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Question Number 115531    Answers: 0   Comments: 0

Question Number 115532    Answers: 0   Comments: 0

Question Number 115520    Answers: 1   Comments: 0

Given matrix A = (((a 1 1)),((1 a 1)),((1 1 a)) ) If B = b.A and B is orthogonal determine value of a and b.

$${Given}\:{matrix}\:{A}\:=\:\begin{pmatrix}{{a}\:\:\:\mathrm{1}\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:{a}\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\mathrm{1}\:\:\:\:{a}}\end{pmatrix}\: \\ $$$${If}\:{B}\:=\:{b}.{A}\:{and}\:{B}\:{is}\:{orthogonal}\: \\ $$$${determine}\:{value}\:{of}\:{a}\:{and}\:{b}. \\ $$

Question Number 115516    Answers: 1   Comments: 1

Find the supremum and the infimum of (x/(sin x)) on the interval (0, (π/2) ]

$${Find}\:{the}\:{supremum}\:{and}\:{the}\:{infimum} \\ $$$${of}\:\frac{{x}}{\mathrm{sin}\:{x}}\:{on}\:{the}\:{interval}\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\:\right] \\ $$

Question Number 115508    Answers: 2   Comments: 0

If determinant (((a a^2 1+a^3 )),((b b^2 1+b^3 )),((c c^2 1+c^3 )))= 0 a≠b≠c → { ((a =?)),((b=? )),((c=?)) :}

$${If}\begin{vmatrix}{{a}\:\:\:\:{a}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{1}+{a}^{\mathrm{3}} }\\{{b}\:\:\:\:{b}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{1}+{b}^{\mathrm{3}} }\\{{c}\:\:\:\:{c}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{1}+{c}^{\mathrm{3}} }\end{vmatrix}=\:\mathrm{0} \\ $$$${a}\neq{b}\neq{c}\:\rightarrow\begin{cases}{{a}\:=?}\\{{b}=?\:}\\{{c}=?}\end{cases} \\ $$

Question Number 115507    Answers: 2   Comments: 0

.... ...matematical analysis... prove that ::: a>0 :: [((i : ∫_(0 ) ^( ∞) ((sin^2 (ax))/x^(3/2) ) dx= (√(πa)))),((ii: ∫_0 ^( ∞) ((sin^3 (ax))/( (√x))) dx = ((−1+3(√(3 )))/4) (√((π/(6a)) )) )) ] ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:....\:\:\:...{matematical}\:{analysis}...\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}>\mathrm{0}\:::\:\:\:\begin{bmatrix}{{i}\::\:\:\int_{\mathrm{0}\:} ^{\:\infty} \frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:{dx}=\:\sqrt{\pi{a}}}\\{{ii}:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({ax}\right)}{\:\sqrt{{x}}}\:{dx}\:=\:\frac{−\mathrm{1}+\mathrm{3}\sqrt{\mathrm{3}\:}}{\mathrm{4}}\:\sqrt{\frac{\pi}{\mathrm{6}{a}}\:\:}\:}\end{bmatrix} \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 115499    Answers: 0   Comments: 0

form a double integral to represent the area of a plane fiqure bounded by the polar centre

$${form}\:{a}\:{double}\:{integral}\:{to}\:{represent} \\ $$$${the}\:{area}\:{of}\:{a}\:{plane}\:{fiqure}\:{bounded}\:{by} \\ $$$${the}\:{polar}\:{centre} \\ $$

Question Number 115498    Answers: 5   Comments: 2

∫ sec x dx ? ∫ (√(x−(√x))) dx ? lim_(n→∞) (((((1+(√(1+n^2 ))))^(1/(n )) )(((2+(√(4+n^2 ))))^(1/(n )) )(((3+(√(9+n^2 ))))^(1/(n )) )...(((n+(√(2n^2 ))))^(1/(n )) ))/n)?

