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AlgebraQuestion and Answers: Page 57

Question Number 201728    Answers: 1   Comments: 0

cos^2 4x ∙ sin^2 4x = 0,25 for equation [0;90] how many roots are there in the piece?

$$\mathrm{cos}^{\mathrm{2}} \:\mathrm{4x}\:\centerdot\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{4x}\:=\:\mathrm{0},\mathrm{25}\:\mathrm{for}\:\mathrm{equation} \\ $$$$\left[\mathrm{0};\mathrm{90}\right]\:\mathrm{how}\:\mathrm{many}\:\mathrm{roots}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{piece}? \\ $$

Question Number 201702    Answers: 1   Comments: 1

Find: ∫_1 ^( 3) dx ∫_x ^( x^3 ) (x − y) dy

$$\mathrm{Find}: \\ $$$$\int_{\mathrm{1}} ^{\:\mathrm{3}} \:\mathrm{dx}\:\int_{\boldsymbol{\mathrm{x}}} ^{\:\boldsymbol{\mathrm{x}}^{\mathrm{3}} } \:\left(\mathrm{x}\:−\:\mathrm{y}\right)\:\mathrm{dy} \\ $$

Question Number 201689    Answers: 1   Comments: 0

Question Number 201679    Answers: 5   Comments: 0

Question Number 201629    Answers: 0   Comments: 3

Question Number 201615    Answers: 1   Comments: 0

x,y,z ∈ R a,b,c>0 prove that: (x^2 /a) + (y^2 /b) + (z^2 /c) ≥ (((x + y + z)^2 )/(a + b + c))

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{a}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{b}}\:+\:\frac{\mathrm{z}^{\mathrm{2}} }{\mathrm{c}}\:\geqslant\:\frac{\left(\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\right)^{\mathrm{2}} }{\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}} \\ $$

Question Number 201613    Answers: 2   Comments: 0

x,y,z ∈ R { ((xy + yz + zx = 3)),((x + y + z = 5)) :} → max(z) = ?

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{xy}\:+\:\mathrm{yz}\:+\:\mathrm{zx}\:=\:\mathrm{3}}\\{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{5}}\end{cases}\:\:\:\:\:\rightarrow\:\:\:\:\mathrm{max}\left(\boldsymbol{\mathrm{z}}\right)\:=\:? \\ $$

Question Number 201547    Answers: 1   Comments: 0

Question Number 201557    Answers: 2   Comments: 0

5 ∙ 555...5_( 50) find the sum of the digits of the product.

$$\mathrm{5}\:\centerdot\:\underset{\:\mathrm{50}} {\underbrace{\mathrm{555}...\mathrm{5}}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{product}. \\ $$

Question Number 201477    Answers: 1   Comments: 0

how to prove that (3d_3 +4d_2 +3d_1 )^2 ≤5(d_1 ^2 +d_2 ^2 +d_3 ^2 +(d_2 +d_1 )^2 +(d_3 +d_2 )^2 +(d_1 +d_2 +d_3 )^2 )

$$\mathrm{how}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\left(\mathrm{3d}_{\mathrm{3}} +\mathrm{4d}_{\mathrm{2}} +\mathrm{3d}_{\mathrm{1}} \right)^{\mathrm{2}} \leqslant\mathrm{5}\left(\mathrm{d}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{d}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{d}_{\mathrm{3}} ^{\mathrm{2}} +\left(\mathrm{d}_{\mathrm{2}} +\mathrm{d}_{\mathrm{1}} \right)^{\mathrm{2}} +\left(\mathrm{d}_{\mathrm{3}} +\mathrm{d}_{\mathrm{2}} \right)^{\mathrm{2}} +\left(\mathrm{d}_{\mathrm{1}} +\mathrm{d}_{\mathrm{2}} +\mathrm{d}_{\mathrm{3}} \right)^{\mathrm{2}} \right) \\ $$

Question Number 201464    Answers: 2   Comments: 0

if f(2) = 3 and f(4) = 5 find ∫_2 ^( 4) f(x) ∙ f^′ (x) dx = ?

$$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:\mathrm{3}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{4}\right)\:=\:\mathrm{5} \\ $$$$\mathrm{find}\:\:\:\int_{\mathrm{2}} ^{\:\mathrm{4}} \:\mathrm{f}\left(\mathrm{x}\right)\:\centerdot\:\mathrm{f}\:^{'} \left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 201441    Answers: 0   Comments: 0

Let f(x) and g(x) be given by f(x)= (1/x) +(1/(x−2)) +(1/(x−4)) + ... +(1/(x−2018)) and g(x)=(1/(x−1)) +(1/(x−3)) +(1/(x−5)) +...+ (1/(x−2017)). Prove that ∣ f(x)−g(x)∣ >2 for any non−integer real number x satisfying 0<x<2018.

