a^5 +1

$${find}\:{factor}: \\ $$$${a}^{\mathrm{10}} +{a}^{\mathrm{5}} +\mathrm{1} \\ $$

Question Number 8355 Answers: 2 Comments: 2

if a+b+c=0 then (((a−b)/c)+((b−c)/a)+((c−a)/b))((a/(b−c))+(b/(c−a))+(c/(a−b)))=?

$${if}\:{a}+{b}+{c}=\mathrm{0}\:{then} \\ $$$$ \\ $$$$\left(\frac{{a}−{b}}{{c}}+\frac{{b}−{c}}{{a}}+\frac{{c}−{a}}{{b}}\right)\left(\frac{{a}}{{b}−{c}}+\frac{{b}}{{c}−{a}}+\frac{{c}}{{a}−{b}}\right)=? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 8353 Answers: 1 Comments: 0

(√(240+2(√(14319))))=?

$$\sqrt{\mathrm{240}+\mathrm{2}\sqrt{\mathrm{14319}}}=? \\ $$

Question Number 8351 Answers: 1 Comments: 1

x^2 =816−64(√(77)) x=?[write x like a(√(c ))+b(√d])

$$ \\ $$$$\:\:\:{x}^{\mathrm{2}} =\mathrm{816}−\mathrm{64}\sqrt{\mathrm{77}} \\ $$$$\:\:\:{x}=?\left[{write}\:{x}\:{like}\:{a}\sqrt{{c}\:}+{b}\sqrt{\left.{d}\right]}\right. \\ $$$$ \\ $$

Question Number 8350 Answers: 1 Comments: 0

solve: (5/(3x+2))+(8/(4x+2))=((33)/(9x+8))

$${solve}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\frac{\mathrm{5}}{\mathrm{3}{x}+\mathrm{2}}+\frac{\mathrm{8}}{\mathrm{4}{x}+\mathrm{2}}=\frac{\mathrm{33}}{\mathrm{9}{x}+\mathrm{8}} \\ $$

Question Number 8345 Answers: 1 Comments: 0

y=(x+2)^2 −3 Translation T_1 = ((a),(b) ) y′=x^2 a=? b=?

$${y}=\left({x}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{3} \\ $$$${Translation}\:{T}_{\mathrm{1}} =\begin{pmatrix}{{a}}\\{{b}}\end{pmatrix} \\ $$$${y}'={x}^{\mathrm{2}} \\ $$$${a}=?\:{b}=? \\ $$

Question Number 8325 Answers: 0 Comments: 0

In Δ ABC,∠A=90° , AD⊥BC, DE⊥AC, AF⊥FG,GH⊥FC. (a)How many triangles are there? (b)If AB=((16)/9) , ∠B=60°,find the length of GH.

$${In}\:\Delta\:{ABC},\angle{A}=\mathrm{90}°\:,\:{AD}\bot{BC},\:{DE}\bot{AC}, \\ $$$${AF}\bot{FG},{GH}\bot{FC}. \\ $$$$\left({a}\right){How}\:{many}\:{triangles}\:{are}\:{there}? \\ $$$$\left({b}\right){If}\:{AB}=\frac{\mathrm{16}}{\mathrm{9}}\:,\:\angle{B}=\mathrm{60}°,{find}\:{the}\:{length} \\ $$$$\:\:\:\:\:\:{of}\:{GH}. \\ $$

Question Number 8175 Answers: 0 Comments: 0

Prove (4^n /n)<^(2n) C_n for all n≥4 and n∈Z^+

$$\mathrm{Prove} \\ $$$$\frac{\mathrm{4}^{{n}} }{{n}}<\:^{\mathrm{2}{n}} {C}_{{n}} \:\mathrm{for}\:\mathrm{all}\:{n}\geqslant\mathrm{4}\:\mathrm{and}\:{n}\in\mathbb{Z}^{+} \\ $$

Question Number 8168 Answers: 0 Comments: 3

Find the coefficient of in the expansion of (1+x)(1+x^2 )(1+x^3 )...(1+x^n ).

$${Find}\:{the}\:{coefficient}\:{of}\:{in}\:{the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{3}} \right)...\left(\mathrm{1}+{x}^{{n}} \right). \\ $$

Question Number 8129 Answers: 0 Comments: 0

An ellipse having focii at (3 3)and (−4 4) and passing through origin has e??

$${An}\:{ellipse}\:{having}\:{focii}\:{at}\:\left(\mathrm{3}\:\mathrm{3}\right){and}\:\left(−\mathrm{4}\:\mathrm{4}\right)\:{and}\:{passing}\:{through}\:{origin}\:{has}\:{e}?? \\ $$

