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Question Number 31569 Answers: 1 Comments: 0

(a,(1/a)),(b,(1/b)),(c,(1/c)),(d,(1/d)) are four distinct points on a circle of radius is 4 units then abcd is equal to ?

$$\left({a},\frac{\mathrm{1}}{{a}}\right),\left({b},\frac{\mathrm{1}}{{b}}\right),\left({c},\frac{\mathrm{1}}{{c}}\right),\left({d},\frac{\mathrm{1}}{{d}}\right)\:{are}\:{four} \\ $$$${distinct}\:{points}\:{on}\:{a}\:{circle}\:{of}\:{radius} \\ $$$${is}\:\mathrm{4}\:{units}\:{then}\:{abcd}\:{is}\:{equal}\:{to}\:? \\ $$

Question Number 31543 Answers: 0 Comments: 2

∫((x^3 +x+1)/(x^2 +1)) dx

$$\int\frac{{x}^{\mathrm{3}} +{x}+\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 31502 Answers: 0 Comments: 3

find f(x)= ∫_0 ^1 ln(1+xt^2 )dt with x>0. 2) give thevalue of ∫_0 ^1 ln(1+t^2 )dt and ∫_0 ^1 ln(1+2t^2 )dt.

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right){dt}\:\:{with}\:{x}>\mathrm{0}. \\ $$$$\left.\mathrm{2}\right)\:{give}\:{thevalue}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right){dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right){dt}. \\ $$

Question Number 31369 Answers: 0 Comments: 0

Question Number 31340 Answers: 0 Comments: 0

Deduce the power series of sin^2 x. Hence show that if x is small then (sin^2 x − x^2 cosx)/x^4 =(1/6) − (x^2 /(360))

$${Deduce}\:{the}\:{power}\:{series}\:{of}\:{sin}^{\mathrm{2}} {x}. \\ $$$${Hence}\:{show}\:{that}\:{if}\:{x}\:{is}\:{small}\:{then} \\ $$$$\left({sin}^{\mathrm{2}} {x}\:−\:{x}^{\mathrm{2}} {cosx}\right)/{x}^{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{6}}\:−\:\frac{{x}^{\mathrm{2}} }{\mathrm{360}} \\ $$

Question Number 31314 Answers: 0 Comments: 0

let x={(1/n)}_(n=1) ^∞ and y={(1/(n+1))}_(n=1) ^∞ be a sequence of real numbers and l_(2 ) ={x=(x_1 ,x_2 ,x_3 ,...):Σ_(n=1) ^∞ ∣xi∣^2 <∞} a linear space. (1) verify that x and y are in l_2 . (2) compute the inner product of x and y on l_2 please help me solve this question.

$${let}\:{x}=\left\{\frac{\mathrm{1}}{{n}}\right\}_{{n}=\mathrm{1}} ^{\infty} {and}\:{y}=\left\{\frac{\mathrm{1}}{{n}+\mathrm{1}}\right\}_{{n}=\mathrm{1}} ^{\infty} {be}\: \\ $$$${a}\:{sequence}\:{of}\:{real}\:{numbers}\:{and} \\ $$$${l}_{\mathrm{2}\:} =\left\{{x}=\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} ,...\right):\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mid{xi}\mid^{\mathrm{2}} <\infty\right\} \\ $$$${a}\:{linear}\:{space}.\: \\ $$$$\left(\mathrm{1}\right)\:{verify}\:{that}\:{x}\:{and}\:{y}\:{are}\:{in}\:{l}_{\mathrm{2}} . \\ $$$$\left(\mathrm{2}\right)\:{compute}\:{the}\:{inner}\:{product}\:{of}\:{x}\: \\ $$$${and}\:{y}\:{on}\:{l}_{\mathrm{2}} \\ $$$$\boldsymbol{{please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\: \\ $$$$\boldsymbol{{que}}{s}\boldsymbol{{tion}}. \\ $$

