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AlgebraQuestion and Answers: Page 310

Question Number 56697 Answers: 2 Comments: 0

Find the shortest distance between the lines L = (1, 4, 2) + N(1, 3, 2) and r = (−1, 1, −1) + λ(1, 2, −1)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{lines} \\ $$$$\:\:\:\mathrm{L}\:\:=\:\:\left(\mathrm{1},\:\mathrm{4},\:\mathrm{2}\right)\:+\:\mathrm{N}\left(\mathrm{1},\:\mathrm{3},\:\mathrm{2}\right)\:\:\:\mathrm{and} \\ $$$$\:\:\:\mathrm{r}\:\:=\:\:\left(−\mathrm{1},\:\mathrm{1},\:−\mathrm{1}\right)\:+\:\lambda\left(\mathrm{1},\:\mathrm{2},\:−\mathrm{1}\right) \\ $$

Question Number 56696 Answers: 0 Comments: 2

Find the perpendicular distance from (1, 7, 1) to 3x − 2y + 2z = 6

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{distance}\:\mathrm{from}\:\:\left(\mathrm{1},\:\mathrm{7},\:\mathrm{1}\right)\:\:\mathrm{to}\:\:\mathrm{3x}\:−\:\mathrm{2y}\:+\:\mathrm{2z}\:\:=\:\:\mathrm{6} \\ $$

Question Number 56685 Answers: 1 Comments: 1

show that α^4 +β^4 = (α^2 +β^2 )^2 −2α^2 β^2

$$\mathrm{show}\:\mathrm{that}\:\alpha^{\mathrm{4}} +\beta^{\mathrm{4}} \:=\:\left(\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} \right)^{\mathrm{2}} \:−\mathrm{2}\alpha^{\mathrm{2}} \beta^{\mathrm{2}} \\ $$

Question Number 56643 Answers: 4 Comments: 2

Question Number 56641 Answers: 1 Comments: 2

x^5 +ax^4 +cx^2 +dx+e=0 let x=((rt+s)/(t+p)) . Find r,s,p such that equation gets transformed to λt^5 +Dt+E=0.

$$\mathrm{x}^{\mathrm{5}} +\mathrm{ax}^{\mathrm{4}} +\mathrm{cx}^{\mathrm{2}} +\mathrm{dx}+\mathrm{e}=\mathrm{0} \\ $$$$\mathrm{let}\:\mathrm{x}=\frac{\mathrm{rt}+\mathrm{s}}{\mathrm{t}+\mathrm{p}}\:\:.\:\mathrm{Find}\:\mathrm{r},\mathrm{s},\mathrm{p}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{equation}\:\mathrm{gets}\:\mathrm{transformed} \\ $$$$\mathrm{to}\:\:\:\:\:\lambda\mathrm{t}^{\mathrm{5}} +\mathrm{Dt}+\mathrm{E}=\mathrm{0}. \\ $$

Question Number 56602 Answers: 1 Comments: 1

Sum the series: sin^2 (α) + sin^2 (2α) + sin^2 (3α) + ... + sin^2 (nα)

$$\mathrm{Sum}\:\mathrm{the}\:\mathrm{series}:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{sin}^{\mathrm{2}} \left(\alpha\right)\:+\:\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}\alpha\right)\:+\:\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}\alpha\right)\:+\:...\:+\:\mathrm{sin}^{\mathrm{2}} \left(\mathrm{n}\alpha\right) \\ $$

Question Number 56575 Answers: 1 Comments: 2

Question Number 56525 Answers: 4 Comments: 2

Question Number 56479 Answers: 1 Comments: 0

Please is there any way to reduce a polynomial of 4th degree and solve. Or probably a polynomial of nth power to smaller power.

$$\mathrm{Please}\:\mathrm{is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{way}\:\mathrm{to}\:\mathrm{reduce}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of}\:\:\mathrm{4th}\:\mathrm{degree} \\ $$$$\mathrm{and}\:\mathrm{solve}.\:\:\mathrm{Or}\:\mathrm{probably}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of}\:\:\:\mathrm{nth}\:\mathrm{power}\:\mathrm{to}\:\mathrm{smaller} \\ $$$$\mathrm{power}.\: \\ $$

