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AlgebraQuestion and Answers: Page 5

Question Number 224886 Answers: 1 Comments: 0

Find for acute θ, sin θ and cos θ in terms of 0<k<1, if ((sin θ(1−cos θ))/(cos θ(1−sin θ)))=k.

$${Find}\:{for}\:{acute}\:\theta,\:\mathrm{sin}\:\theta\:{and}\:\mathrm{cos}\:\theta \\ $$$${in}\:{terms}\:{of}\:\mathrm{0}<{k}<\mathrm{1}, \\ $$$${if}\:\:\:\:\frac{\mathrm{sin}\:\theta\left(\mathrm{1}−\mathrm{cos}\:\theta\right)}{\mathrm{cos}\:\theta\left(\mathrm{1}−\mathrm{sin}\:\theta\right)}={k}. \\ $$

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(2x^3 +x−3)^3 =3−x^2

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

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Prove that: sin(54°) = (((√5) + 1)/4)

$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\mathrm{sin}\left(\mathrm{54}°\right)\:=\:\:\frac{\sqrt{\mathrm{5}}\:+\:\mathrm{1}}{\mathrm{4}} \\ $$

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Question Number 224614 Answers: 1 Comments: 0

A binary operation ∗ is defined on a set real numbers, R by x∗y = 2x + 2y −((xy)/3) . find: (i) The inverse of x under the operation ∗ (ii) Truth set when m∗7=−2∗m

$${A}\:{binary}\:{operation}\:\ast\:{is}\:{defined}\:{on}\:{a}\:{set} \\ $$$${real}\:{numbers},\:{R}\:{by} \\ $$$${x}\ast{y}\:=\:\mathrm{2}{x}\:+\:\mathrm{2}{y}\:−\frac{{xy}}{\mathrm{3}}\:. \\ $$$${find}: \\ $$$$\left({i}\right)\:{The}\:{inverse}\:{of}\:{x}\:{under}\:{the}\:{operation}\:\ast \\ $$$$\left({ii}\right)\:{Truth}\:{set}\:{when}\:{m}\ast\mathrm{7}=−\mathrm{2}\ast{m} \\ $$

Question Number 224591 Answers: 1 Comments: 0

p is a prime number prove that if p^2 +8 is prime ⇒ ⇒ p^3 +4 is also prime

$$\mathrm{p}\:{is}\:{a}\:{prime}\:{number} \\ $$$${prove}\:{that}\:{if}\:\mathrm{p}^{\mathrm{2}} +\mathrm{8}\:{is}\:{prime}\:\Rightarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:{p}^{\mathrm{3}} +\mathrm{4}\:{is}\:{also}\:{prime} \\ $$

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Question Number 224539 Answers: 3 Comments: 0

a^x =m, a^y =n ,a^2 =(m^y n^x )^z prove xyz=1

$${a}^{{x}} ={m},\:{a}^{{y}} ={n}\:,{a}^{\mathrm{2}} =\left({m}^{{y}} {n}^{{x}} \right)^{{z}} \\ $$$${prove}\:{xyz}=\mathrm{1} \\ $$

Question Number 224538 Answers: 2 Comments: 0

32^(4r^2 −8) =1 then find r=?

$$\mathrm{32}^{\mathrm{4r}^{\mathrm{2}} −\mathrm{8}} =\mathrm{1}\:\:\:\:\mathrm{then}\:\mathrm{find}\:\mathrm{r}=? \\ $$

Question Number 224455 Answers: 1 Comments: 0

If a,b,c ≠ 0 what is the difference between the maximum snd minimum value of S = 1 + ((∣a∣)/a) + ((2∣b∣)/b) + ((3∣ab∣)/(ab)) − ((4∣c∣)/c) ?

$${If}\:{a},{b},{c}\:\neq\:\mathrm{0}\:{what}\:{is}\:{the}\:{difference} \\ $$$${between}\:{the}\:{maximum}\:{snd}\:{minimum}\: \\ $$$${value}\:{of} \\ $$$${S}\:=\:\mathrm{1}\:+\:\frac{\mid{a}\mid}{{a}}\:+\:\frac{\mathrm{2}\mid{b}\mid}{{b}}\:+\:\frac{\mathrm{3}\mid{ab}\mid}{{ab}}\:−\:\frac{\mathrm{4}\mid{c}\mid}{{c}}\:? \\ $$

Question Number 224453 Answers: 0 Comments: 6

Question Number 224392 Answers: 2 Comments: 0

−∞<a<b<∞ and 0<λ<1 x_1 = a , x_2 = b x_(n+2) = λx_n + (1−λ)x_(n+1) ∀ n ∈ N find x_(n ) = ?

$$\:\:\:\:\:\:\:−\infty<{a}<{b}<\infty\:\:{and}\:\mathrm{0}<\lambda<\mathrm{1}\:\: \\ $$$$\:\:\:\:\:\:\:{x}_{\mathrm{1}} \:=\:{a}\:,\:{x}_{\mathrm{2}} \:=\:{b} \\ $$$$\:\:\:\:\:\:\:\:{x}_{{n}+\mathrm{2}} \:=\:\lambda{x}_{{n}} \:+\:\left(\mathrm{1}−\lambda\right){x}_{{n}+\mathrm{1}} \:\:\forall\:{n}\:\in\:\mathbb{N} \\ $$$$\:\:\mathrm{find}\:\:{x}_{{n}\:} \:=\:?\: \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$

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a,b,c > 0 prove that Σ a^2 ≥ Σ ab + (1/8) Σ (∣a−c∣ + ∣b−c∣)^2

$$\mathrm{a},\mathrm{b},\mathrm{c}\:>\:\mathrm{0} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\Sigma\:\mathrm{a}^{\mathrm{2}} \:\geqslant\:\Sigma\:\mathrm{ab}\:+\:\frac{\mathrm{1}}{\mathrm{8}}\:\Sigma\:\left(\mid\mathrm{a}−\mathrm{c}\mid\:+\:\mid\mathrm{b}−\mathrm{c}\mid\right)^{\mathrm{2}} \\ $$

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(√x)+(√y)=7 (√(x+y))=5 Find (x,y).

$$\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}=\mathrm{7} \\ $$$$\sqrt{\mathrm{x}+\mathrm{y}}=\mathrm{5} \\ $$$$\mathrm{Find}\:\left(\mathrm{x},\mathrm{y}\right). \\ $$

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