Question Number 31306 Answers: 1 Comments: 0

Complete the square in y^2 +8y+9k and hence find the value of k that makes it a perfect square.

$$\mathrm{Complete}\:\mathrm{the}\:\mathrm{square}\:\mathrm{in}\:\mathrm{y}^{\mathrm{2}} \:+\mathrm{8y}+\mathrm{9k}\:\mathrm{and}\:\mathrm{hence}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{that}\:\mathrm{makes}\:\mathrm{it}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$

Question Number 31281 Answers: 1 Comments: 0

find three nos in AP whose product is equal to the square of their sum.

$${find}\:{three}\:{nos}\:{in}\:{AP}\:{whose}\:{product} \\ $$$${is}\:{equal}\:{to}\:{the}\:{square}\:{of}\:{their}\:{sum}. \\ $$

Question Number 31085 Answers: 0 Comments: 1

calculate ∫∫_(x^2 +y^2 −2x≤0) xdxdy.

$${calculate}\:\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:−\mathrm{2}{x}\leqslant\mathrm{0}} {xdxdy}. \\ $$

Question Number 31017 Answers: 1 Comments: 0

solve (√(1+tan^2 x/1+cot^2 x= tanx))

$${solve}\:\sqrt{\mathrm{1}+{tan}^{\mathrm{2}} {x}/\mathrm{1}+{cot}^{\mathrm{2}} {x}=\:\:\:{tanx}} \\ $$

Question Number 30771 Answers: 0 Comments: 0

let I_n = ∫_0 ^(π/4) (dx/(cos^(2n+1) )) (n∈N) 1) find a and b fromR /∀x∈[0,(π/4)] (1/(cosx))=((acosx)/(1−sinx)) +((bcosx)/(1+sinx)) .find I_0 2) verify the relation (1/(cos^(2n+3) x))=(1/(cos^(2n+1) x)) +((sinx sinx)/(cos^(2n+3) )) .find the relation of recurrence between I_n and I_(n+1) .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} }\:\:\:\:\left({n}\in{N}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{and}\:{b}\:{fromR}\:/\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$$$\frac{\mathrm{1}}{{cosx}}=\frac{{acosx}}{\mathrm{1}−{sinx}}\:+\frac{{bcosx}}{\mathrm{1}+{sinx}}\:\:.{find}\:\:{I}_{\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:{verify}\:{the}\:{relation} \\ $$$$\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} {x}}=\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}\:+\frac{{sinx}\:{sinx}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} }\:.{find}\:{the}\:{relation} \\ $$$${of}\:{recurrence}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}+\mathrm{1}} \:\:. \\ $$

Question Number 30558 Answers: 0 Comments: 0

find I= ∫_(1/2) ^1 arctan((√(1−x^2 )) dx .

$${find}\:{I}=\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:{arctan}\left(\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:\:\:{dx}\:\:.\right. \\ $$

Question Number 30424 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ (x^n /(3n+2)) for ∣x∣<1 then find Σ_(n=0) ^∞ (((−1)^n )/((3n+2)2^n )) .

$${find}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{{n}} }{\mathrm{3}{n}+\mathrm{2}}\:\:\:{for}\:\:\mid{x}\mid<\mathrm{1}\:\:{then}\:{find}\: \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{3}{n}+\mathrm{2}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 30017 Answers: 1 Comments: 0

find the elements of the ellipse given the following equation 1.(x^2 /(25))+(y^2 /4)=1 2.x^2 +9y^2 =9

$$\mathrm{find}\:\mathrm{the}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ellipse}\:\mathrm{given}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation} \\ $$$$\mathrm{1}.\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{25}}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{4}}=\mathrm{1} \\ $$$$\mathrm{2}.\mathrm{x}^{\mathrm{2}} +\mathrm{9y}^{\mathrm{2}} =\mathrm{9} \\ $$

Question Number 29880 Answers: 0 Comments: 3