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Question Number 155583 Answers: 2 Comments: 0

Question Number 155576 Answers: 7 Comments: 3

the polynomial 4x^3 +ax^2 +bx+9, where a and b consant, is denoted by f(x). when f(x) is divide by (x−2) the remainder is r and when divided by (x−3) the remaider is 6r. its further given that (x+3) is a factor of f(x). Show that b−a=14 and hence find a and b

$${the}\:{polynomial}\:\mathrm{4}{x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+\mathrm{9}, \\ $$$${where}\:\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:{consant},\:{is}\:{denoted}\:{by} \\ $$$$\:{f}\left({x}\right).\:{when}\:{f}\left({x}\right)\:{is}\:{divide}\:{by}\:\left({x}−\mathrm{2}\right)\:{the} \\ $$$${remainder}\:{is}\:\boldsymbol{{r}}\:{and}\:{when}\:{divided}\:{by} \\ $$$$\:\left({x}−\mathrm{3}\right)\:{the}\:{remaider}\:{is}\:\mathrm{6}\boldsymbol{{r}}.\:{its}\:{further} \\ $$$$\:{given}\:{that}\:\left({x}+\mathrm{3}\right)\:{is}\:{a}\:{factor}\:{of}\:{f}\left({x}\right). \\ $$$$\:{Show}\:{that}\:\boldsymbol{{b}}−\boldsymbol{{a}}=\mathrm{14} \\ $$$$\:{and}\:{hence}\:{find}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}} \\ $$

Question Number 155574 Answers: 1 Comments: 0

The mean and standard deviation of 20 observation are found to be 10 and 2 respectively .On rechecking it was found that an observation 8 was incorrect.Calculate the incorrect mean and standard deviation (a)If the wrong iterm was ommited (b) If it is replaced by 12

$$\mathrm{The}\:\mathrm{mean}\:\mathrm{and}\:\mathrm{standard}\:\mathrm{deviation} \\ $$$$\mathrm{of}\:\mathrm{20}\:\mathrm{observation}\:\:\mathrm{are}\:\mathrm{found}\:\mathrm{to}\:\mathrm{be}\: \\ $$$$\mathrm{10}\:\mathrm{and}\:\mathrm{2}\:\mathrm{respectively}\:.\mathrm{On}\:\mathrm{rechecking} \\ $$$$\mathrm{it}\:\mathrm{was}\:\mathrm{found}\:\mathrm{that}\:\:\mathrm{an}\:\mathrm{observation} \\ $$$$\mathrm{8}\:\mathrm{was}\:\mathrm{incorrect}.\mathrm{Calculate}\:\mathrm{the}\:\mathrm{incorrect} \\ $$$$\mathrm{mean}\:\mathrm{and}\:\mathrm{standard}\:\mathrm{deviation} \\ $$$$\left(\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{I}}\mathrm{f}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{wrong}}\:\boldsymbol{\mathrm{iterm}}\:\boldsymbol{\mathrm{was}}\:\:\boldsymbol{\mathrm{ommited}} \\ $$$$\left(\boldsymbol{\mathrm{b}}\right)\:\boldsymbol{\mathrm{If}}\:\boldsymbol{\mathrm{it}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{replaced}}\:\boldsymbol{\mathrm{by}}\:\mathrm{12} \\ $$

Question Number 155573 Answers: 0 Comments: 1

Question Number 155572 Answers: 2 Comments: 0

Find Z^4 =1. Hence show that 1+w+w^2 +w^3 =0

$$\mathrm{Find}\:\mathrm{Z}^{\mathrm{4}} =\mathrm{1}. \\ $$$$\mathrm{Hence}\:\mathrm{show}\:\mathrm{that}\:\mathrm{1}+\mathrm{w}+\mathrm{w}^{\mathrm{2}} +\mathrm{w}^{\mathrm{3}} =\mathrm{0} \\ $$

Question Number 155571 Answers: 2 Comments: 0

Find the cube root of one .Hence show that the sum of the root is equal to zero

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\mathrm{one}\:.\mathrm{Hence} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{root}\:\mathrm{is}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{zero} \\ $$

Question Number 155568 Answers: 2 Comments: 0

for a,b,c,d,e ∈R and a+b+c+d+e=5 find the minimum value of a^2 +2b^2 +3c^2 +4d^2 +5e^2 =?

$${for}\:{a},{b},{c},{d},{e}\:\in{R}\:{and}\:{a}+{b}+{c}+{d}+{e}=\mathrm{5} \\ $$$${find}\:{the}\:{minimum}\:{value}\:{of}\: \\ $$$${a}^{\mathrm{2}} +\mathrm{2}{b}^{\mathrm{2}} +\mathrm{3}{c}^{\mathrm{2}} +\mathrm{4}{d}^{\mathrm{2}} +\mathrm{5}{e}^{\mathrm{2}} =? \\ $$

Question Number 155562 Answers: 1 Comments: 2

Question Number 155561 Answers: 1 Comments: 0

Question Number 155560 Answers: 0 Comments: 0

Question Number 155553 Answers: 0 Comments: 0

∫_(−∞) ^( ∞) ((sin(x^2 )cos(x^3 ))/((ln((sin(x)cos(x))^2 ))^2 +1)) dx

$$\: \\ $$$$\int_{−\infty} ^{\:\infty} \:\frac{\mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left({x}^{\mathrm{3}} \right)}{\left(\mathrm{ln}\left(\left(\mathrm{sin}\left({x}\right)\mathrm{cos}\left({x}\right)\right)^{\mathrm{2}} \right)\right)^{\mathrm{2}} +\mathrm{1}}\:\:{dx}\:\: \\ $$$$\: \\ $$