Question Number 8127 Answers: 1 Comments: 3

For ∣x∣<1, we have that (1+x)^(1/2) =1+(1/2)x+((((1/2))((1/2)−1))/(2!))x^2 +(((1/2)((1/2)−1)((1/2)−2))/(3!))x^3 +... (1+x)^(1/2) =1+Σ_(r=1) ^∞ ((Π_(k=0) ^(r−1) (0.5−k))/(r!))x^r . Let g(r)=Π_(k=0) ^(r−1) (0.5−k). Is it true that for x=(1/2)i⇒∣x∣=0.5<1 (1+(1/2)i)^(1/2) =1+Σ_(r=1) ^∞ ((g(r))/(r!))×(1/2^r )i^r ? (i=(√(−1)))

$${For}\:\mid{x}\mid<\mathrm{1},\:{we}\:{have}\:{that} \\ $$$$\left(\mathrm{1}+{x}\right)^{\mathrm{1}/\mathrm{2}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}+\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} +\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2}\right)}{\mathrm{3}!}{x}^{\mathrm{3}} +... \\ $$$$\left(\mathrm{1}+{x}\right)^{\mathrm{1}/\mathrm{2}} =\mathrm{1}+\underset{{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\underset{{k}=\mathrm{0}} {\overset{{r}−\mathrm{1}} {\prod}}\left(\mathrm{0}.\mathrm{5}−{k}\right)}{{r}!}{x}^{{r}} . \\ $$$${Let}\:{g}\left({r}\right)=\underset{{k}=\mathrm{0}} {\overset{{r}−\mathrm{1}} {\prod}}\left(\mathrm{0}.\mathrm{5}−{k}\right). \\ $$$${Is}\:{it}\:{true}\:{that}\:{for}\:{x}=\frac{\mathrm{1}}{\mathrm{2}}{i}\Rightarrow\mid{x}\mid=\mathrm{0}.\mathrm{5}<\mathrm{1} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{i}\right)^{\mathrm{1}/\mathrm{2}} =\mathrm{1}+\underset{{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{g}\left({r}\right)}{{r}!}×\frac{\mathrm{1}}{\mathrm{2}^{{r}} }{i}^{{r}} \:\:? \\ $$$$\left({i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 8035 Answers: 0 Comments: 2

prove >> a^n +b^n =c^n [n>2] it has no integer roots

$${prove}\:>>\:{a}^{{n}} +{b}^{{n}} ={c}^{{n}} \:\:\left[{n}>\mathrm{2}\right] \\ $$$${it}\:{has}\:{no}\:{integer}\:{roots} \\ $$$$ \\ $$

Question Number 8032 Answers: 1 Comments: 0

find the real root: 99x^3 +297x^2 +594x−7867=0

$${find}\:{the}\:{real}\:{root}: \\ $$$$\mathrm{99}{x}^{\mathrm{3}} +\mathrm{297}{x}^{\mathrm{2}} +\mathrm{594}{x}−\mathrm{7867}=\mathrm{0} \\ $$

Question Number 8031 Answers: 1 Comments: 0

(√2) ≈((19601)/(13860))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\sqrt{\mathrm{2}}\:\approx\frac{\mathrm{19601}}{\mathrm{13860}} \\ $$

Question Number 8027 Answers: 1 Comments: 1

prove→ any prime number>2 can be written into( x^2 −y^(2 ) ) where (x,y)∈N

$${prove}\rightarrow\:{any}\:{prime}\:{number}>\mathrm{2}\: \\ $$$${can}\:{be}\:{written}\:{into}\left(\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}\:} \right)\:{where} \\ $$$$\left({x},{y}\right)\in{N} \\ $$

Question Number 8026 Answers: 1 Comments: 0

Find the factor of (3^(200) +4)

$${Find}\:{the}\:{factor}\:{of}\:\left(\mathrm{3}^{\mathrm{200}} +\mathrm{4}\right) \\ $$

Question Number 8025 Answers: 1 Comments: 1

find factor of ≫ (2^(4n+2) +1) at the same way expand 2^(58) +1

$${find}\:{factor}\:{of}\:\gg \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{2}^{\mathrm{4}{n}+\mathrm{2}} +\mathrm{1}\right) \\ $$$${at}\:{the}\:{same}\:{way}\:{expand}\:\mathrm{2}^{\mathrm{58}} +\mathrm{1} \\ $$$$ \\ $$

Question Number 8022 Answers: 1 Comments: 1

prove that ∀x∈N (2x−1)^2 +(2x^2 −2x)^2 is a proper square number.

$$ \\ $$$${prove}\:{that}\:\forall{x}\in\boldsymbol{{N}}\: \\ $$$$\left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{x}\right)^{\mathrm{2}} \:\:{is}\:{a}\:{proper} \\ $$$${square}\:{number}. \\ $$

Question Number 7945 Answers: 1 Comments: 4

Question Number 7935 Answers: 1 Comments: 2

Find U_x , U_(xy ) , U_(yy) Given that : U = x^3 y − siny

$${Find}\:\:{U}_{{x}} \:,\:\:{U}_{{xy}\:} \:,\:\:{U}_{{yy}} \:\: \\ $$$${Given}\:{that}\::\:\: \\ $$$${U}\:=\:{x}^{\mathrm{3}} {y}\:−\:{siny} \\ $$

Question Number 7887 Answers: 1 Comments: 0

Find the first four terms of the power series expansion of ((sinx)/(1 − x))

$${Find}\:{the}\:{first}\:{four}\:{terms}\:{of}\:{the}\:{power}\:{series}\: \\ $$$${expansion}\:{of}\:\:\:\:\:\frac{{sinx}}{\mathrm{1}\:−\:{x}}\:\:\: \\ $$

Question Number 7702 Answers: 0 Comments: 1

Question Number 7701 Answers: 0 Comments: 1

(x−2y

$$\left({x}−\mathrm{2}{y}\right. \\ $$$$ \\ $$

Question Number 7667 Answers: 1 Comments: 0

lim_(x→α) [x−x^2 log (1+(1/x))]

$$\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\left[{x}−{x}^{\mathrm{2}} \mathrm{log}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)\right] \\ $$

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