Question and Answers Forum

AlgebraQuestion and Answers: Page 327

Question Number 33569 Answers: 1 Comments: 0

Given f(x) = x^3 + ax^2 + bx + c with a, b, c ∈ R, the roots are x_1 , x_2 , x_3 ∈ R Let λ is an positive integer that satisfied x_2 − x_1 = λ x_3 > (1/2)(x_1 + x_2 ) What is the max value of ((2a^3 + 27c − 9ab)/λ^3 ) ?

$$\mathrm{Given}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c} \\ $$$$\mathrm{with}\:{a},\:{b},\:{c}\:\in\:\mathbb{R},\:\mathrm{the}\:\mathrm{roots}\:\mathrm{are}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} \:\in\:\mathbb{R} \\ $$$$\mathrm{Let}\:\lambda\:\mathrm{is}\:\mathrm{an}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{satisfied} \\ $$$${x}_{\mathrm{2}} \:−\:{x}_{\mathrm{1}} \:=\:\lambda \\ $$$${x}_{\mathrm{3}} \:>\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}_{\mathrm{1}} \:+\:{x}_{\mathrm{2}} \right) \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{max}\:\mathrm{value}\:\mathrm{of}\:\:\frac{\mathrm{2}{a}^{\mathrm{3}} \:+\:\mathrm{27}{c}\:−\:\mathrm{9}{ab}}{\lambda^{\mathrm{3}} }\:? \\ $$

Question Number 33518 Answers: 0 Comments: 0

α^4 +β^(4 ) solve please

$$\alpha^{\mathrm{4}} +\beta^{\mathrm{4}\:} {solve}\:{please} \\ $$

Question Number 33515 Answers: 0 Comments: 1

expand α^4 +β^(β ) please

$${expand}\:\alpha^{\mathrm{4}} +\beta^{\beta\:\:} {please} \\ $$

Question Number 33496 Answers: 1 Comments: 0

prove that e^(iπ) +1=0

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{e}^{\mathrm{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$

Question Number 33473 Answers: 1 Comments: 3

e^(iπ ) = −1 squaring both sides e^(2πi) = 1 = e^0 comparing powers 2πi = 0 π = 0 or i = 0 ???

$$\:{e}^{{i}\pi\:} =\:−\mathrm{1} \\ $$$${squaring}\:{both}\:{sides} \\ $$$${e}^{\mathrm{2}\pi{i}} \:=\:\mathrm{1}\:=\:{e}^{\mathrm{0}} \\ $$$${comparing}\:{powers} \\ $$$$\mathrm{2}\pi{i}\:=\:\mathrm{0} \\ $$$$\:\pi\:=\:\mathrm{0}\:{or}\:{i}\:=\:\mathrm{0}\:??? \\ $$

Question Number 33468 Answers: 1 Comments: 0

The set of integers that satisfies 5>∣n−2∣≥∣n+1∣ is

$${The}\:{set}\:{of}\:{integers}\:{that}\:{satisfies} \\ $$$$\mathrm{5}>\mid{n}−\mathrm{2}\mid\geqslant\mid{n}+\mathrm{1}\mid\:{is} \\ $$

Question Number 33307 Answers: 0 Comments: 0

let z=x+iy with x≠0 prove?that ∣ ((e^z −1)/z) ∣≤∣ ((e^x −1)/x) ∣

$${let}\:{z}={x}+{iy}\:\:{with}\:{x}\neq\mathrm{0}\:{prove}?{that} \\ $$$$\mid\:\frac{{e}^{{z}} \:−\mathrm{1}}{{z}}\:\mid\leqslant\mid\:\frac{{e}^{{x}} \:−\mathrm{1}}{{x}}\:\mid \\ $$

Question Number 33240 Answers: 0 Comments: 3

Question Number 33200 Answers: 1 Comments: 0

Solve 2^(3n+2) −7×2^(2n+2) −31×2^n −8=0, n∈R. I need some help with this

$$\mathrm{Solve}\:\mathrm{2}^{\mathrm{3n}+\mathrm{2}} \:−\mathrm{7}×\mathrm{2}^{\mathrm{2n}+\mathrm{2}} \:−\mathrm{31}×\mathrm{2}^{\mathrm{n}} \:−\mathrm{8}=\mathrm{0},\:\mathrm{n}\in\boldsymbol{\mathrm{R}}. \\ $$$$\mathrm{I}\:\mathrm{need}\:\mathrm{some}\:\mathrm{help}\:\mathrm{with}\:\mathrm{this} \\ $$

Question Number 33193 Answers: 1 Comments: 0

Q. If α is a root of the equation x^3 −3x−1=0, prove that the other roots are 2−α^2 and α^2 −α−2. Please help.

$$\mathrm{Q}.\:\:\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{3}} −\mathrm{3}{x}−\mathrm{1}=\mathrm{0}, \\ $$$$\:\:\:\:\:\:\:\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{other}\:\mathrm{roots}\:\mathrm{are} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{2}−\alpha^{\mathrm{2}} \:\mathrm{and}\:\alpha^{\mathrm{2}} −\alpha−\mathrm{2}. \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Please}\:\mathrm{help}. \\ $$

Question Number 33186 Answers: 1 Comments: 0

Find the exact value of sinθ if cosθ=(1/(57)) and θ is obtuse

$${Find}\:{the}\:{exact}\:{value}\:{of}\:{sin}\theta\:{if} \\ $$$${cos}\theta=\frac{\mathrm{1}}{\mathrm{57}}\:{and}\:\theta\:{is}\:{obtuse} \\ $$

