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Question Number 216055    Answers: 1   Comments: 0

lim_(x→0) ((sin^2 2x)/( ((cos x))^(1/3) −((cos x))^(1/4) )) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{cos}\:\mathrm{x}}}\:=? \\ $$

Question Number 216050    Answers: 1   Comments: 0

x,y,z ∈ N lcd(x;y)=72 lcd(x;z)=600 lcd(y;z)=900 Find: 1.(x;y;z)=? 2.(x;y;z)=? 3.(x;y;z)=? ... a)15 b)16 c)24 d)27 e)64

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{N} \\ $$$$\mathrm{lcd}\left(\mathrm{x};\mathrm{y}\right)=\mathrm{72} \\ $$$$\mathrm{lcd}\left(\mathrm{x};\mathrm{z}\right)=\mathrm{600} \\ $$$$\mathrm{lcd}\left(\mathrm{y};\mathrm{z}\right)=\mathrm{900} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{1}.\left(\mathrm{x};\mathrm{y};\mathrm{z}\right)=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}.\left(\mathrm{x};\mathrm{y};\mathrm{z}\right)=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}.\left(\mathrm{x};\mathrm{y};\mathrm{z}\right)=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:... \\ $$$$\left.\mathrm{a}\left.\right)\left.\mathrm{1}\left.\mathrm{5}\left.\:\:\:\:\:\mathrm{b}\right)\mathrm{16}\:\:\:\:\:\mathrm{c}\right)\mathrm{24}\:\:\:\:\:\mathrm{d}\right)\mathrm{27}\:\:\:\:\:\mathrm{e}\right)\mathrm{64} \\ $$

Question Number 216046    Answers: 2   Comments: 0

x = bz + cy, y = cx + az and z = bx + ay then prove that a^2 + b^2 + c^2 + 2abc = 1.

$${x}\:=\:{bz}\:+\:{cy},\:{y}\:=\:{cx}\:+\:{az}\:\mathrm{and}\:{z}\:=\:{bx}\:+\:{ay} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:+\:\mathrm{2}{abc}\:=\:\mathrm{1}. \\ $$

Question Number 216042    Answers: 1   Comments: 2

(i) ∫sec^5 θdθ (ii) ∫ (((√(tan θ)) dθ)/(cos θ))

$$\left({i}\right)\:\:\:\int\mathrm{sec}\:^{\mathrm{5}} \theta{d}\theta \\ $$$$\left({ii}\right)\:\:\int\:\frac{\sqrt{\mathrm{tan}\:\theta}\:{d}\theta}{\mathrm{cos}\:\theta} \\ $$

Question Number 216033    Answers: 2   Comments: 0

Question Number 216032    Answers: 2   Comments: 0

lim_(x→0) ((1−cos x (√(cos 2x)))/x^2 ) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}\:\sqrt{\mathrm{cos}\:\mathrm{2x}}}{\mathrm{x}^{\mathrm{2}} }\:=? \\ $$

Question Number 216016    Answers: 1   Comments: 0

Question Number 216014    Answers: 1   Comments: 0

lim_(Δx→cos(π/2)) ((sin^3 (Δx+x)−sin^3 x)/(2^(−1) ∙Δx))=?

$$\underset{\Delta{x}\rightarrow{cos}\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{sin}^{\mathrm{3}} \left(\Delta{x}+{x}\right)−{sin}^{\mathrm{3}} {x}}{\mathrm{2}^{−\mathrm{1}} \centerdot\Delta{x}}=? \\ $$

Question Number 216010    Answers: 2   Comments: 0

Solve for x and y ax^2 + bxy + cy^2 = bx^2 + cxy + ay^2 = d.

$$\mathrm{Solve}\:\mathrm{for}\:{x}\:\mathrm{and}\:{y} \\ $$$${ax}^{\mathrm{2}} \:+\:{bxy}\:+\:{cy}^{\mathrm{2}} \:=\:{bx}^{\mathrm{2}} \:+\:{cxy}\:+\:{ay}^{\mathrm{2}} \:=\:{d}. \\ $$

Question Number 215999    Answers: 2   Comments: 0

if the fraction ((m^2 +25m)/(m+1)) is reductible. how many values does m take if is a 2 digit number? thanks

$${if}\:{the}\:{fraction}\:\frac{{m}^{\mathrm{2}} +\mathrm{25}{m}}{{m}+\mathrm{1}}\:\:{is}\:{reductible}.\:{how}\:{many}\:{values}\:{does}\:{m}\:\:{take}\:{if}\:{is}\:{a}\:\mathrm{2}\:{digit}\:\:{number}?\:{thanks} \\ $$

Question Number 215995    Answers: 1   Comments: 0

∫∫∫_D (√(x^2 +y^2 +z^2 )) dv = ? D = x^2 +y^2 +z^2 <z

$$\int\int\underset{{D}} {\int}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }\:\mathrm{dv}\:=\:? \\ $$$$\mathrm{D}\:=\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} <{z} \\ $$

Question Number 215994    Answers: 1   Comments: 0

Solve for z∈C: ∣z^z ∣=1

$$\mathrm{Solve}\:\mathrm{for}\:{z}\in\mathbb{C}:\:\:\:\:\:\mid{z}^{{z}} \mid=\mathrm{1} \\ $$

