Question and Answers Forum

Father reduced the quantity of food bought for the family by ((25)/2)% . When he found that the cost of living has increased by 15%. What is the fraction increase in the family′s food bill now ?

$$\mathrm{Father}\:\mathrm{reduced}\:\mathrm{the}\:\mathrm{quantity}\:\mathrm{of}\:\mathrm{food}\:\mathrm{bought}\:\mathrm{for} \\ $$$$\mathrm{the}\:\mathrm{family}\:\mathrm{by}\:\frac{\mathrm{25}}{\mathrm{2}}\%\:.\:\mathrm{When}\:\mathrm{he}\:\mathrm{found}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{cost}\:\mathrm{of}\:\mathrm{living}\:\mathrm{has}\:\mathrm{increased}\:\mathrm{by}\:\mathrm{15\%}.\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{fraction}\:\mathrm{increase}\:\mathrm{in}\:\mathrm{the}\:\mathrm{family}'\mathrm{s}\:\mathrm{food} \\ $$$$\mathrm{bill}\:\mathrm{now}\:? \\ $$

Question Number 8762 Answers: 1 Comments: 0

∫x^2 (2x + 1)^(1/2) dx

$$\int\mathrm{x}^{\mathrm{2}} \left(\mathrm{2x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} \:\mathrm{dx} \\ $$

Question Number 8756 Answers: 0 Comments: 0

show that every sphere through the circle x^2 +y^2 −2ax+r^2 =0,z=0 ,z=0 cuts orthogonally every sphere through the circle x^2 +z^2 =r^2 , y=o .

$${show}\:{that}\:{every}\:{sphere}\:{through}\:{the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{ax}+{r}^{\mathrm{2}} =\mathrm{0},{z}=\mathrm{0} \\ $$$$,{z}=\mathrm{0}\:\:\:\:\:\:\:{cuts}\:{orthogonally}\:{every}\:{sphere}\:{through}\:{the}\:{circle}\: \\ $$$${x}^{\mathrm{2}} +{z}^{\mathrm{2}} ={r}^{\mathrm{2}} ,\:{y}={o}\:. \\ $$

Question Number 8755 Answers: 0 Comments: 0

find the equation of the sphere which touches the plane 3x+2y−z+2=0 at the point (1,−2,1) and cuts orthogonally the the sphere x^2 +y^2 +z^2 −4x+6y+4=0

$${find}\:{the}\:{equation}\:{of}\:{the}\:{sphere}\:{which}\:{touches}\:{the}\:{plane}\: \\ $$$$\mathrm{3}{x}+\mathrm{2}{y}−{z}+\mathrm{2}=\mathrm{0}\:{at}\:{the}\:{point}\:\left(\mathrm{1},−\mathrm{2},\mathrm{1}\right)\:{and}\:{cuts}\:{orthogonally}\:{the} \\ $$$${the}\:{sphere}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{6}{y}+\mathrm{4}=\mathrm{0} \\ $$

Question Number 8750 Answers: 1 Comments: 2

wx + 2z = 3 ............ (i) 3x − y + 4z = 4 ........... (ii) 6x + 2wy = − 4 ........... (iii) find w, x, y, z

$$\mathrm{wx}\:+\:\mathrm{2z}\:=\:\mathrm{3}\:\:\:............\:\left(\mathrm{i}\right) \\ $$$$\mathrm{3x}\:−\:\mathrm{y}\:+\:\mathrm{4z}\:=\:\mathrm{4}\:\:\:...........\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{6x}\:+\:\mathrm{2wy}\:=\:−\:\mathrm{4}\:\:\:...........\:\left(\mathrm{iii}\right) \\ $$$$ \\ $$$$\mathrm{find}\:\:\mathrm{w},\:\mathrm{x},\:\mathrm{y},\:\mathrm{z} \\ $$

Question Number 8764 Answers: 0 Comments: 2

(a) Find the sum given by S_n = (1/(1.3)) + (1/(3.5)) + (1/(5.7)) + ... + (1/((2n − 1)(2n + 1))) (b) find the limit of S_n as n → ∞

$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{given}\:\mathrm{by} \\ $$$$\mathrm{S}_{\mathrm{n}} \:=\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{5}.\mathrm{7}}\:+\:...\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of}\:\:\:\mathrm{S}_{\mathrm{n}} \:\:\mathrm{as}\:\:\mathrm{n}\:\rightarrow\:\infty \\ $$

Question Number 8763 Answers: 1 Comments: 2

∫x(√(3x + 1)) dx

$$\int\mathrm{x}\sqrt{\mathrm{3x}\:+\:\mathrm{1}}\:\:\mathrm{dx} \\ $$

Question Number 8865 Answers: 1 Comments: 0

(1^2 /(1×3))+(2^2 /(3×5))+...+((n2)/((2n−1)(2n+1)))=((n(n+1)/(2(2n+1)))

$$\frac{\mathrm{1}^{\mathrm{2}} }{\mathrm{1}×\mathrm{3}}+\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}×\mathrm{5}}+...+\frac{{n}\mathrm{2}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}=\frac{{n}\left({n}+\mathrm{1}\right.}{\mathrm{2}\left(\mathrm{2}{n}+\mathrm{1}\right)} \\ $$

Question Number 8740 Answers: 0 Comments: 2

x_(1,1) =1 x_(n+1,m) =x_(n,m) +m x_(n,m+1) =nx_(n,m) −1 x_(2,2) =?

$${x}_{\mathrm{1},\mathrm{1}} =\mathrm{1} \\ $$$${x}_{{n}+\mathrm{1},{m}} ={x}_{{n},{m}} +{m} \\ $$$${x}_{{n},{m}+\mathrm{1}} ={nx}_{{n},{m}} −\mathrm{1} \\ $$$${x}_{\mathrm{2},\mathrm{2}} =? \\ $$

