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Question Number 27015    Answers: 0   Comments: 1

find the simple intrest on rs 840 for 3 year at at 5% per annum?

$${find}\:{the}\:{simple}\:{intrest}\:{on}\:{rs}\:\mathrm{840}\:{for}\:\mathrm{3}\:{year}\:{at}\: \\ $$$${at}\:\mathrm{5\%}\:{per}\:{annum}? \\ $$

Question Number 27003    Answers: 0   Comments: 2

∫_(1/8) ^(1/2) ⌊ln ⌈(1/x)⌉⌋ dx

$$\underset{\mathrm{1}/\mathrm{8}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\lfloor\mathrm{ln}\:\lceil\frac{\mathrm{1}}{{x}}\rceil\rfloor\:{dx} \\ $$

Question Number 26999    Answers: 1   Comments: 0

calculate Π_(k=1) ^n cos((a/2^k )) and0<a<π then find the value of lim_(n−>∝) Σ_(k=1) ^n ln(cos((a/2^k ))).

$${calculate}\:\prod_{{k}=\mathrm{1}} ^{{n}} {cos}\left(\frac{{a}}{\mathrm{2}^{{k}} }\right)\:\:{and}\mathrm{0}<{a}<\pi\:\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$${lim}_{{n}−>\propto} \:\sum_{{k}=\mathrm{1}} ^{{n}} {ln}\left({cos}\left(\frac{{a}}{\mathrm{2}^{{k}} }\right)\right). \\ $$

Question Number 26998    Answers: 0   Comments: 0

smlify X= Π_(p=2) ^n ((p^3 −1)/(p^3 +1)) by using 1,j,j^2 and j=e^((i2π)/3) .

$${smlify}\:{X}=\:\:\prod_{{p}=\mathrm{2}} ^{{n}} \frac{{p}^{\mathrm{3}} −\mathrm{1}}{{p}^{\mathrm{3}} \:+\mathrm{1}}\:{by}\:{using}\:\mathrm{1},{j},{j}^{\mathrm{2}} {and}\:{j}={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} . \\ $$

Question Number 26997    Answers: 0   Comments: 3

let give ξ ∈C and ξ^n =1 (ξ is the n^(me) root of 1) simplify A= 1+ξ^p +ξ^(2p) +... +ξ^((n−1)p) and B= 1+2ξ +3ξ^2 +...+nξ^(n−1) .

$${let}\:{give}\:\xi\:\in\mathbb{C}\:{and}\:\xi^{{n}} =\mathrm{1}\:\left(\xi\:{is}\:{the}\:{n}^{{me}} \:{root}\:{of}\:\mathrm{1}\right) \\ $$$${simplify}\:\:{A}=\:\mathrm{1}+\xi^{{p}} +\xi^{\mathrm{2}{p}} +...\:+\xi^{\left({n}−\mathrm{1}\right){p}} \\ $$$${and}\:{B}=\:\mathrm{1}+\mathrm{2}\xi\:+\mathrm{3}\xi^{\mathrm{2}} +...+{n}\xi^{{n}−\mathrm{1}} . \\ $$

Question Number 27153    Answers: 0   Comments: 2

find the value of Π_(k=1) ^(n−1) sin(((kπ)/(2n)) ) .

$${find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{sin}\left(\frac{{k}\pi}{\mathrm{2}{n}}\:\right)\:. \\ $$

Question Number 26993    Answers: 0   Comments: 5

Question Number 26983    Answers: 1   Comments: 0

Find (dy/dx) x^y =y^x

$$\mathrm{Find}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:\:\: \\ $$$$\:\:\:\:\mathrm{x}^{\mathrm{y}} =\mathrm{y}^{\mathrm{x}} \\ $$

Question Number 26982    Answers: 1   Comments: 0

Question Number 26981    Answers: 1   Comments: 0

If the value of y satisfying the equations xsin^3 y + 3xsin y cos^2 y = 63 and xcos^3 y + 3xcosy sin^2 y = 62 simultaneously. Then tan y is equal to

$$\mathrm{If}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{y}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equations} \\ $$$${x}\mathrm{sin}^{\mathrm{3}} {y}\:+\:\mathrm{3}{x}\mathrm{sin}\:{y}\:\mathrm{cos}^{\mathrm{2}} {y}\:=\:\mathrm{63}\:\mathrm{and}\:{x}\mathrm{cos}^{\mathrm{3}} {y} \\ $$$$+\:\mathrm{3}{x}\mathrm{cos}{y}\:\mathrm{sin}^{\mathrm{2}} {y}\:=\:\mathrm{62}\:\mathrm{simultaneously}. \\ $$$$\mathrm{Then}\:\mathrm{tan}\:{y}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 26959    Answers: 3   Comments: 0

Two balls, each of radius R, equal mass and density are placed in contact, then the force of gravitation between them is proportional to (1) F ∝ (1/R^2 ) (2) F ∝ R (3) F ∝ R^4 (4) F ∝ (1/R)

$$\mathrm{Two}\:\mathrm{balls},\:\mathrm{each}\:\mathrm{of}\:\mathrm{radius}\:{R},\:\mathrm{equal}\:\mathrm{mass} \\ $$$$\mathrm{and}\:\mathrm{density}\:\mathrm{are}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{contact},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{force}\:\mathrm{of}\:\mathrm{gravitation}\:\mathrm{between}\:\mathrm{them} \\ $$$$\mathrm{is}\:\mathrm{proportional}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:{F}\:\propto\:\frac{\mathrm{1}}{{R}^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:{F}\:\propto\:{R} \\ $$$$\left(\mathrm{3}\right)\:{F}\:\propto\:{R}^{\mathrm{4}} \\ $$$$\left(\mathrm{4}\right)\:{F}\:\propto\:\frac{\mathrm{1}}{{R}} \\ $$

