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AlgebraQuestion and Answers: Page 223

Question Number 133318 Answers: 1 Comments: 0

(a) Are there any graphs with 5 vertices which have vertices of degrees 1,2,3,4 and 5?

$$\left(\mathrm{a}\right)\:\mathrm{Are}\:\mathrm{there}\:\mathrm{any}\:\mathrm{graphs}\:\mathrm{with}\:\mathrm{5} \\ $$$$\mathrm{vertices}\:\mathrm{which}\:\mathrm{have}\:\mathrm{vertices}\:\mathrm{of}\: \\ $$$$\mathrm{degrees}\:\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:\mathrm{and}\:\mathrm{5}?\: \\ $$

Question Number 133279 Answers: 0 Comments: 0

hi, everybody ! with n ∈ N, prove that : ∃ n_0 ∈ N / ∀ n ≥ n_0 , n^2 ≤ 2^n .

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{everybody}}\:! \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{n}}\:\in\:\mathbb{N}, \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\::\:\exists\:\boldsymbol{\mathrm{n}}_{\mathrm{0}} \:\in\:\mathbb{N}\:/\:\forall\:\boldsymbol{\mathrm{n}}\:\geqslant\:\boldsymbol{\mathrm{n}}_{\mathrm{0}} \:,\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:\leqslant\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} . \\ $$

Question Number 133201 Answers: 2 Comments: 0

(√(81))

$$\sqrt{\mathrm{81}} \\ $$

Question Number 133180 Answers: 1 Comments: 3

Question Number 133072 Answers: 1 Comments: 0

Show that (5/(2−(3)^(1/4) )) is in F_2 by expressing the number in form a_1 +b_1 (√k_1 ) where a_1 ,b_1 , k_1 are in F_1

$$\mathrm{Show}\:\mathrm{that}\:\frac{\mathrm{5}}{\mathrm{2}−\sqrt[{\mathrm{4}}]{\mathrm{3}}}\:\mathrm{is}\:\mathrm{in}\:\mathrm{F}_{\mathrm{2}} \:\mathrm{by}\:\mathrm{expressing} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{in}\:\mathrm{form}\:{a}_{\mathrm{1}} +{b}_{\mathrm{1}} \sqrt{{k}_{\mathrm{1}} }\:\mathrm{where} \\ $$$${a}_{\mathrm{1}} ,{b}_{\mathrm{1}} ,\:{k}_{\mathrm{1}} \:{are}\:{in}\:{F}_{\mathrm{1}} \\ $$

Question Number 133053 Answers: 1 Comments: 0

When the polynomial f(x) is divided by (x−2) the remainder is 4 and when it is divided (x−3) the remainder is 7. Given that f(x) may be written in the formf(x)=(x−2)(x−3)Q(x)+ax+b, find the remainder when f(x) is divided by (x−2)(x−3). If also f(x) is a cubic function in which the coefficient of x^3 is unity and f(1)=1, determine Q(x).

$$\mathrm{When}\:\mathrm{the}\:\mathrm{polynomial}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left(\mathrm{x}−\mathrm{2}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\left(\mathrm{x}−\mathrm{3}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{7}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{f}\left({x}\right) \\ $$$$\mathrm{may}\:\mathrm{be}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{formf}\left({x}\right)=\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)\mathrm{Q}\left(\mathrm{x}\right)+\mathrm{ax}+\mathrm{b}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right).\:\mathrm{If}\:\mathrm{also}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{cubic}\:\mathrm{function} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} \:\mathrm{is}\:\mathrm{unity}\:\mathrm{and} \\ $$$$\mathrm{f}\left(\mathrm{1}\right)=\mathrm{1},\:\mathrm{determine}\:\mathrm{Q}\left(\mathrm{x}\right). \\ $$

Question Number 133009 Answers: 1 Comments: 1

Question Number 132961 Answers: 1 Comments: 0

Question Number 132900 Answers: 0 Comments: 1

a pinhole camera is placed 300cm in front of a building so that the image is formed on a screen 5cm from the pin hole. if the image is 2.5cm high. the height of the building will be?

$${a}\:{pinhole}\:{camera}\:{is}\:{placed}\:\mathrm{300}{cm} \\ $$$${in}\:{front}\:{of}\:{a}\:{building}\:{so}\:{that}\:{the}\:{image} \\ $$$${is}\:{formed}\:{on}\:{a}\:{screen}\:\mathrm{5}{cm}\:{from}\:{the} \\ $$$${pin}\:{hole}.\:{if}\:{the}\:{image}\:{is}\:\mathrm{2}.\mathrm{5}{cm}\:{high}. \\ $$$${the}\:{height}\:{of}\:{the}\:{building}\:{will}\:{be}? \\ $$

Question Number 132697 Answers: 3 Comments: 2

Find the condition that one root of ax^2 +bx+c = 0 ,a≠ 0 is square of the other .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{that}\:\mathrm{one} \\ $$$$\mathrm{root}\:\mathrm{of}\:{ax}^{\mathrm{2}} +{bx}+{c}\:=\:\mathrm{0}\:,{a}\neq\:\mathrm{0} \\ $$$$\mathrm{is}\:\mathrm{square}\:\mathrm{of}\:\mathrm{the}\:\mathrm{other}\:. \\ $$

Question Number 132630 Answers: 0 Comments: 0