Question Number 155547 Answers: 1 Comments: 0

let a;b;c>0 and a+b+c=3 find min value of the expression: S = abc + (a-1)^2 + (b-1)^2 + (c-1)^2

$$\mathrm{let}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3} \\ $$$$\mathrm{find}\:\mathrm{min}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}: \\ $$$$\mathrm{S}\:=\:\mathrm{abc}\:+\:\left(\mathrm{a}-\mathrm{1}\right)^{\mathrm{2}} \:+\:\left(\mathrm{b}-\mathrm{1}\right)^{\mathrm{2}} \:+\:\left(\mathrm{c}-\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 155545 Answers: 1 Comments: 6

Solve in R ((5x)/( (√(5x^2 + 4)) + 7(√x))) + ((x + 2)/( (√(x^2 - 3x - 18)) + 2(√x)))

$$\mathrm{Solve}\:\mathrm{in}\:\mathbb{R} \\ $$$$\frac{\mathrm{5x}}{\:\sqrt{\mathrm{5x}^{\mathrm{2}} \:+\:\mathrm{4}}\:+\:\mathrm{7}\sqrt{\mathrm{x}}}\:+\:\frac{\mathrm{x}\:+\:\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{3x}\:-\:\mathrm{18}}\:+\:\mathrm{2}\sqrt{\mathrm{x}}} \\ $$

Question Number 155542 Answers: 1 Comments: 4

find the tylor series expantion of ((z^2 −1)/((z+1)(z+3)))

$${find}\:{the}\:{tylor}\:{series}\:{expantion}\:{of}\:\frac{{z}^{\mathrm{2}} −\mathrm{1}}{\left({z}+\mathrm{1}\right)\left({z}+\mathrm{3}\right)} \\ $$

Question Number 155541 Answers: 0 Comments: 0

form a partial equation from (x^2 /2)+(y^2 /2)=z^2

$${form}\:{a}\:{partial}\:{equation}\:{from}\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{y}^{\mathrm{2}} }{\mathrm{2}}={z}^{\mathrm{2}} \\ $$

Question Number 155540 Answers: 1 Comments: 0

f(5−4x)=27x^(2000) −2187x^(204) −x^4 +3x+8 f(3x−7)=?

$$\:\:\mathrm{f}\left(\mathrm{5}−\mathrm{4x}\right)=\mathrm{27x}^{\mathrm{2000}} −\mathrm{2187x}^{\mathrm{204}} −\mathrm{x}^{\mathrm{4}} +\mathrm{3x}+\mathrm{8} \\ $$$$\mathrm{f}\left(\mathrm{3x}−\mathrm{7}\right)=? \\ $$

Question Number 155537 Answers: 1 Comments: 0

Question Number 155533 Answers: 1 Comments: 0

coefficiient of x^( 60) = ? P = (x−1)(x^( 2) −1)(x^( 3) −1)...(x^( 15) −1)

$$ \\ $$$$\:\:\:{coefficiient}\:{of}\:\:\:\:{x}^{\:\mathrm{60}} \:=\:? \\ $$$$ \\ $$$$\:\:\:\mathrm{P}\:=\:\left({x}−\mathrm{1}\right)\left({x}^{\:\mathrm{2}} −\mathrm{1}\right)\left({x}^{\:\mathrm{3}} −\mathrm{1}\right)...\left({x}^{\:\mathrm{15}} −\mathrm{1}\right) \\ $$$$ \\ $$

Question Number 155531 Answers: 0 Comments: 4

Question Number 155524 Answers: 0 Comments: 1

how can convert the interval of the intigral ∫_0 ^( 1) f(x) dx to the interval ∫_0 ^( ∞) f(x) dx ?

$$\boldsymbol{{how}}\:\boldsymbol{{can}}\:\boldsymbol{{convert}}\:\boldsymbol{{the}}\:\boldsymbol{{interval}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{intigral}}\: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:\boldsymbol{{to}}\:\boldsymbol{{the}}\:\boldsymbol{{interval}}\:\int_{\mathrm{0}} ^{\:\infty} \:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:? \\ $$

Question Number 155527 Answers: 2 Comments: 0

If f(tan^2 (θ/2))= (2/(1+cos θ)) , find f(sin (θ/2)).

$$\mathrm{If}\:{f}\left(\mathrm{tan}^{\mathrm{2}} \:\frac{\theta}{\mathrm{2}}\right)=\:\frac{\mathrm{2}}{\mathrm{1}+\mathrm{cos}\:\theta}\:,\:\mathrm{find}\:{f}\left(\mathrm{sin}\:\frac{\theta}{\mathrm{2}}\right). \\ $$

Question Number 155510 Answers: 1 Comments: 0

Question Number 155506 Answers: 1 Comments: 0

Question Number 155500 Answers: 2 Comments: 0

Question Number 155497 Answers: 1 Comments: 0

Question Number 155496 Answers: 1 Comments: 0

Given I_n =∫_(nπ) ^((n+1)π) e^(−x) sinx dx , n∈N. 1. Find a relation between I_(n+1) and I_n .

$${Given}\:{I}_{{n}} =\underset{{n}\pi} {\overset{\left({n}+\mathrm{1}\right)\pi} {\int}}{e}^{−{x}} {sinx}\:{dx}\:,\:{n}\in\mathbb{N}. \\ $$$$\mathrm{1}.\:{Find}\:{a}\:{relation}\:{between}\:{I}_{{n}+\mathrm{1}} {and}\:{I}_{{n}} . \\ $$

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