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AlgebraQuestion and Answers: Page 71

Question Number 191663 Answers: 4 Comments: 0

2^a +4^b +8^c =328 find a,b and c when(a,b,c)is natual number

$$\mathrm{2}^{{a}} +\mathrm{4}^{{b}} +\mathrm{8}^{{c}} =\mathrm{328} \\ $$$${find}\:{a},{b}\:{and}\:{c} \\ $$$${when}\left({a},{b},{c}\right){is}\:{natual}\:{number} \\ $$

Question Number 191632 Answers: 0 Comments: 1

∫_0 ^∞ (√(tan θ)) dθ

$$\int_{\mathrm{0}} ^{\infty} \sqrt{\mathrm{tan}\:\theta}\:{d}\theta \\ $$

Question Number 191631 Answers: 1 Comments: 0

∫_0 ^∞ x^(1/2) e^(−x^2 ) dx

$$\int_{\mathrm{0}} ^{\infty} {x}^{\frac{\mathrm{1}}{\mathrm{2}}} {e}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 191624 Answers: 1 Comments: 0

Question Number 191623 Answers: 2 Comments: 0

Question Number 191615 Answers: 1 Comments: 0

a + b + c = 0. Prove that, (a/(a^2 − bc)) + (b/(b^2 − ca)) + (c/(c^2 − ab)) = 0.

$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0}.\:\mathrm{Prove}\:\mathrm{that}, \\ $$$$\frac{{a}}{{a}^{\mathrm{2}} \:−\:{bc}}\:+\:\frac{{b}}{{b}^{\mathrm{2}} \:−\:{ca}}\:+\:\frac{{c}}{{c}^{\mathrm{2}} \:−\:{ab}}\:=\:\mathrm{0}. \\ $$

Question Number 191614 Answers: 0 Comments: 1

((α^(100) +β^(100) )/(α^(100) −β^(100) )) = (((−w)^(100) +(−w^2 )^(100) )/((−w)^(100) −(−w^2 )^(100) )) = ((w^(100) +w^(200) )/(w^(100) −w^(200) )) = ((1+w^(100) )/(1−w^(100 ) )) = ((1+w)/(1−w)) = (2/(2w)) = (1/w) =

$$\frac{\alpha^{\mathrm{100}} +\beta^{\mathrm{100}} }{\alpha^{\mathrm{100}} −\beta^{\mathrm{100}} }\:=\: \\ $$$$\frac{\left(−{w}\right)^{\mathrm{100}} +\left(−{w}^{\mathrm{2}} \right)^{\mathrm{100}} }{\left(−{w}\right)^{\mathrm{100}} −\left(−{w}^{\mathrm{2}} \right)^{\mathrm{100}} } \\ $$$$=\:\frac{{w}^{\mathrm{100}} +{w}^{\mathrm{200}} }{{w}^{\mathrm{100}} −{w}^{\mathrm{200}} } \\ $$$$=\:\frac{\mathrm{1}+{w}^{\mathrm{100}} }{\mathrm{1}−{w}^{\mathrm{100}\:} } \\ $$$$=\:\frac{\mathrm{1}+{w}}{\mathrm{1}−{w}}\:=\:\frac{\mathrm{2}}{\mathrm{2}{w}}\:=\:\frac{\mathrm{1}}{{w}}\:=\: \\ $$

Question Number 191610 Answers: 1 Comments: 0

Q: if x+(1/x)=2cos(θ) prove it x^n +(1/x^n )=2cos(nθ)

$${Q}:\:{if}\:\:{x}+\frac{\mathrm{1}}{{x}}=\mathrm{2}{cos}\left(\theta\right)\:\:{prove}\:{it}\:\:{x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }=\mathrm{2}{cos}\left({n}\theta\right) \\ $$

Question Number 191589 Answers: 1 Comments: 1

a^x = bc, b^y = ca, c^z = ab. Prove that, (x/(1 + x)) + (y/(1 + y)) + (z/(1 + z)) = 2. (Without using log) a ≠ b ≠ c

$${a}^{{x}} \:=\:{bc},\:{b}^{{y}} \:=\:{ca},\:{c}^{{z}} \:=\:{ab}. \\ $$$$\mathrm{Prove}\:\mathrm{that},\:\frac{{x}}{\mathrm{1}\:+\:{x}}\:+\:\frac{{y}}{\mathrm{1}\:+\:{y}}\:+\:\frac{{z}}{\mathrm{1}\:+\:{z}}\:=\:\mathrm{2}. \\ $$$$\left(\mathrm{Without}\:\mathrm{using}\:\mathrm{log}\right) \\ $$$${a}\:\neq\:{b}\:\neq\:{c} \\ $$

Question Number 191569 Answers: 0 Comments: 0

Question Number 191555 Answers: 0 Comments: 1

Solve (√x) + y = 11 x + (√y) = 7

$$\mathrm{Solve} \\ $$$$\sqrt{{x}}\:+\:{y}\:=\:\mathrm{11} \\ $$$${x}\:+\:\sqrt{{y}}\:=\:\mathrm{7} \\ $$

Question Number 191553 Answers: 4 Comments: 1

If a^2 + a + 1 = 0 then find a^5 + a^4 + 1.

$$\mathrm{If}\:{a}^{\mathrm{2}} \:+\:{a}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find}\:{a}^{\mathrm{5}} \:+\:{a}^{\mathrm{4}} \:+\:\mathrm{1}. \\ $$

