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AlgebraQuestion and Answers: Page 222

Question Number 134067 Answers: 1 Comments: 0

Calculate lim_(n→∞) Π_(k=2) ^n ((k^3 −1)/(k^3 +1)) = ?

$$\:\mathrm{Calculate}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{k}=\mathrm{2}} {\overset{\mathrm{n}} {\prod}}\:\frac{\mathrm{k}^{\mathrm{3}} −\mathrm{1}}{\mathrm{k}^{\mathrm{3}} +\mathrm{1}}\:=\:? \\ $$

Question Number 134061 Answers: 1 Comments: 0

If Σ a_n is a convergent series of nonnegative terms,what can be said about Σ a_n .a_(n+1) ? (a) always converges (b) always diverges (c) may converges or diverge

$$\mathrm{If}\:\Sigma\:\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{series}\:\mathrm{of} \\ $$$$\mathrm{nonnegative}\:\mathrm{terms},\mathrm{what}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{said}\:\mathrm{about}\:\Sigma\:\mathrm{a}_{\mathrm{n}} .\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{always}\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{always}\:\mathrm{diverges} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{may}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverge} \\ $$

Question Number 134060 Answers: 2 Comments: 0

If p>1 and q>1 what can be said about the convergence of Σ_(n=2) ^∞ (1/(n^p .(ln n)^q )) ? (a) always converges (b) always diverges (c) may converges or diverges

$$\mathrm{If}\:\mathrm{p}>\mathrm{1}\:\mathrm{and}\:\mathrm{q}>\mathrm{1}\:\mathrm{what}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{said}\:\mathrm{about}\:\mathrm{the}\:\mathrm{convergence}\: \\ $$$$\mathrm{of}\:\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{p}} .\left(\mathrm{ln}\:\mathrm{n}\right)^{\mathrm{q}} }\:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{always}\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{always}\:\mathrm{diverges} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{may}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges} \\ $$

Question Number 134052 Answers: 1 Comments: 0

Question Number 134034 Answers: 1 Comments: 0

how many zeros has the number 1000! at the end? and what is the last digit before these zeros?

$${how}\:{many}\:{zeros}\:{has}\:{the}\:{number} \\ $$$$\mathrm{1000}!\:{at}\:{the}\:{end}?\:{and}\:{what}\:{is}\:{the} \\ $$$${last}\:{digit}\:{before}\:{these}\:{zeros}? \\ $$

Question Number 134005 Answers: 1 Comments: 0

solve x^3 −2⌊x⌋=5

$${solve}\:{x}^{\mathrm{3}} −\mathrm{2}\lfloor{x}\rfloor=\mathrm{5} \\ $$

Question Number 133996 Answers: 2 Comments: 0

show that (√2)<log_2 3<(√3)

$${show}\:{that}\:\sqrt{\mathrm{2}}<\mathrm{log}_{\mathrm{2}} \:\mathrm{3}<\sqrt{\mathrm{3}} \\ $$

Question Number 133991 Answers: 1 Comments: 0

∣ 3x−∣ 4x+2 ∣∣ ≥ 4 > ∣ 5x+8 ∣

$$\:\mid\:\mathrm{3x}−\mid\:\mathrm{4x}+\mathrm{2}\:\mid\mid\:\geqslant\:\mathrm{4}\:>\:\mid\:\mathrm{5x}+\mathrm{8}\:\mid\: \\ $$

Question Number 133903 Answers: 0 Comments: 3

Question Number 133885 Answers: 1 Comments: 2

Consider the equations of two intersecting straight lines { ((ax+by+c=0)),((a_1 x+b_1 y+c_1 =0)) :} Find the equation of straight line passing through a given point (x_0 ,y_0 ) and the intersection point of the given straight lines.

$$\:\mathrm{Consider}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{intersecting}\:\mathrm{straight}\:\mathrm{lines} \\ $$$$\begin{cases}{{ax}+{by}+{c}=\mathrm{0}}\\{{a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} =\mathrm{0}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\mathrm{passing}\:\mathrm{through}\:\mathrm{a}\:\mathrm{given}\:\mathrm{point} \\ $$$$\left(\mathrm{x}_{\mathrm{0}} ,\mathrm{y}_{\mathrm{0}} \right)\:\mathrm{and}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{point} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{given}\:\mathrm{straight}\:\mathrm{lines}. \\ $$

Question Number 133793 Answers: 0 Comments: 1

Question Number 133981 Answers: 2 Comments: 0

If x = 5+2(√6) then ((x−1)/( (√x))) =?

$$\mathrm{If}\:{x}\:=\:\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}\:\mathrm{then}\:\frac{{x}−\mathrm{1}}{\:\sqrt{{x}}}\:=? \\ $$

Question Number 133783 Answers: 0 Comments: 1

To complete a job, 24 worker are needed in 35 days. After they worked for 8 days , half of the workers stopped working . In order for work to be completed , an additional time of .... days needed (a) 54 (b) 38 (c) 28 (d) 14 (e) 19

$$\mathrm{To}\:\mathrm{complete}\:\mathrm{a}\:\mathrm{job},\:\mathrm{24}\:\mathrm{worker}\:\mathrm{are} \\ $$$$\mathrm{needed}\:\mathrm{in}\:\mathrm{35}\:\mathrm{days}.\:\mathrm{After}\:\mathrm{they}\:\mathrm{worked} \\ $$$$\mathrm{for}\:\mathrm{8}\:\mathrm{days}\:,\:\mathrm{half}\:\mathrm{of}\:\mathrm{the}\:\mathrm{workers}\: \\ $$$$\mathrm{stopped}\:\mathrm{working}\:.\:\mathrm{In}\:\mathrm{order}\:\mathrm{for} \\ $$$$\mathrm{work}\:\mathrm{to}\:\mathrm{be}\:\mathrm{completed}\:,\:\mathrm{an}\:\mathrm{additional} \\ $$$$\mathrm{time}\:\mathrm{of}\:....\:\mathrm{days}\:\mathrm{needed} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{54}\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{38}\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{28}\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{14} \\ $$$$\left(\mathrm{e}\right)\:\mathrm{19}\: \\ $$

Question Number 133738 Answers: 3 Comments: 0