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AlgebraQuestion and Answers: Page 250
Question Number 98205 Answers: 1 Comments: 0
$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{nth}}\:\:\boldsymbol{\mathrm{term}}\:\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:\:=\:\:\mathrm{1},\:\:\:\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}} \:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:\:+\:\:\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:−\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\:−\:\:\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}\right)^{\mathrm{2}} }\:\:\:\:\left(\boldsymbol{\mathrm{n}}\:\:=\:\:\mathrm{1},\:\:\mathrm{2},\:\:\mathrm{3},\:\:...\right) \\ $$
Question Number 98192 Answers: 2 Comments: 1
Question Number 98293 Answers: 2 Comments: 0
Question Number 98130 Answers: 0 Comments: 0
Question Number 98077 Answers: 0 Comments: 4
Question Number 98073 Answers: 1 Comments: 2
$$\mathrm{solve}\:\sqrt{\mathrm{6}−\mathrm{x}}\:=\:\mathrm{6}−\mathrm{x}^{\mathrm{2}} \\ $$
Question Number 98067 Answers: 2 Comments: 0
Question Number 97968 Answers: 1 Comments: 1
$$ \\ $$
Question Number 99300 Answers: 1 Comments: 0
$$\mathrm{Find}\:\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}\boldsymbol{{n}}\right)!}=? \\ $$
Question Number 97936 Answers: 1 Comments: 1
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\sqrt{\mathrm{45}−\sqrt{\mathrm{2000}}\:}\:\:+\:\:\sqrt{\mathrm{45}+\sqrt{\mathrm{2000}}}\: \\ $$
Question Number 97920 Answers: 0 Comments: 4
Question Number 97919 Answers: 1 Comments: 2
Question Number 97818 Answers: 1 Comments: 0
$$\boldsymbol{\mathrm{if}}\:\:\:\:\:\boldsymbol{\mathrm{y}}^{\mathrm{2}} \:\:=\:\:\boldsymbol{\mathrm{ax}}^{\mathrm{2}} \:+\:\boldsymbol{\mathrm{bx}}\:+\:\:\boldsymbol{\mathrm{c}} \\ $$$$\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{3}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{3}} }\:\:+\:\:\mathrm{3}\:\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}} }\:\:\:=\:\:\:\mathrm{0} \\ $$
Question Number 97808 Answers: 0 Comments: 1
Question Number 97694 Answers: 1 Comments: 7
Question Number 97675 Answers: 1 Comments: 0
Question Number 97637 Answers: 0 Comments: 1
$$\mathrm{Given}\:\mathrm{p},\mathrm{q}\in\mathbb{R}_{+} ^{\ast} −\left\{−\mathrm{1}\right\}/\frac{\mathrm{1}}{\mathrm{p}}+\frac{\mathrm{1}}{\mathrm{q}}=\mathrm{1}\:\mathrm{show}\:\mathrm{that}; \\ $$$$\forall\mathrm{a},\mathrm{b}\:\in\mathbb{R}\:\mathrm{ab}\leqslant\frac{\mathrm{a}^{\mathrm{p}} }{\mathrm{p}}+\frac{\mathrm{b}^{\mathrm{q}} }{\mathrm{q}} \\ $$
Question Number 97564 Answers: 1 Comments: 0
Question Number 97512 Answers: 0 Comments: 2
$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the} \\ $$$$\mathrm{quadratic}\:\mathrm{equation}\:\left(\mathrm{1}−\mathrm{2k}\right)\mathrm{x}^{\mathrm{2}} −\mathrm{6kx}−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{kx}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}=\mathrm{0}\:\mathrm{have}\:\mathrm{atleast} \\ $$$$\mathrm{one}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{common}\:\mathrm{are}\:\_\_\_ \\ $$
Question Number 97501 Answers: 0 Comments: 2
$$\mathrm{The}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\mathrm{the}\:\mathrm{expression}\:\mathrm{y}\:=\:\mathrm{5log}^{\mathrm{2}} \:_{\mathrm{3}} \left(\mathrm{n}\right)\:− \\ $$$$\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{n}^{\mathrm{12}} \right)+\mathrm{9}\:,\:\mathrm{has}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{is}\:\_\_\_ \\ $$
Question Number 97496 Answers: 1 Comments: 0
$${if}\:\:\:\:\:\:{f}\left(\frac{\mathrm{2}{x}+\mathrm{5}}{{x}−\mathrm{3}}\right)=\mathrm{3}{x}+\mathrm{5}\:\:\:{find}\:\:\:{f}\left({x}\right) \\ $$$$ \\ $$$${please}\:{solve}\:{it} \\ $$
Question Number 97485 Answers: 1 Comments: 3
Question Number 97428 Answers: 2 Comments: 5
Question Number 97386 Answers: 0 Comments: 2
Question Number 97306 Answers: 0 Comments: 2
$$\mathrm{if}\:\:\:\:\:\:\:\:\:\mathrm{sin14}=\mathrm{x} \\ $$$$\mathrm{then} \\ $$$$\mathrm{cos}^{\mathrm{2}} \mathrm{22}−\mathrm{cos}^{\mathrm{2}} \mathrm{8}=? \\ $$
Question Number 97303 Answers: 0 Comments: 0
$$\boldsymbol{\mathrm{G}}\mathrm{iven}\:\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} \:=\:\mathrm{0}\:,\:\mathrm{y}_{\mathrm{1}} \:+\:\mathrm{y}_{\mathrm{2}} +\mathrm{y}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{x}_{\mathrm{1}} \mathrm{y}_{\mathrm{1}} +\:\mathrm{x}_{\mathrm{2}} \mathrm{y}_{\mathrm{2}} \:+\:\mathrm{x}_{\mathrm{3}} \mathrm{y}_{\mathrm{3}} \:=\:\mathrm{0}\:.\:\mathrm{The}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} ^{\mathrm{2}} }\:+\:\frac{\mathrm{y}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{y}_{\mathrm{1}} ^{\mathrm{2}} \:+\mathrm{y}_{\mathrm{2}} ^{\mathrm{2}} \:+\mathrm{y}_{\mathrm{3}} ^{\mathrm{2}} }\:=\:?\: \\ $$
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