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AlgebraQuestion and Answers: Page 355

Question Number 22057 Answers: 0 Comments: 0

∫f(x)dx=((d/dx))^2 f(x)=???

$$\int{f}\left({x}\right){dx}=\left(\frac{{d}}{{dx}}\right)^{\mathrm{2}} \\ $$$${f}\left({x}\right)=??? \\ $$

Question Number 22047 Answers: 2 Comments: 1

If x > 0 and the 4^(th) term in the expansion of (2 + (3/8)x)^(10) has maximum value then find the range of x.

$$\mathrm{If}\:{x}\:>\:\mathrm{0}\:\mathrm{and}\:\mathrm{the}\:\mathrm{4}^{\mathrm{th}} \:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{2}\:+\:\frac{\mathrm{3}}{\mathrm{8}}{x}\right)^{\mathrm{10}} \:\mathrm{has}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{x}. \\ $$

Question Number 22014 Answers: 0 Comments: 0

The number of solution(s) of the equation ∣log_e (∣x∣)∣ + ∣x∣ = 10 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solution}\left(\mathrm{s}\right)\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid\mathrm{log}_{{e}} \left(\mid{x}\mid\right)\mid\:+\:\mid{x}\mid\:=\:\mathrm{10}\:\mathrm{is} \\ $$

Question Number 21990 Answers: 1 Comments: 1

i still search about a general and complete solution about this determine x in N where 7 divise 2^x +3^x note = it is just an exercise in secondary so dont go away... maybe we must use separation of cases methode....

$${i}\:{still}\:{search}\:{about}\:{a}\:{general}\:{and}\: \\ $$$${complete}\:{solution}\:{about}\:{this} \\ $$$${determine}\:{x}\:{in}\:{N}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$${note}\:=\:{it}\:{is}\:{just}\:{an}\:{exercise}\:{in}\:{secondary} \\ $$$${so}\:{dont}\:{go}\:{away}... \\ $$$${maybe}\:{we}\:{must}\:{use}\:{separation}\:{of}\:{cases} \\ $$$${methode}.... \\ $$

Question Number 21940 Answers: 3 Comments: 1

A polynomial function f(x) satisfies f(x)f((1/x)) = 2f(x) + 2f((1/x)); x ≠ 0 and f(2) = 18, then f(3) is equal to

$$\mathrm{A}\:\mathrm{polynomial}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{satisfies} \\ $$$${f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{2}{f}\left({x}\right)\:+\:\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right);\:{x}\:\neq\:\mathrm{0}\:\mathrm{and} \\ $$$${f}\left(\mathrm{2}\right)\:=\:\mathrm{18},\:\mathrm{then}\:{f}\left(\mathrm{3}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 21877 Answers: 2 Comments: 3

x=^3 (√(7+5(√2)))+^3 (√(7−5(√2))) 1. According to a video, x=2 2. According to WolframAlpha, x≈0.2071+0.3587i for “principal root” and x=2(√2) for “real-valued root” 3. According to google, x≈2.8284 Please help and explain! :)

$${x}=^{\mathrm{3}} \sqrt{\mathrm{7}+\mathrm{5}\sqrt{\mathrm{2}}}+^{\mathrm{3}} \sqrt{\mathrm{7}−\mathrm{5}\sqrt{\mathrm{2}}} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{a}\:\mathrm{video},\:{x}=\mathrm{2} \\ $$$$\: \\ $$$$\mathrm{2}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{WolframAlpha}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{x}\approx\mathrm{0}.\mathrm{2071}+\mathrm{0}.\mathrm{3587}{i}\:\:\mathrm{for}\:``\mathrm{principal}\:\mathrm{root}'' \\ $$$$\mathrm{and}\:\:\:{x}=\mathrm{2}\sqrt{\mathrm{2}}\:\:\mathrm{for}\:``\mathrm{real}-\mathrm{valued}\:\mathrm{root}'' \\ $$$$\: \\ $$$$\mathrm{3}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{google},\:{x}\approx\mathrm{2}.\mathrm{8284} \\ $$$$\: \\ $$$$\left.\mathrm{Please}\:\mathrm{help}\:\mathrm{and}\:\mathrm{explain}!\:\:\::\right) \\ $$

Question Number 21874 Answers: 2 Comments: 0

Find the remainder if 2^(2006) is divided by 17

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:\:\:\mathrm{2}^{\mathrm{2006}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{17} \\ $$

Question Number 21867 Answers: 0 Comments: 0

The number of points in the cartesian plane with integral coordinates satisfying the inequalities ∣x∣ ≤ 4, ∣y∣ ≤ 4 and ∣x − y∣ ≤ 4 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{points}\:\mathrm{in}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{plane}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coordinates} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{inequalities}\:\mid{x}\mid\:\leqslant\:\mathrm{4},\:\mid{y}\mid\:\leqslant \\ $$$$\mathrm{4}\:\mathrm{and}\:\mid{x}\:−\:{y}\mid\:\leqslant\:\mathrm{4}\:\mathrm{is} \\ $$

