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Question Number 119059 Answers: 1 Comments: 0

Question Number 119057 Answers: 3 Comments: 0

show that ∣x+y∣≤∣x∣+∣y∣

$${show}\:{that} \\ $$$$\mid{x}+{y}\mid\leqslant\mid{x}\mid+\mid{y}\mid \\ $$$$ \\ $$

Question Number 119055 Answers: 2 Comments: 0

Show by recurence that: ∀ n∈N^∗ Σ_(k=1) ^n (1/(k(k+1)(k+2)))=((n(n+3))/(4(n+1)(n+2)))

$$ \\ $$$${Show}\:{by}\:{recurence}\:{that}:\:\forall\:{n}\in\mathbb{N}^{\ast} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)\left({k}+\mathrm{2}\right)}=\frac{{n}\left({n}+\mathrm{3}\right)}{\mathrm{4}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$

Question Number 119054 Answers: 2 Comments: 0

Show by recurence that: 5^n ≥1+4n ; n∈N

$${Show}\:{by}\:{recurence}\:{that}: \\ $$$$\mathrm{5}^{{n}} \geqslant\mathrm{1}+\mathrm{4}{n}\:;\:{n}\in\mathbb{N} \\ $$

Question Number 119052 Answers: 3 Comments: 0

Question Number 119048 Answers: 1 Comments: 0

Show that : ⌊x⌋ = ⌊((⌊nx⌋)/n)⌋ where n∈N^∗ x∈R

$$\mathrm{Show}\:\mathrm{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\lfloor{x}\rfloor\:=\:\lfloor\frac{\lfloor{nx}\rfloor}{{n}}\rfloor\:\:\:\:\mathrm{where}\:{n}\in\mathbb{N}^{\ast} \:\:\:{x}\in\mathbb{R} \\ $$$$ \\ $$

Question Number 119038 Answers: 2 Comments: 0

∫_(−π/4) ^(+π/4) ((√(1+tan x))/( (√(1−tan x))))dx

$$\underset{−\pi/\mathrm{4}} {\overset{+\pi/\mathrm{4}} {\int}}\frac{\sqrt{\mathrm{1}+\mathrm{tan}\:{x}}}{\:\sqrt{\mathrm{1}−\mathrm{tan}\:{x}}}{dx} \\ $$

Question Number 119035 Answers: 1 Comments: 0

Among the positive integers less than 1200, how many of them are relatively prime to 60?

$$\mathrm{Among}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{less}\:\mathrm{than}\:\mathrm{1200}, \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{of}\:\mathrm{them}\:\mathrm{are}\:\mathrm{relatively}\:\mathrm{prime} \\ $$$$\mathrm{to}\:\mathrm{60}? \\ $$

Question Number 119032 Answers: 1 Comments: 0

Question Number 119025 Answers: 0 Comments: 0

Question Number 119094 Answers: 1 Comments: 0

Why the Fermat formula about prime numbers is for n = 5 and n = 6 fail? f(n) = 2^2^n + 1

$${Why}\:{the}\:{Fermat}\:{formula}\:{about}\:{prime}\:{numbers}\:{is}\:{for}\:{n}\:=\:\mathrm{5}\:{and}\:{n}\:=\:\mathrm{6}\:{fail}? \\ $$$$ \\ $$$${f}\left({n}\right)\:=\:\mathrm{2}^{\mathrm{2}^{{n}} } \:+\:\mathrm{1} \\ $$

Question Number 119021 Answers: 1 Comments: 0

Question Number 119015 Answers: 1 Comments: 1

x = ((sin 1°+sin 2°+sin 3°+...+sin 45°)/(cos 1°+cos 2°+cos 3°+...+cos 45°)) x = ?

$${x}\:=\:\frac{\mathrm{sin}\:\mathrm{1}°+\mathrm{sin}\:\mathrm{2}°+\mathrm{sin}\:\mathrm{3}°+...+\mathrm{sin}\:\mathrm{45}°}{\mathrm{cos}\:\mathrm{1}°+\mathrm{cos}\:\mathrm{2}°+\mathrm{cos}\:\mathrm{3}°+...+\mathrm{cos}\:\mathrm{45}°} \\ $$$${x}\:=\:? \\ $$

Question Number 119013 Answers: 1 Comments: 1

Question Number 119008 Answers: 1 Comments: 1

Question Number 119007 Answers: 2 Comments: 0

Given point A=(7,26) and B=(12,12) , find all points P such that ∣AP ∣ = ∣ BP ∣ and ∠APB = 90° .

$${Given}\:{point}\:{A}=\left(\mathrm{7},\mathrm{26}\right)\:{and}\:{B}=\left(\mathrm{12},\mathrm{12}\right)\:,\:{find} \\ $$$${all}\:{points}\:{P}\:{such}\:{that}\:\mid{AP}\:\mid\:=\:\mid\:{BP}\:\mid\:{and}\:\angle{APB}\:=\:\mathrm{90}°\:. \\ $$

Question Number 119006 Answers: 2 Comments: 0

Question Number 119097 Answers: 2 Comments: 0

Find the remainder when (1!×1)+(2!×2)+(3!×3)+...+(100!×100) is divided by 101.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\: \\ $$$$\left(\mathrm{1}!×\mathrm{1}\right)+\left(\mathrm{2}!×\mathrm{2}\right)+\left(\mathrm{3}!×\mathrm{3}\right)+...+\left(\mathrm{100}!×\mathrm{100}\right) \\ $$$$\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{101}. \\ $$

Question Number 119096 Answers: 0 Comments: 0

Question Number 119002 Answers: 1 Comments: 0

(dy/dx) = ((xy)/(x^2 +1)) and y((√(15))) = 2

$$\frac{{dy}}{{dx}}\:=\:\frac{{xy}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{and}\:{y}\left(\sqrt{\mathrm{15}}\right)\:=\:\mathrm{2} \\ $$

Question Number 119000 Answers: 1 Comments: 1

lim_(x→∞) x^(3/2) sin 2x =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\frac{\mathrm{3}}{\mathrm{2}}} \:\mathrm{sin}\:\mathrm{2}{x}\:=? \\ $$

Question Number 118997 Answers: 3 Comments: 0

Question Number 118990 Answers: 2 Comments: 0

Question Number 118984 Answers: 1 Comments: 0

z=x+iy and z^5 =1 Show that 4x(y^4 −x^4 )=1 Pls help

$$\mathrm{z}=\mathrm{x}+\mathrm{iy} \\ $$$$\mathrm{and}\:\mathrm{z}^{\mathrm{5}} =\mathrm{1} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{4x}\left(\mathrm{y}^{\mathrm{4}} −\mathrm{x}^{\mathrm{4}} \right)=\mathrm{1} \\ $$$$\mathrm{Pls}\:\mathrm{help} \\ $$

Question Number 118975 Answers: 2 Comments: 0

(3x−5y) dx + (x+y) dy = 0

$$\:\left(\mathrm{3}{x}−\mathrm{5}{y}\right)\:{dx}\:+\:\left({x}+{y}\right)\:{dy}\:=\:\mathrm{0} \\ $$

Question Number 118969 Answers: 3 Comments: 0

Given f(x) = ((sin x+cos x)/(sin x−cos x)) find the value of f ′′(x) + f ′(x) + 1 .

$$\:{Given}\:{f}\left({x}\right)\:=\:\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}−\mathrm{cos}\:{x}} \\ $$$${find}\:{the}\:{value}\:{of}\: \\ $$$$\:{f}\:''\left({x}\right)\:+\:{f}\:'\left({x}\right)\:+\:\mathrm{1}\:. \\ $$

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