Question Number 191552 Answers: 2 Comments: 0

If x^2 − 3x + 1 = 0 then find the value of (x^2 + x + (1/x) + (1/x^2 ))^2

$$\mathrm{If}\:{x}^{\mathrm{2}} \:−\:\mathrm{3}{x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\left({x}^{\mathrm{2}} \:+\:{x}\:+\:\frac{\mathrm{1}}{{x}}\:+\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} \\ $$

Question Number 191546 Answers: 1 Comments: 0

If m + 1 = (√n) + 3 then find the value of (1/2)(((m^3 − 6m^2 + 12m −8)/( (√n))) − n)

$$\mathrm{If}\:{m}\:+\:\mathrm{1}\:=\:\sqrt{{n}}\:+\:\mathrm{3}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{m}^{\mathrm{3}} \:−\:\mathrm{6}{m}^{\mathrm{2}} \:+\:\mathrm{12}{m}\:−\mathrm{8}}{\:\sqrt{{n}}}\:−\:{n}\right) \\ $$

Question Number 191536 Answers: 2 Comments: 0

Factorize (2/(2x − 1)) −5 + (3/(3x − 1))

$$\mathrm{Factorize} \\ $$$$\frac{\mathrm{2}}{\mathrm{2}{x}\:−\:\mathrm{1}}\:−\mathrm{5}\:+\:\frac{\mathrm{3}}{\mathrm{3}{x}\:−\:\mathrm{1}} \\ $$

Question Number 191530 Answers: 0 Comments: 3

Question Number 191529 Answers: 2 Comments: 0

If ((x − y)/(x(√y) + y(√x))) = (1/( (√x))) ; (x > 0 and y > 0) then Find the value of (x/y) .

$$\mathrm{If}\:\frac{{x}\:−\:{y}}{{x}\sqrt{{y}}\:+\:{y}\sqrt{{x}}}\:=\:\frac{\mathrm{1}}{\:\sqrt{{x}}}\:;\:\left({x}\:>\:\mathrm{0}\:\mathrm{and}\:{y}\:>\:\mathrm{0}\right)\:\mathrm{then} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{{x}}{{y}}\:. \\ $$

Question Number 191527 Answers: 2 Comments: 0

x + y = 1 and x^2 + y^2 = 2. Find the value of x^(11) + y^(11) .

$${x}\:+\:{y}\:=\:\mathrm{1}\:\mathrm{and}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:\mathrm{2}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{x}^{\mathrm{11}} \:+\:{y}^{\mathrm{11}} . \\ $$

Question Number 191525 Answers: 1 Comments: 0

Q : Show that the numbers (√(3 )) , 2 & (√8) cannot be terms of an arithmetic sequence.

$${Q}\::\:{Show}\:{that}\:{the}\:{numbers}\:\sqrt{\mathrm{3}\:}\:,\:\mathrm{2}\:\&\:\sqrt{\mathrm{8}}\:{cannot}\:{be}\:{terms}\:{of}\:{an}\:{arithmetic}\:{sequence}. \\ $$

Question Number 191595 Answers: 1 Comments: 0

a = ((xy)/(x + y)) , b = ((xz)/(x + z)) and c = ((yz)/(y + z)) . Represent x in a, b, c form. [x, y, z ≠ 0]

$${a}\:=\:\frac{{xy}}{{x}\:+\:{y}}\:,\:{b}\:=\:\frac{{xz}}{{x}\:+\:{z}}\:\mathrm{and}\:{c}\:=\:\frac{{yz}}{{y}\:+\:{z}}\:. \\ $$$$\mathrm{Represent}\:{x}\:\mathrm{in}\:{a},\:{b},\:{c}\:\mathrm{form}.\:\left[{x},\:{y},\:{z}\:\neq\:\mathrm{0}\right] \\ $$

Question Number 191512 Answers: 1 Comments: 0

Q: the equation ⌊ cos(4x )⌋=m.cos(2x) has no solution . x∈ (0, (π/2) ) find the acceptable real values for ”m”.

$$ \\ $$$$\:\:\:\:\:{Q}:\:\:\:\:\:\:\:{the}\:{equation}\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\lfloor\:\mathrm{cos}\left(\mathrm{4}{x}\:\right)\rfloor={m}.\mathrm{cos}\left(\mathrm{2}{x}\right) \\ $$$$\:\:\:{has}\:{no}\:\:{solution}\:.\:\:{x}\in\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:{find}\:{the}\:{acceptable} \\ $$$$\:\:\:\:\:\:\:{real}\:{values}\:{for}\:\:\:\:''{m}''. \\ $$

Question Number 191508 Answers: 0 Comments: 0

Question Number 191486 Answers: 1 Comments: 0

a(√a) + b(√b) = 183 and b(√a) + a(√b) = 182 What is the value of (9/5) (a + b) ?

$${a}\sqrt{{a}}\:+\:{b}\sqrt{{b}}\:=\:\mathrm{183}\:\mathrm{and}\:{b}\sqrt{{a}}\:+\:{a}\sqrt{{b}}\:=\:\mathrm{182} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{9}}{\mathrm{5}}\:\left({a}\:+\:{b}\right)\:? \\ $$

Question Number 191473 Answers: 0 Comments: 0

Question Number 191456 Answers: 0 Comments: 0

Question Number 191454 Answers: 1 Comments: 2

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