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AlgebraQuestion and Answers: Page 46

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Question Number 204621 Answers: 3 Comments: 0

a , b , c ∈ R^+ If (√a) + (√b) + (√c) = 1 Prove that: a + b + c ≥ (1/3)

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:\in\:\mathbb{R}^{+} \\ $$$$\mathrm{If}\:\:\:\sqrt{\mathrm{a}}\:+\:\sqrt{\mathrm{b}}\:+\:\sqrt{\mathrm{c}}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 204610 Answers: 1 Comments: 0

If , f(x) = { (( 2^(2x) − log_3 ( x+3 ) ; x ≥5)),(( f (1+ x ) −4 ; x < 5)) :} ⇒ f (0 )= ?

$$ \\ $$$$\:\:\:{If}\:,\:\:\:{f}\left({x}\right)\:=\:\begin{cases}{\:\mathrm{2}^{\mathrm{2}{x}} −\:{log}_{\mathrm{3}} \:\left(\:{x}+\mathrm{3}\:\right)\:\:\:\:;\:\:\:{x}\:\geqslant\mathrm{5}}\\{\:{f}\:\left(\mathrm{1}+\:{x}\:\right)\:\:−\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:;\:\:{x}\:<\:\mathrm{5}}\end{cases}\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{f}\:\left(\mathrm{0}\:\right)=\:? \\ $$$$ \\ $$

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Question Number 204583 Answers: 1 Comments: 0

For z = a − bi If (∣z∣ − z)∙(∣z∣ + z^(−) ) = 4bi Find ∣z∣ = ?

$$\mathrm{For}\:\:\:\mathrm{z}\:=\:\mathrm{a}\:−\:\mathrm{bi} \\ $$$$\mathrm{If}\:\:\:\left(\mid\mathrm{z}\mid\:−\:\mathrm{z}\right)\centerdot\left(\mid\mathrm{z}\mid\:+\:\overline {\mathrm{z}}\right)\:=\:\mathrm{4bi} \\ $$$$\mathrm{Find}\:\:\:\mid\mathrm{z}\mid\:=\:? \\ $$

Question Number 204545 Answers: 2 Comments: 0

If a = (1/2^2 ) + (1/3^2 ) + ... + (1/(100^2 )) b = 0,99 Prove that: a < b

$$\mathrm{If} \\ $$$$\mathrm{a}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:\:+\:\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{100}^{\mathrm{2}} } \\ $$$$\mathrm{b}\:=\:\mathrm{0},\mathrm{99} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}\:<\:\mathrm{b} \\ $$

Question Number 204512 Answers: 2 Comments: 0

solve for x∈C 3^(2ix) −3^(ix) 2+5=0

$$\mathrm{solve}\:\mathrm{for}\:{x}\in\mathbb{C} \\ $$$$\mathrm{3}^{\mathrm{2i}{x}} −\mathrm{3}^{\mathrm{i}{x}} \mathrm{2}+\mathrm{5}=\mathrm{0} \\ $$

Question Number 204511 Answers: 1 Comments: 0

solve for x x^2 −10⌊x⌋+((57)/4)=0

$$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$$${x}^{\mathrm{2}} −\mathrm{10}\lfloor{x}\rfloor+\frac{\mathrm{57}}{\mathrm{4}}=\mathrm{0} \\ $$

Question Number 204510 Answers: 1 Comments: 0

solve for x≠y∧y≠z∧z≠x (exact solutions required) (√((−3+4i)x))=y (√((−3+4i)y))=z (√((−3+4i)z))=x

$$\mathrm{solve}\:\mathrm{for}\:{x}\neq{y}\wedge{y}\neq{z}\wedge{z}\neq{x} \\ $$$$\left(\mathrm{exact}\:\mathrm{solutions}\:\mathrm{required}\right) \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){x}}={y} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){y}}={z} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){z}}={x} \\ $$

Question Number 204544 Answers: 1 Comments: 0

Prove that: (1/3^3 ) + (1/4^3 ) + ... + (1/n^3 ) < (1/(12))

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{3}} }\:\:+\:\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} }\:\:<\:\:\frac{\mathrm{1}}{\mathrm{12}} \\ $$

Question Number 204500 Answers: 1 Comments: 0

Given that I = ∫∫_R (x^2 +y^2 )^(5/2) dxdy where R is the region x^2 +y^2 ≤ a^2 use a suitable transformation to evaluate I

$$\mathrm{Given}\:\mathrm{that}\:{I}\:=\:\int\int_{{R}} \left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\frac{\mathrm{5}}{\mathrm{2}}} {dxdy}\:\mathrm{where}\:{R} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{region}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:\leqslant\:{a}^{\mathrm{2}} \\ $$$$\mathrm{use}\:\mathrm{a}\:\mathrm{suitable}\:\mathrm{transformation}\:\mathrm{to}\:\mathrm{evaluate}\:{I} \\ $$

Question Number 204499 Answers: 1 Comments: 0

1. Find the directional derivative of F(x,y,z) = 2xy−z^2 at the point (2,−1,1) in a direction towards (3,1,−1) in what direction is the directional derivative maximum? what is the value of this maximum?

$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{directional}\:\mathrm{derivative}\:\mathrm{of}\: \\ $$$${F}\left({x},{y},{z}\right)\:=\:\mathrm{2}{xy}−{z}^{\mathrm{2}} \:\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{2},−\mathrm{1},\mathrm{1}\right)\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{direction}\:\mathrm{towards}\:\left(\mathrm{3},\mathrm{1},−\mathrm{1}\right)\: \\ $$$$\mathrm{in}\:\mathrm{what}\:\mathrm{direction}\:\mathrm{is}\:\mathrm{the}\:\mathrm{directional}\:\mathrm{derivative} \\ $$$$\mathrm{maximum}?\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{this}\:\mathrm{maximum}? \\ $$

Question Number 204480 Answers: 2 Comments: 1

x + (1/x) = 2.05 x = ?

$${x}\:+\:\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{2}.\mathrm{05} \\ $$$${x}\:=\:? \\ $$

Question Number 204437 Answers: 1 Comments: 0

∫_0 ^(Π/2) sin(t)ln(sint)dt

$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} {sin}\left({t}\right){ln}\left({sint}\right){dt} \\ $$

Question Number 204461 Answers: 2 Comments: 0

Solve for x (x − 12)(x − 13) = ((34)/(33^2 ))

$$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$$\left({x}\:−\:\mathrm{12}\right)\left({x}\:−\:\mathrm{13}\right)\:=\:\frac{\mathrm{34}}{\mathrm{33}^{\mathrm{2}} } \\ $$

Question Number 204421 Answers: 3 Comments: 0

x^3 +y^3 = 35 (1/x)+(1/y) =(5/6) solve for all possible values of x and y

$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} \:=\:\mathrm{35} \\ $$$$\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}\:=\frac{\mathrm{5}}{\mathrm{6}} \\ $$$${solve}\:{for}\:{all}\:{possible}\:{values}\:{of}\:{x}\:{and}\:{y} \\ $$$$ \\ $$

Question Number 204417 Answers: 2 Comments: 0

Solve for z∈C e^z =ln z

$$\mathrm{Solve}\:\mathrm{for}\:{z}\in\mathbb{C} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{e}^{{z}} =\mathrm{ln}\:{z} \\ $$

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