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AllQuestion and Answers: Page 605

Question Number 157942 Answers: 0 Comments: 1

Question Number 157913 Answers: 1 Comments: 0

5^2 ∙ 5^4 ∙ 5^6 ∙ ... ∙ 5^(2x) = 0,04^(-28) find x=?

$$\mathrm{5}^{\mathrm{2}} \:\centerdot\:\mathrm{5}^{\mathrm{4}} \:\centerdot\:\mathrm{5}^{\mathrm{6}} \:\centerdot\:...\:\centerdot\:\mathrm{5}^{\mathrm{2}\boldsymbol{\mathrm{x}}} \:=\:\mathrm{0},\mathrm{04}^{-\mathrm{28}} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 157908 Answers: 1 Comments: 0

(a_n ) in numerical series 7+9+11+13+...+(2n+1)=an^2 +bn+c find a+b+c=?

$$\left(\mathrm{a}_{\boldsymbol{\mathrm{n}}} \right)\:\mathrm{in}\:\mathrm{numerical}\:\mathrm{series} \\ $$$$\mathrm{7}+\mathrm{9}+\mathrm{11}+\mathrm{13}+...+\left(\mathrm{2n}+\mathrm{1}\right)=\mathrm{an}^{\mathrm{2}} +\mathrm{bn}+\mathrm{c} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}=? \\ $$

Question Number 157906 Answers: 0 Comments: 4

f(x)=(px+1)(2x+q+4) function is a single function, find p+q+pq=?

$$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{px}+\mathrm{1}\right)\left(\mathrm{2x}+\mathrm{q}+\mathrm{4}\right)\:\mathrm{function}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{single}\:\mathrm{function},\:\mathrm{find}\:\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}}+\boldsymbol{\mathrm{pq}}=? \\ $$$$ \\ $$

Question Number 157926 Answers: 2 Comments: 0

if the line px+qy=r tangents the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1, then 1) prove a^2 p^2 +b^2 q^2 =r^2 2) find the coordinates of the touching point.

$${if}\:{the}\:{line}\:{px}+{qy}={r}\:{tangents}\:{the} \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1},\:{then}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:\boldsymbol{{a}}^{\mathrm{2}} \boldsymbol{{p}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} \boldsymbol{{q}}^{\mathrm{2}} =\boldsymbol{{r}}^{\mathrm{2}} \: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{coordinates}\:{of}\:{the}\: \\ $$$$\:\:\:\:\:{touching}\:{point}. \\ $$

Question Number 157925 Answers: 1 Comments: 0

Question Number 157947 Answers: 1 Comments: 0

What are the coordinates of the points on the curve x^2 −y^2 =16 which nearest to (0,6)?

$${What}\:{are}\:{the}\:{coordinates}\:{of}\:{the} \\ $$$${points}\:{on}\:{the}\:{curve}\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{16} \\ $$$${which}\:{nearest}\:{to}\:\left(\mathrm{0},\mathrm{6}\right)? \\ $$

Question Number 157891 Answers: 2 Comments: 0

Question Number 157887 Answers: 1 Comments: 2

Question Number 157884 Answers: 1 Comments: 0

(1/(sin(10°))) - 4 sin(70°) = ?

$$\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{10}°\right)}\:-\:\mathrm{4}\:\mathrm{sin}\left(\mathrm{70}°\right)\:=\:? \\ $$

Question Number 157883 Answers: 1 Comments: 0

lim_(n→∞) ((1/(n^2 +1)) + (2/(n^2 +1)) + ... + ((n-1)/(n^2 +1))) = ?

$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\:+\:\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\:+\:...\:+\:\frac{\mathrm{n}-\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\right)\:=\:? \\ $$

Question Number 157880 Answers: 0 Comments: 1

Question Number 157873 Answers: 0 Comments: 0

if: 𝛂 =∫_( 0) ^( ∞) ∫_( 0) ^( ∞) ((x)^(1/3) /(1 + (x)^(1/3) )) e^(-𝛑y(1+x^2 +(1/x^2 ))) dydx find: (√(19683𝛂^6 - 94041𝛂^4 + 105786𝛂^2 ))

$$\mathrm{if}:\:\boldsymbol{\alpha}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{x}}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:\mathrm{e}^{-\boldsymbol{\pi\mathrm{y}}\left(\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)} \:\mathrm{dydx} \\ $$$$\mathrm{find}:\:\sqrt{\mathrm{19683}\boldsymbol{\alpha}^{\mathrm{6}} \:-\:\mathrm{94041}\boldsymbol{\alpha}^{\mathrm{4}} \:+\:\mathrm{105786}\boldsymbol{\alpha}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 157871 Answers: 0 Comments: 0

if x;y>0 then prove that: (x/(x^2 -x+1)) + (y/(y^2 -y+1)) + ((xy)/(x^2 y^2 -xy+1)) ≤ ≤ (x^2 /(x^2 -x+1)) + (y^2 /(y^2 -y+1)) + (1/(x^2 y^2 -xy+1))

$$\mathrm{if}\:\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{x}^{\mathrm{2}} -\mathrm{x}+\mathrm{1}}\:+\:\frac{\mathrm{y}}{\mathrm{y}^{\mathrm{2}} -\mathrm{y}+\mathrm{1}}\:+\:\frac{\mathrm{xy}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} -\mathrm{xy}+\mathrm{1}}\:\leqslant \\ $$$$\leqslant\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} -\mathrm{x}+\mathrm{1}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{y}^{\mathrm{2}} -\mathrm{y}+\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} -\mathrm{xy}+\mathrm{1}} \\ $$

