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AllQuestion and Answers: Page 389

Question Number 182066 Answers: 2 Comments: 0

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Question Number 182050 Answers: 1 Comments: 0

A= [(a,b,c),((−2),3,6),(0,(−2),5) ]and B= [(1,2,4),(0,3,9),((−1),2,2) ] A×B= [((−1),3,(−1)),((−8),d,(31)),((−5),4,e) ]find the missing value

$${A}=\begin{bmatrix}{{a}}&{{b}}&{{c}}\\{−\mathrm{2}}&{\mathrm{3}}&{\mathrm{6}}\\{\mathrm{0}}&{−\mathrm{2}}&{\mathrm{5}}\end{bmatrix}{and}\:{B}=\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{4}}\\{\mathrm{0}}&{\mathrm{3}}&{\mathrm{9}}\\{−\mathrm{1}}&{\mathrm{2}}&{\mathrm{2}}\end{bmatrix} \\ $$$${A}×{B}=\begin{bmatrix}{−\mathrm{1}}&{\mathrm{3}}&{−\mathrm{1}}\\{−\mathrm{8}}&{{d}}&{\mathrm{31}}\\{−\mathrm{5}}&{\mathrm{4}}&{{e}}\end{bmatrix}{find}\:{the}\:{missing}\:{value} \\ $$

Question Number 182120 Answers: 2 Comments: 0

Question Number 182045 Answers: 2 Comments: 1

∫_(π/6) ^(π/3) (1/(1+(tanx)^(2013) )) dx = ?

$$ \\ $$$$\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \:\frac{\mathrm{1}}{\mathrm{1}+\left({tanx}\right)^{\mathrm{2013}} }\:{dx}\:=\:? \\ $$$$ \\ $$

Question Number 182043 Answers: 0 Comments: 0

For a, b, c, d ∈ R a+b+c+d=0 ab, ac, ad, bc, bd, cd ≠0 Prove the inequality: ((ab)/((a+b)^2 ))+((ac)/((a+c)^2 ))+((ad)/((a+d)^2 ))+((bc)/((b+c)^2 ))+((bd)/((b+d)^2 ))+((cd)/((c+d)^2 ))≤−(3/2) When is equality reached?

$${For}\:{a},\:{b},\:{c},\:{d}\:\in\:\mathbb{R} \\ $$$${a}+{b}+{c}+{d}=\mathrm{0} \\ $$$${ab},\:{ac},\:{ad},\:{bc},\:{bd},\:{cd}\:\neq\mathrm{0} \\ $$$${Prove}\:{the}\:{inequality}: \\ $$$$\frac{{ab}}{\left({a}+{b}\right)^{\mathrm{2}} }+\frac{{ac}}{\left({a}+{c}\right)^{\mathrm{2}} }+\frac{{ad}}{\left({a}+{d}\right)^{\mathrm{2}} }+\frac{{bc}}{\left({b}+{c}\right)^{\mathrm{2}} }+\frac{{bd}}{\left({b}+{d}\right)^{\mathrm{2}} }+\frac{{cd}}{\left({c}+{d}\right)^{\mathrm{2}} }\leq−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${When}\:{is}\:{equality}\:{reached}? \\ $$

Question Number 182039 Answers: 1 Comments: 0

if f′′′(x)=f(x) find what is f(x)

$${if}\:{f}'''\left({x}\right)={f}\left({x}\right) \\ $$$${find}\:{what}\:{is}\:{f}\left({x}\right)\: \\ $$

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Question Number 182026 Answers: 3 Comments: 0

h(x)= determinant (((sin x cos x tan x)),((cos 2x sin 2x tan 2x)),(( x^3 (1/4)x^4 xsin x))) h′(0) =?

$$\:\:\mathrm{h}\left(\mathrm{x}\right)=\:\begin{vmatrix}{\mathrm{sin}\:\mathrm{x}\:\:\:\:\:\mathrm{cos}\:\mathrm{x}\:\:\:\:\:\:\:\mathrm{tan}\:\mathrm{x}}\\{\mathrm{cos}\:\mathrm{2x}\:\:\mathrm{sin}\:\mathrm{2x}\:\:\:\:\:\mathrm{tan}\:\mathrm{2x}}\\{\:\:\:\mathrm{x}^{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{4}}\mathrm{x}^{\mathrm{4}} \:\:\:\:\:\:\mathrm{xsin}\:\mathrm{x}}\end{vmatrix} \\ $$$$\:\:\mathrm{h}'\left(\mathrm{0}\right)\:=? \\ $$