Question Number 8734 Answers: 0 Comments: 2

Evaluate the integral ∫[a(b^• .a + b.a^• ) + a^• (b.a) − 2(a^• .a)b − b^• ∣a∣^2 ]dt In which a^• ,b^• are the derivatives of a,b with respect to t

$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int\left[\mathrm{a}\left(\overset{\bullet} {\mathrm{b}}.\mathrm{a}\:+\:\mathrm{b}.\overset{\bullet} {\mathrm{a}}\right)\:+\:\overset{\bullet} {\mathrm{a}}\left(\mathrm{b}.\mathrm{a}\right)\:−\:\mathrm{2}\left(\overset{\bullet} {\mathrm{a}}.\mathrm{a}\right)\mathrm{b}\:−\:\overset{\bullet} {\mathrm{b}}\mid\mathrm{a}\mid^{\mathrm{2}} \right]\mathrm{dt} \\ $$$$\mathrm{In}\:\mathrm{which}\:\:\overset{\bullet} {\mathrm{a}}\:,\overset{\bullet} {\mathrm{b}}\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{derivatives}\:\mathrm{of}\:\:\mathrm{a},\mathrm{b}\:\mathrm{with}\: \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{t} \\ $$

Question Number 8728 Answers: 0 Comments: 2

∫cos x/4−x^2 ∫((cos x)/(4−x^2 ))

$$\int\mathrm{cos}\:{x}/\mathrm{4}−{x}^{\mathrm{2}} \\ $$$$\int\frac{\mathrm{cos}\:{x}}{\mathrm{4}−{x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 8721 Answers: 0 Comments: 2

If z∈C satisfies ∣z^3 +z^(−3) ∣≤2 then maximum possible value of∣z+z^(−1) ∣is?

$$\mathrm{If}\:\mathrm{z}\in\mathrm{C}\:\mathrm{satisfies} \\ $$$$\mid\mathrm{z}^{\mathrm{3}} +\mathrm{z}^{−\mathrm{3}} \mid\leqslant\mathrm{2} \\ $$$${t}\mathrm{hen}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\mid\mathrm{z}+\mathrm{z}^{−\mathrm{1}} \mid\mathrm{is}? \\ $$

Question Number 8720 Answers: 1 Comments: 5

Question Number 8757 Answers: 0 Comments: 1

A balloon is inflated such that every point expands at a units/second. An ant runs from one point A to another point B. If the ant moves b units/second, what will influence if or not the ant will ever reach point B?

$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{is}\:\mathrm{inflated}\:\mathrm{such}\:\mathrm{that}\:\mathrm{every} \\ $$$$\mathrm{point}\:\mathrm{expands}\:\mathrm{at}\:{a}\:\mathrm{units}/\mathrm{second}. \\ $$$$ \\ $$$$\mathrm{An}\:\mathrm{ant}\:\mathrm{runs}\:\mathrm{from}\:\mathrm{one}\:\mathrm{point}\:\boldsymbol{{A}}\:\mathrm{to}\:\mathrm{another} \\ $$$$\mathrm{point}\:\boldsymbol{{B}}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ant}\:\mathrm{moves}\:{b}\:\mathrm{units}/\mathrm{second}, \\ $$$$\mathrm{what}\:\mathrm{will}\:\mathrm{influence}\:\mathrm{if}\:\mathrm{or}\:\mathrm{not}\:\mathrm{the}\:\mathrm{ant}\:\mathrm{will} \\ $$$$\mathrm{ever}\:\mathrm{reach}\:\mathrm{point}\:\boldsymbol{{B}}? \\ $$

Question Number 8707 Answers: 1 Comments: 0

Find an integer x that satisfies the equation x^5 −101x^3 −999x^2 +100900=0

$${Find}\:{an}\:{integer}\:{x}\:{that}\:{satisfies}\:{the}\:{equation}\: \\ $$$${x}^{\mathrm{5}} −\mathrm{101}{x}^{\mathrm{3}} −\mathrm{999}{x}^{\mathrm{2}} +\mathrm{100900}=\mathrm{0} \\ $$

Question Number 8704 Answers: 1 Comments: 0

Solving for A. U(z) = U_b +((2A)/(h+1))(ρ×g×sin(α))^n [H^(n+1) −(H−Z)^(n+1) ]

$${Solving}\:{for}\:{A}. \\ $$$${U}\left({z}\right)\:=\:{U}_{{b}} +\frac{\mathrm{2}{A}}{{h}+\mathrm{1}}\left(\rho×{g}×{sin}\left(\alpha\right)\right)^{{n}} \left[{H}^{{n}+\mathrm{1}} −\left({H}−{Z}\right)^{{n}+\mathrm{1}} \right] \\ $$

Question Number 8695 Answers: 2 Comments: 0

solving for B? U_b = U_s − (2/(n+1))(((ρ×g×sinα)/B))^n H^(n+1)

$${solving}\:{for}\:{B}? \\ $$$${U}_{{b}} \:=\:{U}_{{s}} \:−\:\frac{\mathrm{2}}{{n}+\mathrm{1}}\left(\frac{\rho×{g}×{sin}\alpha}{{B}}\right)^{{n}} {H}^{{n}+\mathrm{1}} \\ $$

Question Number 8691 Answers: 1 Comments: 1

∫_0 ^( ∞) x^(−ln(x)) dx

$$\int_{\mathrm{0}} ^{\:\infty} {x}^{−\mathrm{ln}\left({x}\right)} {dx} \\ $$

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