Question Number 21795 Answers: 1 Comments: 3

help x∈N determine x where 7 divise 2^x +3^x

$${help} \\ $$$${x}\in{N} \\ $$$${determine}\:{x}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$$ \\ $$

Question Number 21733 Answers: 1 Comments: 0

If p is one of roots from x^2 − 2x + 6 = 0 then p^4 + 16p is equal to ...

$$\mathrm{If}\:{p}\:\mathrm{is}\:\mathrm{one}\:\mathrm{of}\:\:\mathrm{roots}\:\mathrm{from}\:{x}^{\mathrm{2}} \:−\:\mathrm{2}{x}\:+\:\mathrm{6}\:=\:\mathrm{0} \\ $$$$\mathrm{then}\:{p}^{\mathrm{4}} \:+\:\mathrm{16}{p}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21654 Answers: 0 Comments: 0

Find the minimum value of Q that satisfy: ∣xy(x^2 − y^2 ) + yz(y^2 − z^2 ) + zx(z^2 − x^2 )∣ ≤ Q(x^2 + y^2 + z^2 )^2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{Q}\:\mathrm{that}\:\mathrm{satisfy}: \\ $$$$\mid{xy}\left({x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \right)\:+\:{yz}\left({y}^{\mathrm{2}} \:−\:{z}^{\mathrm{2}} \right)\:+\:{zx}\left({z}^{\mathrm{2}} \:−\:{x}^{\mathrm{2}} \right)\mid\:\leqslant\:{Q}\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{2}} \\ $$

Question Number 21653 Answers: 0 Comments: 0

Find all pair of solutions (x,y) that satisfy the equation: ((7x^2 − 13xy + 7y^2 ))^(1/3) = ∣x − y∣ + 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{solutions}\:\left({x},{y}\right)\:\mathrm{that}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{13}{xy}\:+\:\mathrm{7}{y}^{\mathrm{2}} }\:=\:\mid{x}\:−\:{y}\mid\:+\:\mathrm{1} \\ $$

Question Number 21650 Answers: 0 Comments: 0

The three distinct successive terms of an A.P are the first,second and fourth terms of a G.P. If the sum to infinity of a G.P is 3+(√5) , find the first term.

$${The}\:{three}\:{distinct}\:{successive}\:{terms}\:{of}\:{an}\:{A}.{P}\:{are} \\ $$$${the}\:{first},{second}\:{and}\:{fourth}\:{terms}\:{of}\:{a}\:{G}.{P}.\:{If}\:{the}\: \\ $$$${sum}\:{to}\:{infinity}\:{of}\:{a}\:{G}.{P}\:{is}\:\mathrm{3}+\sqrt{\mathrm{5}}\:,\:{find}\: \\ $$$${the}\:{first}\:{term}. \\ $$$$ \\ $$

Question Number 21622 Answers: 2 Comments: 0

$$\mathrm{If}\:\mathrm{sec}\:{x}\:+\:\mathrm{tan}\:{x}\:=\:\mathrm{2012} \\ $$$$\mathrm{then}\:\mathrm{2011}\left(\mathrm{cosec}\:{x}\:+\:\mathrm{cot}\:{x}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left({A}\right)\:\mathrm{2011} \\ $$$$\left({B}\right)\:\mathrm{2012} \\ $$$$\left({C}\right)\:\mathrm{2013} \\ $$$$\left({D}\right)\:\frac{\mathrm{2011}}{\mathrm{2013}} \\ $$$$\left({E}\right)\:\frac{\mathrm{2013}}{\mathrm{2012}} \\ $$

Question Number 21643 Answers: 0 Comments: 1

(1/(1 + 1^2 + 1^4 )) + (2/(1 + 2^2 + 2^4 )) + (3/(1 + 3^2 + 3^4 )) + ... + ((2012)/(1 + 2012^2 + 2012^4 ))

$$\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{1}^{\mathrm{2}} \:+\:\mathrm{1}^{\mathrm{4}} }\:+\:\frac{\mathrm{2}}{\mathrm{1}\:+\:\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{2}^{\mathrm{4}} }\:+\:\frac{\mathrm{3}}{\mathrm{1}\:+\:\mathrm{3}^{\mathrm{2}} \:+\:\mathrm{3}^{\mathrm{4}} }\:+\:...\:+\:\frac{\mathrm{2012}}{\mathrm{1}\:+\:\mathrm{2012}^{\mathrm{2}} \:+\:\mathrm{2012}^{\mathrm{4}} } \\ $$

Question Number 21578 Answers: 0 Comments: 1

Find the whole part of A? A=(1/(√2))+(1/(√3))+(1/(√4))+......+(1/(√(9999)))+(1/(√(10000))).

