Question and Answers Forum

AllQuestion and Answers: Page 1017

Question Number 117852 Answers: 3 Comments: 0

Determine the value of (1)(tan ((7π)/(24))+tan ((5π)/(24))).cos (π/(12)) + 2 . (2) (((9−4(√5))/(5x)))^(1/(4 )) .(5(√x) +(√(20x)) )^(0.5) . 2^(−1) = ?

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\mathrm{1}\right)\left(\mathrm{tan}\:\frac{\mathrm{7}\pi}{\mathrm{24}}+\mathrm{tan}\:\frac{\mathrm{5}\pi}{\mathrm{24}}\right).\mathrm{cos}\:\frac{\pi}{\mathrm{12}}\:+\:\mathrm{2}\:. \\ $$$$\left(\mathrm{2}\right)\:\sqrt[{\mathrm{4}\:}]{\frac{\mathrm{9}−\mathrm{4}\sqrt{\mathrm{5}}}{\mathrm{5x}}}\:.\left(\mathrm{5}\sqrt{\mathrm{x}}\:+\sqrt{\mathrm{20x}}\:\right)^{\mathrm{0}.\mathrm{5}} .\:\mathrm{2}^{−\mathrm{1}} \:=\:? \\ $$

Question Number 117848 Answers: 0 Comments: 0

Determine all functions f:R→R such that the equality f([x] y)= f(x) [f(y) ] holds for all x,y ∈R . Here by [x] we denote the greatest integer not exceeding x.

$${Determine}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${such}\:{that}\:{the}\:{equality}\:{f}\left(\left[{x}\right]\:{y}\right)=\:{f}\left({x}\right)\:\left[{f}\left({y}\right)\:\right] \\ $$$${holds}\:{for}\:{all}\:{x},{y}\:\in\mathbb{R}\:.\:{Here}\:\:{by}\:\left[{x}\right]\:{we}\: \\ $$$${denote}\:{the}\:{greatest}\:{integer}\:{not}\:{exceeding}\:{x}. \\ $$$$ \\ $$

Question Number 117844 Answers: 0 Comments: 1

Question Number 117841 Answers: 1 Comments: 0

∫ ln (1−e^(−2x) ) dx =?

$$\int\:\mathrm{ln}\:\left(\mathrm{1}−{e}^{−\mathrm{2}{x}} \right)\:{dx}\:=? \\ $$

Question Number 117838 Answers: 1 Comments: 0

Let be P the set of prime numbers and A=P∪{0,1} Prove that Π_(n∉A) (n/( (√(n^2 −1)))) =(2/π)(√3)

$$\:\:{Let}\:{be}\:{P}\:\:{the}\:{set}\:{of}\:{prime}\:{numbers}\:{and}\: \\ $$$${A}={P}\cup\left\{\mathrm{0},\mathrm{1}\right\} \\ $$$${Prove}\:{that}\:\:\:\underset{{n}\notin{A}} {\prod}\:\frac{{n}}{\:\sqrt{{n}^{\mathrm{2}} −\mathrm{1}}}\:=\frac{\mathrm{2}}{\pi}\sqrt{\mathrm{3}}\: \\ $$

Question Number 117832 Answers: 1 Comments: 0

Let a,b>0 and x∈]0;(π/2)[ Prove ((a/(sinx))+1)((b/(cosx))+1)≥(1+(√(2ab)))^2

$$\left.{Let}\:{a},{b}>\mathrm{0}\:\:{and}\:{x}\in\right]\mathrm{0};\frac{\pi}{\mathrm{2}}\left[\:\right. \\ $$$$\:\:{Prove}\:\:\:\left(\frac{{a}}{{sinx}}+\mathrm{1}\right)\left(\frac{{b}}{{cosx}}+\mathrm{1}\right)\geqslant\left(\mathrm{1}+\sqrt{\mathrm{2}{ab}}\right)^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 117828 Answers: 1 Comments: 0