Question Number 157869 Answers: 1 Comments: 0

partial fraction in below: 1. ((x^2 −15x+41)/((x+2)(x−3)^2 )) 2.((4x^2 −5x+6)/((x+1)(x^2 +4)))

$$\mathrm{partial}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{below}: \\ $$$$\mathrm{1}.\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{15x}+\mathrm{41}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\mathrm{2}.\frac{\mathrm{4x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{6}}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{4}\right)} \\ $$

Question Number 157867 Answers: 1 Comments: 0

Given that point P(a cos θ, b sin θ) is a point on the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1. The tangent to the curve at point P is perpendicular to a straight line which passes through the focus, F (ae,0). If N is the intersection point, show that the equation of the locus of N is x^2 +y^2 =a^2 .

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{point}\:{P}\left({a}\:\mathrm{cos}\:\theta,\:{b}\:\mathrm{sin}\:\theta\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}. \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{at}\:\mathrm{point}\:{P}\:\:\mathrm{is}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{which}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{focus}, \\ $$$${F}\:\left({ae},\mathrm{0}\right).\:\mathrm{If}\:{N}\:\mathrm{is}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{point},\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{N}\:\mathrm{is}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={a}^{\mathrm{2}} . \\ $$

Question Number 157855 Answers: 0 Comments: 2

Question Number 157853 Answers: 0 Comments: 1

Question Number 157851 Answers: 1 Comments: 2

Question Number 157970 Answers: 2 Comments: 0

Find the neext number for this sequence below 1). 1, 3, 6, 10, 15.... 2). 1, 5 ,14 ,30, 55 .... 3). 1,7,17,31, 49 .... 4). 4, 13,28, 49,74... 5). 1,8,27,64,125....

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{neext}\:\mathrm{number}\:\mathrm{for}\:\mathrm{this} \\ $$$$\mathrm{sequence}\:\mathrm{below}\: \\ $$$$\left.\mathrm{1}\right).\:\mathrm{1},\:\:\mathrm{3},\:\mathrm{6},\:\mathrm{10},\:\mathrm{15}.... \\ $$$$\left.\mathrm{2}\right).\:\mathrm{1},\:\mathrm{5}\:,\mathrm{14}\:,\mathrm{30},\:\:\mathrm{55}\:.... \\ $$$$\left.\mathrm{3}\right).\:\mathrm{1},\mathrm{7},\mathrm{17},\mathrm{31},\:\mathrm{49}\:.... \\ $$$$\left.\mathrm{4}\right).\:\:\mathrm{4},\:\mathrm{13},\mathrm{28},\:\mathrm{49},\mathrm{74}... \\ $$$$\left.\mathrm{5}\right).\:\:\mathrm{1},\mathrm{8},\mathrm{27},\mathrm{64},\mathrm{125}.... \\ $$$$ \\ $$

Question Number 157870 Answers: 1 Comments: 0

find the integral: ∫{(3x+1)/(x^2 +4)}dx

$${find}\:{the}\:{integral}: \\ $$$$\int\left\{\left(\mathrm{3}{x}+\mathrm{1}\right)/\left({x}^{\mathrm{2}} +\mathrm{4}\right)\right\}{dx} \\ $$

Question Number 157902 Answers: 1 Comments: 1

Solve for integers: x∙(x + 4) = 5∙(3^y - 1)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{integers}: \\ $$$$\mathrm{x}\centerdot\left(\mathrm{x}\:+\:\mathrm{4}\right)\:=\:\mathrm{5}\centerdot\left(\mathrm{3}^{\boldsymbol{\mathrm{y}}} \:-\:\mathrm{1}\right) \\ $$$$ \\ $$

Question Number 157839 Answers: 1 Comments: 0

find the last four digits of 11^(15999) ?

$${find}\:{the}\:{last}\:{four}\:{digits}\:{of}\: \\ $$$$\mathrm{11}^{\mathrm{15999}} ? \\ $$

Question Number 157835 Answers: 1 Comments: 0

Question Number 157836 Answers: 1 Comments: 0

find the indicated higher order derivative of the following function f(x) = (x^3 +4x−5)^4 , f(x)^(iv)

$${find}\:{the}\:{indicated}\:{higher}\:{order}\:{derivative} \\ $$$${of}\:{the}\:{following}\:{function} \\ $$$${f}\left({x}\right)\:=\:\left({x}^{\mathrm{3}} +\mathrm{4}{x}−\mathrm{5}\right)^{\mathrm{4}} ,\:{f}\left({x}\right)^{{iv}} \\ $$

Question Number 157832 Answers: 1 Comments: 0

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