$$\boldsymbol{\mathrm{Find}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{whole}}\:\:\boldsymbol{\mathrm{part}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{A}}? \\ $$$$\boldsymbol{\mathrm{A}}=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{4}}}+......+\frac{\mathrm{1}}{\sqrt{\mathrm{9999}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{10000}}}. \\ $$

Question Number 21541 Answers: 0 Comments: 0

Question Number 21531 Answers: 1 Comments: 2

Factorise the equation by factor theorem 12x^ 3 + 4x^ 2−3x−1

$${Factorise}\:{the}\:{equation}\:{by}\:{factor}\:{theorem} \\ $$$$\mathrm{12}\hat {{x}}\mathrm{3}\:+\:\mathrm{4}\hat {{x}}\mathrm{2}−\mathrm{3}{x}−\mathrm{1} \\ $$

Question Number 21526 Answers: 1 Comments: 3

Question Number 21498 Answers: 0 Comments: 0

a^(−2 ) + b^3 + c^(−4) = ((433)/(499)) Find a + b + c

$${a}^{−\mathrm{2}\:} +\:{b}^{\mathrm{3}} \:+\:{c}^{−\mathrm{4}} \:=\:\frac{\mathrm{433}}{\mathrm{499}} \\ $$$$\mathrm{Find}\:{a}\:+\:{b}\:+\:{c} \\ $$

Question Number 21493 Answers: 1 Comments: 0

Is always a÷b = (a/b) ?

$$\mathrm{Is}\:\mathrm{always}\:\mathrm{a}\boldsymbol{\div}\mathrm{b}\:=\:\frac{\mathrm{a}}{\mathrm{b}}\:? \\ $$

Question Number 21471 Answers: 2 Comments: 0

If a + b + c = 0, then (((a + b)(b + c)(a + c))/(abc)) is equal to ...

$$\mathrm{If}\:\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then} \\ $$$$\frac{\left({a}\:+\:{b}\right)\left({b}\:+\:{c}\right)\left({a}\:+\:{c}\right)}{{abc}}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21470 Answers: 1 Comments: 1

2^x = 3^y = 6^(−z) Find the value of (((2017)/x) + ((2017)/y) + ((2017)/z))^(2017)

$$\mathrm{2}^{{x}} \:=\:\mathrm{3}^{{y}} \:=\:\mathrm{6}^{−{z}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{2017}}{{x}}\:+\:\frac{\mathrm{2017}}{{y}}\:+\:\frac{\mathrm{2017}}{{z}}\right)^{\mathrm{2017}} \\ $$

Question Number 21463 Answers: 0 Comments: 0

Let A be the collection of functions f : [0, 1] → R which have an infinite number of derivatives. Let A_0 ⊂ A be the subcollection of those functions f with f(0) = 0. Define D : A_0 → A by D(f) = df/dx. Use the mean value theorem to show that D is injective. Use the fundamental theorem of calculus to show that D is surjective.

$$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{collection}\:\mathrm{of}\:\mathrm{functions} \\ $$$${f}\::\:\left[\mathrm{0},\:\mathrm{1}\right]\:\rightarrow\:\mathbb{R}\:\mathrm{which}\:\mathrm{have}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{derivatives}.\:\mathrm{Let}\:{A}_{\mathrm{0}} \:\subset\:{A} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{subcollection}\:\mathrm{of}\:\mathrm{those}\:\mathrm{functions} \\ $$$${f}\:\mathrm{with}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{0}.\:\mathrm{Define}\:{D}\::\:{A}_{\mathrm{0}} \:\rightarrow\:{A} \\ $$$$\mathrm{by}\:{D}\left({f}\right)\:=\:{df}/{dx}.\:\mathrm{Use}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value} \\ $$$$\mathrm{theorem}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{injective}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{fundamental}\:\mathrm{theorem}\:\mathrm{of} \\ $$$$\mathrm{calculus}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{surjective}. \\ $$

Question Number 21422 Answers: 1 Comments: 3

Find all integer values of a such that the quadratic expression (x + a)(x + 1991) + 1 can be factored as a product (x + b)(x + c) where b and c are integers.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:{a}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{expression} \\ $$$$\left({x}\:+\:{a}\right)\left({x}\:+\:\mathrm{1991}\right)\:+\:\mathrm{1}\:\mathrm{can}\:\mathrm{be}\:\mathrm{factored} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{product}\:\left({x}\:+\:{b}\right)\left({x}\:+\:{c}\right)\:\mathrm{where}\:{b}\:\mathrm{and} \\ $$$${c}\:\mathrm{are}\:\mathrm{integers}. \\ $$

Question Number 21357 Answers: 1 Comments: 0

Solve : log_(2x+3) x^2 < 1

$$\mathrm{Solve}\::\:\mathrm{log}_{\mathrm{2}{x}+\mathrm{3}} {x}^{\mathrm{2}} \:<\:\mathrm{1} \\ $$

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