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Question Number 10621 Answers: 1 Comments: 0

Question Number 10620 Answers: 1 Comments: 1

Question Number 10612 Answers: 2 Comments: 1

Question Number 10607 Answers: 1 Comments: 0

Solve for x: 2x^3 + 2x^2 − 5x − 1 = 0

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}: \\ $$$$\mathrm{2x}^{\mathrm{3}} \:+\:\mathrm{2x}^{\mathrm{2}} \:−\:\mathrm{5x}\:−\:\mathrm{1}\:=\:\mathrm{0} \\ $$

Question Number 10597 Answers: 1 Comments: 0

prove that ∣z_1 − z_2 ∣ ≤ ∣z_1 ∣ + ∣z_2 ∣

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\mid\mathrm{z}_{\mathrm{1}} −\:\mathrm{z}_{\mathrm{2}} \mid\:\leqslant\:\mid\mathrm{z}_{\mathrm{1}} \mid\:+\:\mid\mathrm{z}_{\mathrm{2}} \mid \\ $$

Question Number 10583 Answers: 2 Comments: 3

The sum of the 4^(th ) and 6^(th ) terms of an AP is 42. the sum of the third and 9th terms of the proression is 52. Find the first term , the common difference and the sum of the first 10 terms of the progression.

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{4}^{\mathrm{th}\:} \:\mathrm{and}\:\mathrm{6}^{\mathrm{th}\:} \mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{is}\:\mathrm{42}.\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{third}\:\mathrm{and}\:\mathrm{9th}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{proression}\:\mathrm{is}\:\mathrm{52}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{term}\:,\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first} \\ $$$$\mathrm{10}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{progression}. \\ $$

Question Number 10577 Answers: 1 Comments: 0

5^(71) +5^(72) +5^(73) =?

$$\mathrm{5}^{\mathrm{71}} +\mathrm{5}^{\mathrm{72}} +\mathrm{5}^{\mathrm{73}} =? \\ $$

Question Number 10575 Answers: 1 Comments: 1

3(2^2 +1)(2^4 +1)(2^8 +1)(2^(16) +1)(2^(32) +1)(2^(64) +1)(2^(128) +1)=...?

$$\mathrm{3}\left(\mathrm{2}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{4}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{8}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{16}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{32}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{64}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{128}} +\mathrm{1}\right)=...? \\ $$

Question Number 10572 Answers: 1 Comments: 0

factorise x^3 + 1

$$\mathrm{factorise} \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1} \\ $$

Question Number 10571 Answers: 0 Comments: 0

Question Number 10570 Answers: 0 Comments: 0

n(A)=12 n(B)=10 ⇒min(n(A−B−B^′ ))

$${n}\left({A}\right)=\mathrm{12} \\ $$$${n}\left({B}\right)=\mathrm{10} \\ $$$$\Rightarrow{min}\left({n}\left({A}−{B}−{B}^{'} \right)\right) \\ $$

Question Number 10566 Answers: 2 Comments: 0

Question Number 10564 Answers: 1 Comments: 0

Prove that: lim_(ε→0) ((−1+x^ε )/ε) = ln(x)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\epsilon\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{1}+{x}^{\epsilon} }{\epsilon}\:=\:\mathrm{ln}\left({x}\right) \\ $$

Question Number 10563 Answers: 0 Comments: 0

can someone explain to me big K notation? (I don′t know the name) It is related to continuous fractions. e.g. x=b_0 +K_(i=1) ^∞ (a_i /b_i ) e^x =(x^0 /(0!))+(x^1 /(1!))+(x^2 /(2!))+... e^x =Σ_(i=0) ^∞ (x^i /(i!)) =1+(x/(1−((1x)/(2+x−((2x)/(3+x−((3x)/(...)))))))) How do you interperate this in big K notation?