$$\int\:\mathrm{sec}\:{x}\:{dx}\:? \\ $$$$\int\:\sqrt{{x}−\sqrt{{x}}}\:{dx}\:? \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\sqrt[{{n}\:}]{\mathrm{1}+\sqrt{\mathrm{1}+{n}^{\mathrm{2}} }}\right)\left(\sqrt[{{n}\:}]{\mathrm{2}+\sqrt{\mathrm{4}+{n}^{\mathrm{2}} }}\right)\left(\sqrt[{{n}\:}]{\mathrm{3}+\sqrt{\mathrm{9}+{n}^{\mathrm{2}} }}\right)...\left(\sqrt[{{n}\:}]{{n}+\sqrt{\mathrm{2}{n}^{\mathrm{2}} }}\right)}{{n}}? \\ $$

Question Number 115487    Answers: 0   Comments: 3

Question Number 115484    Answers: 1   Comments: 3

find the value of ((111)/(1+1+1)) , ((222)/(2+2+2)) , ((333)/(3+3+3)), ((444)/(4+4+4)) ((555)/(5+5+5)), ((666)/(6+6+6)) , ((777)/(7+7+7)) , ((888)/(8+8+8)) ((999)/(9+9+9))

$${find}\:{the}\:{value}\:{of}\: \\ $$$$\frac{\mathrm{111}}{\mathrm{1}+\mathrm{1}+\mathrm{1}}\:,\:\frac{\mathrm{222}}{\mathrm{2}+\mathrm{2}+\mathrm{2}}\:,\:\frac{\mathrm{333}}{\mathrm{3}+\mathrm{3}+\mathrm{3}},\:\frac{\mathrm{444}}{\mathrm{4}+\mathrm{4}+\mathrm{4}} \\ $$$$\frac{\mathrm{555}}{\mathrm{5}+\mathrm{5}+\mathrm{5}},\:\frac{\mathrm{666}}{\mathrm{6}+\mathrm{6}+\mathrm{6}}\:,\:\frac{\mathrm{777}}{\mathrm{7}+\mathrm{7}+\mathrm{7}}\:,\:\frac{\mathrm{888}}{\mathrm{8}+\mathrm{8}+\mathrm{8}} \\ $$$$\frac{\mathrm{999}}{\mathrm{9}+\mathrm{9}+\mathrm{9}} \\ $$

Question Number 115472    Answers: 1   Comments: 1

Question Number 115489    Answers: 1   Comments: 0

find range for 1. f(x)=((√(x^2 −9))/(x−1)) 2. f(x)=ln ((√(4−9x^2 )))

$${find}\:{range}\:{for} \\ $$$$\mathrm{1}.\:{f}\left({x}\right)=\frac{\sqrt{{x}^{\mathrm{2}} −\mathrm{9}}}{{x}−\mathrm{1}} \\ $$$$\mathrm{2}.\:{f}\left({x}\right)=\mathrm{ln}\:\left(\sqrt{\mathrm{4}−\mathrm{9}{x}^{\mathrm{2}} }\right) \\ $$

Question Number 115459    Answers: 3   Comments: 0

I= ∫_0 ^1 (dx/((1+x^3 )((1+x^3 ))^(1/(3 )) )) ? I=∫_0 ^(π/2) cos^2 x cos^2 (2x) dx = ?

$${I}=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)\sqrt[{\mathrm{3}\:}]{\mathrm{1}+{x}^{\mathrm{3}} }}\:? \\ $$$${I}=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{cos}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{2}{x}\right)\:{dx}\:=\:? \\ $$$$ \\ $$

Question Number 115455    Answers: 1   Comments: 1

cos ((π/(65))).cos (((2π)/(65))).cos (((4π)/(65))).cos (((8π)/(65))).cos (((16π)/(65))).cos (((32π)/(65)))=?

$$\mathrm{cos}\:\left(\frac{\pi}{\mathrm{65}}\right).\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{65}}\right).\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{65}}\right).\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{65}}\right).\mathrm{cos}\:\left(\frac{\mathrm{16}\pi}{\mathrm{65}}\right).\mathrm{cos}\:\left(\frac{\mathrm{32}\pi}{\mathrm{65}}\right)=? \\ $$