Question Number 215992    Answers: 1   Comments: 0

((√3) + (√2))^x + ((√3) − (√2))^x = 10. Solve for x.

$$\left(\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}\right)^{{x}} \:+\:\left(\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{2}}\right)^{{x}} \:=\:\mathrm{10}.\:\mathrm{Solve} \\ $$$$\mathrm{for}\:{x}. \\ $$

Question Number 215991    Answers: 0   Comments: 1

Σ_(x=1) ^y x^n =0 n=?

$$\underset{{x}=\mathrm{1}} {\overset{{y}} {\sum}}{x}^{{n}} =\mathrm{0} \\ $$$${n}=? \\ $$

Question Number 215979    Answers: 0   Comments: 0

u_n = Σ_(k=n+1) ^(2n) (1/k) and v_n = Σ_(k=n) ^(2n−1) (1/k) • show that u_n and v_n are adjacent use ln(x+1) ≤ x and x≤−ln(1−x) and • show that u_n ≤ Σ_(k=n+1) ^(2n) (ln(k)−ln(k−1)) hence deduce that u_n ≤ ln2 • show that v_n ≥ Σ_(k=n) ^(2n−1) (ln(k+1)−ln(k)) hence deduce that v_n ≥ln2

$${u}_{{n}} \:=\:\underset{{k}={n}+\mathrm{1}} {\overset{\mathrm{2}{n}} {\sum}}\frac{\mathrm{1}}{{k}}\:{and}\:{v}_{{n}} \:=\:\underset{{k}={n}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{{k}} \\ $$$$\bullet\:{show}\:{that}\:{u}_{{n}} \:{and}\:{v}_{{n}} \:{are}\:{adjacent} \\ $$$${use}\:{ln}\left({x}+\mathrm{1}\right)\:\leqslant\:{x}\:{and}\:{x}\leqslant−{ln}\left(\mathrm{1}−{x}\right)\:{and} \\ $$$$\bullet\:{show}\:{that}\:{u}_{{n}} \:\leqslant\:\underset{{k}={n}+\mathrm{1}} {\overset{\mathrm{2}{n}} {\sum}}\left({ln}\left({k}\right)−{ln}\left({k}−\mathrm{1}\right)\right) \\ $$$${hence}\:{deduce}\:{that}\:{u}_{{n}} \:\leqslant\:{ln}\mathrm{2} \\ $$$$\bullet\:{show}\:{that}\:{v}_{{n}} \:\geqslant\:\underset{{k}={n}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\left({ln}\left({k}+\mathrm{1}\right)−{ln}\left({k}\right)\right) \\ $$$${hence}\:{deduce}\:{that}\:{v}_{{n}} \geqslant{ln}\mathrm{2} \\ $$

Question Number 215975    Answers: 2   Comments: 0

Question Number 215974    Answers: 1   Comments: 0

∫(√((2sin^(−1) x−x(√(1−x^2 )))^2 +x^4 ))dx

$$\int\sqrt{\left(\mathrm{2sin}^{−\mathrm{1}} {x}−{x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right)^{\mathrm{2}} +{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 215973    Answers: 2   Comments: 0

(a + b) ∝ c and (b + c) ∝ a. Prove that (c + a) ∝ b.

$$\left({a}\:+\:{b}\right)\:\propto\:{c}\:\mathrm{and}\:\left({b}\:+\:{c}\right)\:\propto\:{a}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({c}\:+\:{a}\right)\:\propto\:{b}. \\ $$

Question Number 215963    Answers: 1   Comments: 0

Question Number 215957    Answers: 0   Comments: 2

b^2 −4ac

$${b}^{\mathrm{2}} −\mathrm{4}{ac} \\ $$

Question Number 215958    Answers: 2   Comments: 0

Find the only function that satisfy the expression below: ((dy/dx))^2 = (d^2 y/dx^2 )

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{only}\:\mathrm{function}\:\mathrm{that}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\mathrm{below}: \\ $$$$\:\:\:\:\:\:\:\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{\mathrm{2}} \:\:=\:\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} } \\ $$

Question Number 215952    Answers: 0   Comments: 1

Question Number 215951    Answers: 1   Comments: 0

E_n = 3^E_(n−1) , n≥2 find the unit digit of E_(1000)

$${E}_{{n}} \:=\:\mathrm{3}^{{E}_{{n}−\mathrm{1}} } ,\:{n}\geqslant\mathrm{2} \\ $$$${find}\:{the}\:{unit}\:{digit}\:{of}\:{E}_{\mathrm{1000}} \\ $$

Question Number 215944    Answers: 2   Comments: 0

if m,n,z ∈N so , m<n and z<m then z<n ?

$${if}\:{m},{n},{z}\:\in{N}\:{so}\:,\:{m}<{n}\:{and}\:{z}<{m}\:{then}\:{z}<{n}\:?\: \\ $$

Question Number 215943    Answers: 1   Comments: 0

Question Number 215939    Answers: 1   Comments: 0

for any natural numbers m,n then m=n or m<n or m>n ? prove

$${for}\:{any}\:{natural}\:{numbers}\:{m},{n}\:{then}\:{m}={n}\:{or}\:{m}<{n}\:{or}\:{m}>{n}\:?\:{prove} \\ $$

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