Question Number 182375 Answers: 2 Comments: 1

Question Number 182374 Answers: 2 Comments: 1

Question Number 182012 Answers: 1 Comments: 0

f(x)=9^x −m∙3^x +m+6 ∃x∈R, f(x)+f(−x)=0 find the range of m.

$${f}\left({x}\right)=\mathrm{9}^{{x}} −{m}\centerdot\mathrm{3}^{{x}} +{m}+\mathrm{6} \\ $$$$\exists{x}\in\mathbb{R},\:{f}\left({x}\right)+{f}\left(−{x}\right)=\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{the}\:\:\mathrm{range}\:\mathrm{of}\:{m}. \\ $$

Question Number 182011 Answers: 1 Comments: 1

Is this true for complex number: 4Re(z_1 z_2 ^(−) )=∣z_1 +z_2 ^(−) ∣^2 −∣z_1 −z_2 ^(−) ∣^2

$${Is}\:{this}\:{true}\:{for}\:{complex}\:{number}: \\ $$$$\mathrm{4}\mathscr{R}{e}\left({z}_{\mathrm{1}} \overline {{z}_{\mathrm{2}} }\right)=\mid{z}_{\mathrm{1}} +\overline {{z}_{\mathrm{2}} }\mid^{\mathrm{2}} −\mid{z}_{\mathrm{1}} −\overline {{z}_{\mathrm{2}} }\mid^{\mathrm{2}} \\ $$

Question Number 182006 Answers: 1 Comments: 2

Question Number 182000 Answers: 0 Comments: 1

if a−2b+3c−4d+5e−6f=0, find the maximum of ((∣a+b+c+d+e+f∣)/( (√(a^2 +b^2 +c^2 +d^2 +e^2 +f^2 )))).

$${if}\:\boldsymbol{{a}}−\mathrm{2}\boldsymbol{{b}}+\mathrm{3}\boldsymbol{{c}}−\mathrm{4}\boldsymbol{{d}}+\mathrm{5}\boldsymbol{{e}}−\mathrm{6}\boldsymbol{{f}}=\mathrm{0},\:{find} \\ $$$${the}\:{maximum}\:{of} \\ $$$$\frac{\mid\boldsymbol{{a}}+\boldsymbol{{b}}+\boldsymbol{{c}}+\boldsymbol{{d}}+\boldsymbol{{e}}+\boldsymbol{{f}}\mid}{\:\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{c}}^{\mathrm{2}} +\boldsymbol{{d}}^{\mathrm{2}} +\boldsymbol{{e}}^{\mathrm{2}} +\boldsymbol{{f}}^{\mathrm{2}} }}. \\ $$

Question Number 181998 Answers: 1 Comments: 0

Question Number 181987 Answers: 3 Comments: 0

Question Number 182004 Answers: 3 Comments: 0

f(x)=2^x +3^x −6^x Find f(x)_(max)

$${f}\left({x}\right)=\mathrm{2}^{{x}} +\mathrm{3}^{{x}} −\mathrm{6}^{{x}} \\ $$$$\mathrm{Find}\:{f}\left({x}\right)_{\mathrm{max}} \\ $$

Question Number 182003 Answers: 0 Comments: 0

u_(xx) −u_x u_y −u_(yy) +2u_y −2u_x =e^(2x+3y) +sin(2x+y)+xy

$$\mathrm{u}_{\mathrm{xx}} −\mathrm{u}_{\mathrm{x}} \mathrm{u}_{\mathrm{y}} −\mathrm{u}_{\mathrm{yy}} +\mathrm{2u}_{\mathrm{y}} −\mathrm{2u}_{\mathrm{x}} =\mathrm{e}^{\mathrm{2x}+\mathrm{3y}} +\mathrm{sin}\left(\mathrm{2x}+\mathrm{y}\right)+\mathrm{xy} \\ $$

Question Number 182001 Answers: 2 Comments: 1

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Question Number 181973 Answers: 0 Comments: 1

please how can I get the proofs of the Putnam competition in pdf format?

$$ \\ $$please how can I get the proofs of the Putnam competition in pdf format?

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