Question Number 115449    Answers: 2   Comments: 0

calculate ∫_(−∞) ^∞ (x^2 /((x^2 −x+1)^3 ))dx

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

Question Number 115439    Answers: 0   Comments: 1

Question Number 115438    Answers: 2   Comments: 0

Question Number 115436    Answers: 1   Comments: 0

(√((a)^(1/3) +(b)^(1/3) ))=(1/( (√(b−a×s^3 ))))(((−s^2 ×(a^2 )^(1/3) )/2)+s((ab))^(1/3) +(b^2 )^(1/3) ) what is s?

$$\sqrt{\sqrt[{\mathrm{3}}]{{a}}+\sqrt[{\mathrm{3}}]{{b}}}=\frac{\mathrm{1}}{\:\sqrt{{b}−{a}×{s}^{\mathrm{3}} }}\left(\frac{−{s}^{\mathrm{2}} ×\sqrt[{\mathrm{3}}]{{a}^{\mathrm{2}} }}{\mathrm{2}}+{s}\sqrt[{\mathrm{3}}]{{ab}}+\sqrt[{\mathrm{3}}]{{b}^{\mathrm{2}} }\right) \\ $$$${what}\:{is}\:{s}? \\ $$

Question Number 115429    Answers: 0   Comments: 9

how many ways can you arrange 15 distinct balls into 5 cups if there has to be at least 1 ball in each cup?

$${how}\:{many}\:{ways}\:{can}\:{you}\:{arrange}\:\mathrm{15}\:{distinct} \\ $$$${balls}\:{into}\:\mathrm{5}\:{cups}\:{if}\:{there}\:{has}\:{to}\:{be}\:{at}\:{least} \\ $$$$\mathrm{1}\:{ball}\:{in}\:{each}\:{cup}? \\ $$$$ \\ $$

Question Number 115418    Answers: 3   Comments: 1

Question Number 115417    Answers: 2   Comments: 0

with the use of mathematical induction show that n!>2n^3 , ∀n≥6.

$$\mathrm{with}\:\mathrm{the}\:\mathrm{use}\:\mathrm{of}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{n}!>\mathrm{2n}^{\mathrm{3}} ,\:\forall\mathrm{n}\geqslant\mathrm{6}. \\ $$

Question Number 115464    Answers: 5   Comments: 0

I= ∫_(0 ) ^1 x ln (1+x^2 ) dx ? I=∫ (√(sin x)) .cos^3 x dx ?

$${I}=\:\underset{\mathrm{0}\:} {\overset{\mathrm{1}} {\int}}\:{x}\:\mathrm{ln}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:{dx}\:? \\ $$$${I}=\int\:\sqrt{\mathrm{sin}\:{x}}\:.\mathrm{cos}\:^{\mathrm{3}} {x}\:{dx}\:? \\ $$

Question Number 115408    Answers: 1   Comments: 0

how many 6 digit numbers exist which are divisible by 11 and have no repeating digits?

$${how}\:{many}\:\mathrm{6}\:{digit}\:{numbers}\:{exist} \\ $$$${which}\:{are}\:{divisible}\:{by}\:\mathrm{11}\:{and}\:{have}\:{no} \\ $$$${repeating}\:{digits}? \\ $$

Question Number 115405    Answers: 1   Comments: 0

find the close form of Σ_(n=0) ^∞ (((−1)^n )/((n+1)(n+2)(2n+1)(2n+3)))

$${find}\:{the}\:{close}\:{form}\:{of} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)} \\ $$

Question Number 115404    Answers: 4   Comments: 0

∫_( 0) ^∞ (1/(1+x^4 )) dx =

$$\:\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:= \\ $$

Question Number 115403    Answers: 0   Comments: 0

∫ e^x ((1+n x^(n−1) −x^(2n) )/((1−x)^n (√(1−x^(2n) )))) dx =

$$\int\:{e}^{{x}} \:\frac{\mathrm{1}+{n}\:{x}^{{n}−\mathrm{1}} −{x}^{\mathrm{2}{n}} }{\left(\mathrm{1}−{x}\right)^{{n}} \:\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}{n}} }}\:{dx}\:= \\ $$

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