1)((√3)−1)((√3)+1)=(√3)×(√3)−(√3)−1 =3−(√3)−1 =2−(√3) 2)(2x+(√3))(2x−(√3))=(2x)^2 −2x(√3)+2x(√3)−3 =4x^2 −3

$$\left.\mathrm{1}\right)\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)=\sqrt{\mathrm{3}}×\sqrt{\mathrm{3}}−\sqrt{\mathrm{3}}−\mathrm{1} \\ $$$$=\mathrm{3}−\sqrt{\mathrm{3}}−\mathrm{1} \\ $$$$=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\left(\mathrm{2}{x}+\sqrt{\mathrm{3}}\right)\left(\mathrm{2}{x}−\sqrt{\mathrm{3}}\right)=\left(\mathrm{2}{x}\right)^{\mathrm{2}} −\mathrm{2}{x}\sqrt{\mathrm{3}}+\mathrm{2}{x}\sqrt{\mathrm{3}}−\mathrm{3} \\ $$$$=\mathrm{4}{x}^{\mathrm{2}} −\mathrm{3} \\ $$

Question Number 117827 Answers: 0 Comments: 5

Question Number 117826 Answers: 1 Comments: 0

Prove that the Euler Constant is qlso equal to lim_(x→−1) Γ(x)−(1/(x(x+1)))

$$\:\:{Prove}\:{that}\:{the}\:{Euler}\:{Constant}\:{is}\:{qlso}\:{equal}\:{to} \\ $$$$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\:\:\Gamma\left({x}\right)−\frac{\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)} \\ $$

Question Number 117825 Answers: 1 Comments: 0

Let ABC be a triangle such as 2cosA+3sinB=4 and 3cosB+2sinA=3 Prove that the angle C is right.

$${Let}\:{ABC}\:{be}\:{a}\:{triangle}\:{such}\:{as}\: \\ $$$$\:\mathrm{2}{cosA}+\mathrm{3}{sinB}=\mathrm{4}\:{and}\:\:\mathrm{3}{cosB}+\mathrm{2}{sinA}=\mathrm{3} \\ $$$${Prove}\:{that}\:{the}\:{angle}\:{C}\:{is}\:{right}. \\ $$$$\: \\ $$

Question Number 117820 Answers: 3 Comments: 0

lim_(x→+∞) (x.sin (1/x))^x^2

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left({x}.\mathrm{sin}\:\frac{\mathrm{1}}{{x}}\right)^{{x}^{\mathrm{2}} } \\ $$

Question Number 117818 Answers: 0 Comments: 2

let ABC be a triangle AB=c AC=b BC=a Show that ABC is right ⇔ tan((B/2))=((a+c)/b)

$$\:{let}\:{ABC}\:\:{be}\:{a}\:{triangle}\:{AB}={c}\:\:{AC}={b}\:\:{BC}={a} \\ $$$${Show}\:{that}\:{ABC}\:{is}\:{right}\:\Leftrightarrow\:\:{tan}\left(\frac{{B}}{\mathrm{2}}\right)=\frac{{a}+{c}}{{b}}\: \\ $$

Question Number 117817 Answers: 2 Comments: 1

Question Number 117816 Answers: 1 Comments: 0

find out for n≥1 Π_(k=0) ^(n−1) Γ(1+(k/n))

$$\:\:{find}\:{out}\:\:\:{for}\:{n}\geqslant\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\Gamma\left(\mathrm{1}+\frac{{k}}{{n}}\right) \\ $$$$ \\ $$

Question Number 117811 Answers: 1 Comments: 0

∫_( 0) ^( 𝛑) ln∣sinh(x)∣dx

$$\int_{\:\mathrm{0}} ^{\:\boldsymbol{\pi}} \boldsymbol{\mathrm{ln}}\mid\boldsymbol{\mathrm{sinh}}\left(\boldsymbol{\mathrm{x}}\right)\mid\boldsymbol{\mathrm{dx}} \\ $$

Question Number 117806 Answers: 4 Comments: 2

... nice calculus... i :: 1 +(4/9)+(9/(36))+((16)/(100))+...= ?? ii:: ∫_0 ^( (π/2)) x^2 cot(x) dx=?? m.n.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:{i}\:::\:\:\:\mathrm{1}\:+\frac{\mathrm{4}}{\mathrm{9}}+\frac{\mathrm{9}}{\mathrm{36}}+\frac{\mathrm{16}}{\mathrm{100}}+...=\:?? \\ $$$$\:\:\:\:\:{ii}::\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{2}} {cot}\left({x}\right)\:{dx}=?? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 117800 Answers: 1 Comments: 0