Question Number 33032 Answers: 1 Comments: 5

f:N→R f(1)=2005. and f(1)+f(2)+......+f(n)= n^2 f(n),n>1. Then f(2004)=?

$${f}:{N}\rightarrow{R} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{2005}. \\ $$$${and}\: \\ $$$${f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+......+{f}\left({n}\right)=\:{n}^{\mathrm{2}} \:{f}\left({n}\right),{n}>\mathrm{1}. \\ $$$${Then}\:{f}\left(\mathrm{2004}\right)=? \\ $$

Question Number 32977 Answers: 1 Comments: 0

Prove that ^n C_r +^n C_(r+1) =^(n+1) C_(r+1)

$${Prove}\:{that}\:\:^{{n}} {C}_{{r}} \:\:+\:^{{n}} {C}_{{r}+\mathrm{1}} \:=\:^{{n}+\mathrm{1}} {C}_{{r}+\mathrm{1}} \\ $$$$ \\ $$

Question Number 32889 Answers: 0 Comments: 1

Question Number 32868 Answers: 0 Comments: 0

Question Number 32832 Answers: 0 Comments: 0

Determine n such that 1001n+1 is perfect cube.

$$\mathrm{Determine}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{1001n}+\mathrm{1}\:\mathrm{is} \\ $$$$\mathrm{perfect}\:\mathrm{cube}. \\ $$

Question Number 32788 Answers: 0 Comments: 1

The least positive integral value of ′x′ satisfying : (e^x −2)(sin (x+(π/4)))(x−log_e 2_ )(sinx − cosx)<0

$$\boldsymbol{{T}}{he}\:{least}\:{positive}\:{integral}\:{value}\:{of} \\ $$$$'{x}'\:{satisfying}\:: \\ $$$$\left({e}^{{x}} −\mathrm{2}\right)\left(\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\right)\left({x}−\mathrm{log}_{{e}} \:\underset{} {\mathrm{2}}\right)\left({sinx}\:−\:{cosx}\right)<\mathrm{0} \\ $$

Question Number 32780 Answers: 2 Comments: 0

10−8=(p/9)

$$\mathrm{10}−\mathrm{8}=\frac{{p}}{\mathrm{9}} \\ $$

Question Number 32768 Answers: 1 Comments: 0

Prove that a^2 +b^2 +c^2 ≥ab+bc+ca ∀ a,b,c∈R

$$\mathrm{Prove}\:\mathrm{that}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \geqslant{ab}+{bc}+{ca} \\ $$$$\forall\:{a},{b},{c}\in\mathbb{R} \\ $$

Question Number 32767 Answers: 0 Comments: 2

For a,b,c≥0 if a+b+c=n, determine minimum and maximum values of a^2 +b^2 +c^2 −ab−bc−ca.

$$\mathrm{For}\:{a},{b},{c}\geqslant\mathrm{0}\:\mathrm{if}\:{a}+{b}+{c}={n},\:\mathrm{determine} \\ $$$$\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{values}\:\mathrm{of} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}. \\ $$

Question Number 32682 Answers: 1 Comments: 0

If x_1 and x_(2 ) are roots of the equation acos 2x+bsin x = c and 2sin x_1 sinx_2 = sin x_1 +sinx_2 . Then the value of (b/(c−a)) is ?

$$\boldsymbol{{I}}{f}\:{x}_{\mathrm{1}} \:{and}\:{x}_{\mathrm{2}\:} \:{are}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${acos}\:\mathrm{2}{x}+{bsin}\:{x}\:=\:{c}\:{and}\: \\ $$$$\mathrm{2}{sin}\:{x}_{\mathrm{1}} {sinx}_{\mathrm{2}} =\:{sin}\:{x}_{\mathrm{1}} +{sinx}_{\mathrm{2}} .\:\boldsymbol{{T}}{hen}\: \\ $$$${the}\:{value}\:{of}\:\:\frac{{b}}{{c}−{a}}\:{is}\:? \\ $$

Question Number 32681 Answers: 0 Comments: 2

Total no. of polynomials of the form x^3 +ax^2 +bx+c that are divisible by x^2 +1, where a,b,c∈1,2,3,....,10 is 1) 10 2) 15 3) 5 4) 8

$$\boldsymbol{{T}}{otal}\:{no}.\:{of}\:{polynomials}\:{of}\:{the}\:{form} \\ $$$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}\:\:{that}\:{are}\:{divisible}\:{by}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{1},\:{where}\:{a},{b},{c}\in\mathrm{1},\mathrm{2},\mathrm{3},....,\mathrm{10}\:{is}\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{10} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{15} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{5} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{8} \\ $$

Question Number 32650 Answers: 2 Comments: 0

f(x)=8x−34(√(25−4 (3/2)))

$${f}\left({x}\right)=\mathrm{8}{x}−\mathrm{34}\sqrt{\mathrm{25}−\mathrm{4}\:\frac{\mathrm{3}}{\mathrm{2}}} \\ $$

Question Number 32648 Answers: 1 Comments: 0

Question Number 32647 Answers: 1 Comments: 0

Question Number 32632 Answers: 0 Comments: 2

Pg 322 Pg 323 Pg 324 Pg 325 Pg 326 Pg 327 Pg 328 Pg 329 Pg 330 Pg 331

Contact: info@tinkutara.com