$$\mathrm{can}\:\mathrm{someone}\:\mathrm{explain}\:\mathrm{to}\:\mathrm{me} \\ $$$$\mathrm{big}\:\mathrm{K}\:\mathrm{notation}?\:\left(\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{the}\:\mathrm{name}\right) \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{related}\:\mathrm{to}\:\mathrm{continuous}\:\mathrm{fractions}. \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\:{x}={b}_{\mathrm{0}} +\underset{{i}=\mathrm{1}} {\overset{\infty} {\boldsymbol{\mathrm{K}}}}\frac{{a}_{{i}} }{{b}_{{i}} } \\ $$$$\: \\ $$$${e}^{{x}} =\frac{{x}^{\mathrm{0}} }{\mathrm{0}!}+\frac{{x}^{\mathrm{1}} }{\mathrm{1}!}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+... \\ $$$${e}^{{x}} =\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{i}} }{{i}!} \\ $$$$\:\:\:\:\:=\mathrm{1}+\frac{{x}}{\mathrm{1}−\frac{\mathrm{1}{x}}{\mathrm{2}+{x}−\frac{\mathrm{2}{x}}{\mathrm{3}+{x}−\frac{\mathrm{3}{x}}{...}}}} \\ $$$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{interperate}\:\mathrm{this}\:\mathrm{in} \\ $$$$\mathrm{big}\:\mathrm{K}\:\mathrm{notation}? \\ $$

Question Number 10562 Answers: 0 Comments: 0

x= [(x_1 ),(x_2 ),(( ⋮)),(x_n ) ] y= [(y_1 ),(y_2 ),(( ⋮)),(y_n ) ] x, y ∈ R^n 1. Prove the length of the vector x, denoted ∣∣x∣∣, is equal to (√(x_1 ^2 +x_2 ^2 +...+x_n ^2 )) 2. Determine if ∣∣x−y∣∣=∣∣y−x∣∣

$$\boldsymbol{{x}}=\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{\:\vdots}\\{{x}_{{n}} }\end{bmatrix}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{y}}=\begin{bmatrix}{{y}_{\mathrm{1}} }\\{{y}_{\mathrm{2}} }\\{\:\vdots}\\{{y}_{{n}} }\end{bmatrix}\:\:\:\:\:\:\:\:\boldsymbol{{x}},\:\boldsymbol{{y}}\:\in\:\mathbb{R}^{{n}} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vector}\:\boldsymbol{{x}},\:\mathrm{denoted}\:\mid\mid\boldsymbol{{x}}\mid\mid, \\ $$$$\:\:\:\:\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\sqrt{{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} } \\ $$$$\mathrm{2}.\:\mathrm{Determine}\:\mathrm{if}\:\:\:\mid\mid\boldsymbol{{x}}−\boldsymbol{{y}}\mid\mid=\mid\mid\boldsymbol{{y}}−\boldsymbol{{x}}\mid\mid \\ $$

Question Number 10555 Answers: 0 Comments: 0

why Γ(x)=∫_0 ^1 [−ln(u)]^(x−1) du ? I don′t know how to prove this

$$\mathrm{why}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \left[−\mathrm{ln}\left(\mathrm{u}\right)\right]^{{x}−\mathrm{1}} {du}\:\:? \\ $$$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{how}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{this} \\ $$

Question Number 10553 Answers: 0 Comments: 0

(D^2 +4)y=tan 2x D=d/dx

$$\left(\mathrm{D}^{\mathrm{2}} +\mathrm{4}\right)\mathrm{y}=\mathrm{tan}\:\mathrm{2x}\:\:\:\:\:\:\:\:\:\:\:\mathrm{D}=\mathrm{d}/\mathrm{dx} \\ $$

Question Number 11178 Answers: 0 Comments: 1

If the sum of p terms of an AP is q and the sum of q terms is p, then the sum of p+q terms will be

$$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:{p}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{is}\:{q}\:\mathrm{and}\: \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:{q}\:\mathrm{terms}\:\mathrm{is}\:{p},\:\mathrm{then}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$${p}+{q}\:\:\mathrm{terms}\:\mathrm{will}\:\mathrm{be} \\ $$