Question Number 56386 Answers: 0 Comments: 0

Question Number 56356 Answers: 3 Comments: 1

Find the minimum value of the function F(x) = log_e x − x for x > 0. Hence show that: log_e x ≤ x − 1. For all x > 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{F}\left(\mathrm{x}\right)\:=\:\:\mathrm{log}_{\mathrm{e}} \mathrm{x}\:\:−\:\:\mathrm{x}\:\:\:\mathrm{for}\:\:\:\mathrm{x}\:>\:\mathrm{0}. \\ $$$$\:\:\mathrm{Hence}\:\mathrm{show}\:\mathrm{that}:\:\:\:\mathrm{log}_{\mathrm{e}} \mathrm{x}\:\:\leqslant\:\:\mathrm{x}\:−\:\mathrm{1}.\:\:\mathrm{For}\:\mathrm{all}\:\:\:\mathrm{x}\:>\:\mathrm{0} \\ $$

Question Number 56335 Answers: 2 Comments: 0

If b^x = ((b/k))^y = k^m and b ≠ 1 show that: (1/x) = (1/y) = (1/m)

$$\mathrm{If}\:\:\:\:\:\:\mathrm{b}^{\mathrm{x}} \:\:=\:\:\left(\frac{\mathrm{b}}{\mathrm{k}}\right)^{\mathrm{y}} \:\:=\:\:\mathrm{k}^{\mathrm{m}} \:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\mathrm{b}\:\neq\:\mathrm{1} \\ $$$$\mathrm{show}\:\mathrm{that}:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:=\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:\frac{\mathrm{1}}{\mathrm{m}} \\ $$

Question Number 56316 Answers: 1 Comments: 1

Question Number 56264 Answers: 0 Comments: 3

(x_C −h)^2 +3((s/2)−x_C )^2 = a^2 (x_A −h)^2 +3((s/2)+x_A )^2 = c^2 (x_C −x_A )^2 = b^2 /4 .

$$\left({x}_{{C}} −{h}\right)^{\mathrm{2}} +\mathrm{3}\left(\frac{{s}}{\mathrm{2}}−{x}_{{C}} \right)^{\mathrm{2}} \:=\:{a}^{\mathrm{2}} \\ $$$$\:\left({x}_{{A}} −{h}\right)^{\mathrm{2}} +\mathrm{3}\left(\frac{{s}}{\mathrm{2}}+{x}_{{A}} \right)^{\mathrm{2}} =\:{c}^{\mathrm{2}} \\ $$$$\:\:\left({x}_{{C}} −{x}_{{A}} \right)^{\mathrm{2}} \:=\:{b}^{\mathrm{2}} /\mathrm{4}\:. \\ $$

Question Number 56192 Answers: 1 Comments: 0

find z_1 ,z_2 ∈C (1/(z_1 +z_2 ))=(1/z_1 )+(1/z_2 )

$$\mathrm{find}\:{z}_{\mathrm{1}} ,{z}_{\mathrm{2}} \in\mathbb{C} \\ $$$$\frac{\mathrm{1}}{{z}_{\mathrm{1}} +{z}_{\mathrm{2}} }=\frac{\mathrm{1}}{{z}_{\mathrm{1}} }+\frac{\mathrm{1}}{{z}_{\mathrm{2}} } \\ $$

Question Number 56183 Answers: 1 Comments: 0

find all a,b∈R such that (1/(a+bi))=(1/a)+(i/b)

$$\mathrm{find}\:\mathrm{all}\:{a},{b}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}+{bi}}=\frac{\mathrm{1}}{{a}}+\frac{{i}}{{b}} \\ $$