Question Number 117787 Answers: 2 Comments: 0

Question Number 117781 Answers: 1 Comments: 0

Log (cosβ) = p ⇒ cos β = 10^p ∴ secβ = (1/(cosβ)) = (1/(10^p )) = 10^(−p) ∴ Log (secβ) = Log 10^(−p) = −p Log 10 = −p

$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{Log}\:\left(\mathrm{cos}\beta\right)\:=\:\mathrm{p}\:\:\:\:\:\Rightarrow\:\:\:\mathrm{cos}\:\beta\:=\:\mathrm{10}^{\mathrm{p}} \:\: \\ $$$$\:\:\:\:\:\:\:\therefore\:\:\:\mathrm{sec}\beta\:=\:\:\frac{\mathrm{1}}{\mathrm{cos}\beta}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{10}^{\mathrm{p}} }\:\:=\:\mathrm{10}^{−\mathrm{p}} \\ $$$$\:\:\:\:\:\:\:\therefore\:\:\:\mathrm{Log}\:\left(\mathrm{sec}\beta\right)\:=\:\:\mathrm{Log}\:\mathrm{10}^{−\mathrm{p}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:−\mathrm{p}\:\mathrm{Log}\:\mathrm{10} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:−\mathrm{p} \\ $$

Question Number 117791 Answers: 3 Comments: 0

If k is an integer which satisfies 2sin^2 θ+10cos^2 (θ/2)=7−2k, then k∈? A.{0,1,2,3} B.{−1,0,1,2,3} C.{(−2,−1,0,1,2,3} D.{−3,−2,−1,0}

$$\mathrm{If}\:{k}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer}\:\mathrm{which}\:\mathrm{satisfies}\: \\ $$$$\mathrm{2sin}^{\mathrm{2}} \theta+\mathrm{10cos}^{\mathrm{2}} \frac{\theta}{\mathrm{2}}=\mathrm{7}−\mathrm{2}{k},\:\mathrm{then}\:{k}\in? \\ $$$${A}.\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\}\:\:\:{B}.\left\{−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\} \\ $$$${C}.\left\{\left(−\mathrm{2},−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\}\:\:{D}.\left\{−\mathrm{3},−\mathrm{2},−\mathrm{1},\mathrm{0}\right\}\right. \\ $$

Question Number 117767 Answers: 1 Comments: 1

x^2 +y_ ^2 =a^2 (√(2 )) x^2 +y^2 =a^2 what is intersection Angle=?

$${x}^{\mathrm{2}} +{y}_{} ^{\mathrm{2}} ={a}^{\mathrm{2}} \sqrt{\mathrm{2}\:} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={a}^{\mathrm{2}} \:\:\:\:\:\: \\ $$$$ \\ $$$${what}\:{is}\:{intersection}\:\:{Angle}=?\: \\ $$

Question Number 117759 Answers: 0 Comments: 1

(dθ/dx)=((√(1−(θ^2 /x^2 )))/(sin(θ+x)))

$$\frac{{d}\theta}{{dx}}=\frac{\sqrt{\mathrm{1}−\frac{\theta^{\mathrm{2}} }{{x}^{\mathrm{2}} }}}{{sin}\left(\theta+{x}\right)} \\ $$

Question Number 117754 Answers: 2 Comments: 1

Question Number 117739 Answers: 1 Comments: 0

what is the centre of the circle with radius 4(√2) that can be inscribed in the parabola y=x^2 −16x+128?

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{with}\:\mathrm{radius}\:\mathrm{4}\sqrt{\mathrm{2}}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\: \\ $$$$\mathrm{inscribed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{parabola}\: \\ $$$$\mathrm{y}=\mathrm{x}^{\mathrm{2}} −\mathrm{16x}+\mathrm{128}? \\ $$

Question Number 117728 Answers: 3 Comments: 0

Solution (d^2 y/dx^2 ) + 3(dy/dx) − 4y = x^2

$${Solution}\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{3}\frac{{dy}}{{dx}}\:−\:\mathrm{4}{y}\:=\:{x}^{\mathrm{2}} \\ $$

Question Number 117724 Answers: 2 Comments: 5

∫ ((sin^(−1) (x))/x^2 ) dx =?

$$\int\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Pg 1012 Pg 1013 Pg 1014 Pg 1015 Pg 1016 Pg 1017 Pg 1018 Pg 1019 Pg 1020 Pg 1021

Contact: info@tinkutara.com