Question Number 26947    Answers: 1   Comments: 1

(d/dx)ln (Γ(x+1))=?

$$\frac{{d}}{{dx}}\mathrm{ln}\:\left(\Gamma\left({x}+\mathrm{1}\right)\right)=? \\ $$

Question Number 26946    Answers: 0   Comments: 0

f(x)=x^2 cos((1/x)) when x∈[−(1/π),(1/π)]\{0} and f(x)=0 when x=0. a) find the derivative of f(x) on the interval of [−(1/π),(1/π)]. b) compute minf(x) and maxf(x).

$${f}\left({x}\right)={x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{1}}{{x}}\right)\:{when}\:{x}\in\left[−\frac{\mathrm{1}}{\pi},\frac{\mathrm{1}}{\pi}\right]\backslash\left\{\mathrm{0}\right\} \\ $$$${and}\:{f}\left({x}\right)=\mathrm{0}\:{when}\:{x}=\mathrm{0}. \\ $$$$\left.{a}\right)\:{find}\:{the}\:{derivative}\:{of}\:{f}\left({x}\right)\:{on} \\ $$$${the}\:{interval}\:{of}\:\left[−\frac{\mathrm{1}}{\pi},\frac{\mathrm{1}}{\pi}\right]. \\ $$$$\left.{b}\right)\:{compute}\:{minf}\left({x}\right)\:{and}\:{maxf}\left({x}\right). \\ $$

Question Number 26942    Answers: 1   Comments: 0

Question Number 26941    Answers: 1   Comments: 0

show that the rectangular solid of naximum volume that can be inscribed into a sphere is a cube

$${show}\:{that}\:{the}\:{rectangular}\:{solid}\:\:{of}\: \\ $$$${naximum}\:{volume}\:{that}\:{can}\:{be}\:{inscribed} \\ $$$${into}\:{a}\:{sphere}\:{is}\:{a}\:{cube} \\ $$

Question Number 27027    Answers: 0   Comments: 0

Find max and min of the function: f(x)=x^2 cos((1/x))

$${Find}\:{max}\:{and}\:{min}\:{of}\:{the}\:{function}: \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{1}}{{x}}\right) \\ $$

Question Number 26904    Answers: 1   Comments: 1

y^((2)) +y=sec^3 x

$${y}^{\left(\mathrm{2}\right)} +{y}=\mathrm{sec}\:^{\mathrm{3}} {x} \\ $$

Question Number 26909    Answers: 1   Comments: 0

∫((e^x ((1/x)−(2/x^3 )))/(−2))dx

$$\int\frac{{e}^{{x}} \left(\frac{\mathrm{1}}{{x}}−\frac{\mathrm{2}}{{x}^{\mathrm{3}} }\right)}{−\mathrm{2}}{dx} \\ $$

Question Number 26889    Answers: 1   Comments: 0

Question Number 26888    Answers: 1   Comments: 0

If α, β are the roots of ax^2 +bx+c=0 then find the quadratic equation whose roots are α+β, αβ.

$$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\mathrm{whose}\:\mathrm{roots}\:\mathrm{are}\:\alpha+\beta,\:\:\alpha\beta. \\ $$

Question Number 26864    Answers: 1   Comments: 0

Question Number 27000    Answers: 0   Comments: 1

P is a polynomial havng n roots (x_i )_(1≤i≤n) with x_i ≠ x_j for i≠ j find the values of Σ_(k1) ^(k=n) (1/(x−x_k )) and Σ_(k=1) ^n (1/((x−x_k )^2 )) .

$${P}\:{is}\:{a}\:{polynomial}\:{havng}\:{n}\:{roots}\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \:\:{with}\:{x}_{{i}} \neq\:{x}_{{j}} \:{for}\:{i}\neq\:{j} \\ $$$${find}\:{the}\:{values}\:{of}\:\sum_{{k}\mathrm{1}} ^{{k}={n}} \frac{\mathrm{1}}{{x}−{x}_{{k}} }\:\:{and}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{\left({x}−{x}_{{k}} \right)^{\mathrm{2}} }\:. \\ $$

Question Number 26858    Answers: 2   Comments: 0

Prove that the internal bisector of an angle of a triangle divides the opposite sides in the ratio of the sides containing the angle.

$${Prove}\:{that}\:{the}\:{internal}\:{bisector}\:{of}\:{an}\:{angle}\: \\ $$$${of}\:{a}\:{triangle}\:{divides}\:{the}\:{opposite} \\ $$$${sides}\:{in}\:{the}\:{ratio}\:{of}\:{the}\:{sides}\:{containing} \\ $$$${the}\:{angle}. \\ $$

Question Number 26853    Answers: 1   Comments: 1

Question Number 26839    Answers: 1   Comments: 0

a=3 b=6 a−b=?

$$\mathrm{a}=\mathrm{3}\:\mathrm{b}=\mathrm{6} \\ $$$$\mathrm{a}−\mathrm{b}=? \\ $$

Question Number 26837    Answers: 1   Comments: 0

xy^((2)) =y^((1)) ×ln (y^((1)) /x)

$${xy}^{\left(\mathrm{2}\right)} ={y}^{\left(\mathrm{1}\right)} ×\mathrm{ln}\:\frac{{y}^{\left(\mathrm{1}\right)} }{{x}} \\ $$

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