Question Number 56179 Answers: 1 Comments: 1

draw the graph of f(x)=(√(1−x^2 )) for 0≤x≤1

$$\mathrm{draw}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of} \\ $$$$\mathrm{f}\left({x}\right)=\sqrt{\mathrm{1}−{x}^{\mathrm{2}} } \\ $$$$\mathrm{for}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$

Question Number 56166 Answers: 0 Comments: 1

Question Number 56147 Answers: 1 Comments: 6

if ∫_( 1) ^( 2) f(x) dx = (√( 2 )), then ∫_( 1) ^( 4) (1/((√( x )) )) f(x) dx is ?? please help me Sir. I′ve been trying this for 2 days and getting stuck.

$$\mathrm{if}\:\underset{\:\:\mathrm{1}} {\overset{\:\:\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:=\:\sqrt{\:\mathrm{2}\:},\:\mathrm{then}\:\underset{\:\:\mathrm{1}} {\overset{\:\:\mathrm{4}} {\int}}\:\frac{\mathrm{1}}{\sqrt{\:{x}\:}\:}\:{f}\left({x}\right)\:{dx} \\ $$$$\:\mathrm{is}\:?? \\ $$$$\:\:\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{Sir}.\:\mathrm{I}'\mathrm{ve}\:\mathrm{been}\:\mathrm{trying} \\ $$$$\:\:\mathrm{this}\:\mathrm{for}\:\mathrm{2}\:\mathrm{days}\:\mathrm{and}\:\mathrm{getting}\:\mathrm{stuck}. \\ $$$$ \\ $$$$ \\ $$

Question Number 56145 Answers: 1 Comments: 0

find residu of function f(z)=(e^(1/z) /(z^2 +1)) in z=0

$$\mathrm{find}\:\mathrm{residu}\:\mathrm{of}\:\mathrm{function} \\ $$$${f}\left({z}\right)=\frac{{e}^{\frac{\mathrm{1}}{{z}}} }{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{in}\:{z}=\mathrm{0} \\ $$

Question Number 56144 Answers: 1 Comments: 0

calculate (i−1)^(49) (cos (π/(40))+i sin (π/(40)))^(10)

$$\mathrm{calculate}\:\left({i}−\mathrm{1}\right)^{\mathrm{49}} \left(\mathrm{cos}\:\frac{\pi}{\mathrm{40}}+{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{40}}\right)^{\mathrm{10}} \\ $$

Question Number 56092 Answers: 2 Comments: 0

How to rationalize a denominator in a fraction? Like this. (((x)^(1/3) +2)/((x)^(1/3) −2))

$${How}\:{to}\:{rationalize}\:{a}\:{denominator}\:{in} \\ $$$${a}\:{fraction}?\:{Like}\:{this}. \\ $$$$\frac{\sqrt[{\mathrm{3}}]{{x}}+\mathrm{2}}{\sqrt[{\mathrm{3}}]{{x}}−\mathrm{2}}\: \\ $$

Question Number 56042 Answers: 2 Comments: 0

Question Number 56052 Answers: 1 Comments: 0

Solve following equation for x: x^x^x^n =n with n∈N

$${Solve}\:{following}\:{equation}\:{for}\:{x}: \\ $$$$\boldsymbol{{x}}^{\boldsymbol{{x}}^{\boldsymbol{{x}}^{\boldsymbol{{n}}} } } =\boldsymbol{{n}}\:{with}\:{n}\in\mathbb{N} \\ $$

Question Number 55969 Answers: 1 Comments: 0

solve for x & y x^(logy) = 4 xy = 40

$$\:\:\:\boldsymbol{\mathrm{solve}}\:\:\:\boldsymbol{\mathrm{for}}\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\&\:\:\boldsymbol{\mathrm{y}} \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{logy}}} \:\:\:=\:\:\mathrm{4} \\ $$$$\:\: \\ $$$$\:\:\:\:\boldsymbol{\mathrm{xy}}\:\:\:\:=\:\:\:\:\mathrm{40}\: \\ $$

Question Number 55954 Answers: 1 Comments: 0

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