Question Number 11172 Answers: 1 Comments: 0

Tangent to the curve (x+y)^3 =(x−y+2)^2 at (−1,1).

$$\mathrm{Tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{3}} =\left(\mathrm{x}−\mathrm{y}+\mathrm{2}\right)^{\mathrm{2}} \\ $$$$\mathrm{at}\:\left(−\mathrm{1},\mathrm{1}\right). \\ $$

Question Number 11183 Answers: 1 Comments: 0

Question Number 10547 Answers: 1 Comments: 0

A man can row a boat at 4 km/hr in still water. He rows the boat 2km upstream and 2km back to his starting place in 2 hours. How fast is the stream flowing ?

$$\mathrm{A}\:\mathrm{man}\:\mathrm{can}\:\mathrm{row}\:\mathrm{a}\:\mathrm{boat}\:\mathrm{at}\:\mathrm{4}\:\mathrm{km}/\mathrm{hr}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water}. \\ $$$$\mathrm{He}\:\mathrm{rows}\:\mathrm{the}\:\mathrm{boat}\:\mathrm{2km}\:\mathrm{upstream}\:\mathrm{and}\:\mathrm{2km}\:\mathrm{back}\:\mathrm{to} \\ $$$$\mathrm{his}\:\mathrm{starting}\:\mathrm{place}\:\mathrm{in}\:\mathrm{2}\:\mathrm{hours}.\:\mathrm{How}\:\mathrm{fast}\:\mathrm{is}\:\mathrm{the}\:\mathrm{stream} \\ $$$$\mathrm{flowing}\:? \\ $$

Question Number 11206 Answers: 3 Comments: 0

Question Number 11204 Answers: 0 Comments: 0

Question Number 11203 Answers: 1 Comments: 0

f(x)=((x/(x+1))−(x/(x−1)))^(−1) =−(((x+1)(x−1))/(2x)) g(x)=−(1/2)x why is f(x)≈g(x)?

$${f}\left({x}\right)=\left(\frac{{x}}{{x}+\mathrm{1}}−\frac{{x}}{{x}−\mathrm{1}}\right)^{−\mathrm{1}} =−\frac{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{1}\right)}{\mathrm{2}{x}} \\ $$$${g}\left({x}\right)=−\frac{\mathrm{1}}{\mathrm{2}}{x} \\ $$$$\: \\ $$$$\mathrm{why}\:\mathrm{is}\:{f}\left({x}\right)\approx{g}\left({x}\right)? \\ $$

Question Number 11196 Answers: 0 Comments: 2

Give an example each with justification,of a function defined by ]−1,1[ ,which is 1)one one but not onto. 2)onto but not one one.

$$\mathrm{Give}\:\mathrm{an}\:\mathrm{example}\:\mathrm{each}\:\mathrm{with}\:\mathrm{justification},\mathrm{of}\:\mathrm{a}\:\mathrm{function} \\ $$$$\left.\mathrm{defined}\:\mathrm{by}\:\right]−\mathrm{1},\mathrm{1}\left[\:,\mathrm{which}\:\mathrm{is}\right. \\ $$$$\left.\mathrm{1}\right)\mathrm{one}\:\mathrm{one}\:\mathrm{but}\:\mathrm{not}\:\mathrm{onto}. \\ $$$$\left.\mathrm{2}\right)\mathrm{onto}\:\mathrm{but}\:\mathrm{not}\:\mathrm{one}\:\mathrm{one}. \\ $$

Question Number 10544 Answers: 0 Comments: 1

The number of terms in the expansion of (1+2x+x^2 )^(20) when expanded in descending powers of x, is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}+\mathrm{2}{x}+{x}^{\mathrm{2}} \right)^{\mathrm{20}} \mathrm{when}\:\mathrm{expanded}\:\mathrm{in}\:\mathrm{descending} \\ $$$$\mathrm{powers}\:\mathrm{of}\:{x},\:\mathrm{is} \\ $$

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