$${Let}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right)\:{be}\:{given}\:{by}\: \\ $$$$\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}−\mathrm{2}}\:+\frac{\mathrm{1}}{{x}−\mathrm{4}}\:+\:...\:+\frac{\mathrm{1}}{{x}−\mathrm{2018}} \\ $$$$\:{and}\: \\ $$$$\:\:{g}\left({x}\right)=\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\frac{\mathrm{1}}{{x}−\mathrm{3}}\:+\frac{\mathrm{1}}{{x}−\mathrm{5}}\:+...+\:\frac{\mathrm{1}}{{x}−\mathrm{2017}}. \\ $$$$\:\:{Prove}\:{that}\:\:\mid\:{f}\left({x}\right)−{g}\left({x}\right)\mid\:>\mathrm{2} \\ $$$$\:\:{for}\:{any}\:{non}−{integer}\:{real}\:{number} \\ $$$$\:\:{x}\:{satisfying}\:\mathrm{0}<{x}<\mathrm{2018}.\: \\ $$

Question Number 201430    Answers: 1   Comments: 0

Find: (2/(35)) + (2/(63)) + (2/(99)) + (2/(143)) = ?

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{2}}{\mathrm{35}}\:+\:\frac{\mathrm{2}}{\mathrm{63}}\:+\:\frac{\mathrm{2}}{\mathrm{99}}\:+\:\frac{\mathrm{2}}{\mathrm{143}}\:=\:? \\ $$

Question Number 201408    Answers: 3   Comments: 0

Question Number 201388    Answers: 2   Comments: 0

Question Number 201343    Answers: 1   Comments: 0

Question Number 201463    Answers: 1   Comments: 0

a = constant number: if ∫x ∙ f(x) dx = x^3 - x^2 + 4x - (a/5) find f(2) = ?

$$\mathrm{a}\:=\:\mathrm{constant}\:\mathrm{number}: \\ $$$$\mathrm{if}\:\:\:\int\mathrm{x}\:\centerdot\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4x}\:-\:\frac{\mathrm{a}}{\mathrm{5}} \\ $$$$\mathrm{find}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:? \\ $$

Question Number 201462    Answers: 1   Comments: 0

Find: ∫_0 ^( 𝛑) (√(1 - 4 sin^2 (x/2) cos^2 (x/2))) dx = ?

$$\mathrm{Find}: \\ $$$$\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\sqrt{\mathrm{1}\:-\:\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:\frac{\mathrm{x}}{\mathrm{2}}\:\mathrm{cos}^{\mathrm{2}} \:\frac{\mathrm{x}}{\mathrm{2}}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 201460    Answers: 1   Comments: 0

Find the smallest positive period of the function: y = ∣ tan 2x ∣ + ∣ cot 2x ∣

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{period}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{function}: \\ $$$$\mathrm{y}\:=\:\mid\:\mathrm{tan}\:\mathrm{2x}\:\mid\:\:+\:\:\mid\:\mathrm{cot}\:\mathrm{2x}\:\mid \\ $$

Question Number 201302    Answers: 1   Comments: 0

{ ((x^2 + y^(−2) = 4)),((x^(−2) + y^2 = 5)) :} find: (y/x) = ?

$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{−\mathrm{2}} \:=\:\mathrm{4}}\\{\mathrm{x}^{−\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{5}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\frac{\mathrm{y}}{\mathrm{x}}\:=\:? \\ $$

Question Number 201282    Answers: 1   Comments: 0

Question Number 201253    Answers: 2   Comments: 0

find the sum of n terms of the serice s_n =5+11+19+29+41+..........

$${find}\:{the}\:{sum}\:{of}\:{n}\:{terms}\:{of}\:{the}\:{serice} \\ $$$${s}_{{n}} =\mathrm{5}+\mathrm{11}+\mathrm{19}+\mathrm{29}+\mathrm{41}+.......... \\ $$

Question Number 201237    Answers: 1   Comments: 3

If , a ∣ 5b^( 2) −10b +1 ⇒ [ a , 5b ]_(lcm) = ?

$$ \\ $$$$\:\:\:\:{If}\:\:,\:\:\:{a}\:\mid\:\mathrm{5}{b}^{\:\mathrm{2}} −\mathrm{10}{b}\:+\mathrm{1}\: \\ $$$$\:\:\:\:\:\Rightarrow\:\:\:\left[\:{a}\:\:,\:\mathrm{5}{b}\:\right]_{\mathrm{lc}{m}} \:=\:\:?\: \\ $$$$ \\ $$$$ \\ $$

Question Number 201200    Answers: 3   Comments: 2

Question Number 201190    Answers: 1   Comments: 0

Question Number 201162    Answers: 4   Comments: 0

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