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AllQuestion and Answers: Page 327

Question Number 188521 Answers: 1 Comments: 2

Question Number 188520 Answers: 1 Comments: 0

Question Number 188515 Answers: 1 Comments: 0

in AB^Δ C : a=3 , b=6 , c=7 find the value of : E = (a+b )cos(C) + (b+c)cos(A)+ (a+c )cos(B)=?

$$ \\ $$$$\:\:\:\:\:{in}\:\:{A}\overset{\Delta} {{B}C}\:\::\:\:\:{a}=\mathrm{3}\:\:,\:\:{b}=\mathrm{6}\:\:,\:\:{c}=\mathrm{7} \\ $$$$\:\:\: \\ $$$$\: \\ $$$$\:\:\:\:{find}\:\:{the}\:{value}\:\:{of}\:: \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:{E}\:=\:\left({a}+{b}\:\right){cos}\left({C}\right)\:+\:\left({b}+{c}\right){cos}\left({A}\right)+\:\left({a}+{c}\:\right){cos}\left({B}\right)=?\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 188512 Answers: 0 Comments: 0

x^3 +y^3 +z^3 −3xyz= (x+y+z) (x^2 +y^2 +z^2 −xy−yz−zx) if x+y+z = 0, then x^3 +y^3 +z^3 −3xyz = 0×(x^2 +y^2 +z^2 −xy−yz−zx) = 0

$${x}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{3}} +\boldsymbol{{z}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{xyz}}= \\ $$$$\left(\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\right)\:\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} −\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}\right) \\ $$$$\boldsymbol{{if}}\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\:=\:\mathrm{0},\:{then} \\ $$$${x}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{3}} +\boldsymbol{{z}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{xyz}} \\ $$$$=\:\mathrm{0}×\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} −\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}\right) \\ $$$$=\:\mathrm{0} \\ $$

Question Number 188511 Answers: 0 Comments: 0

evaluate ∫_0 ^π (dx/(a+bcosx )) , a > 0 and deduce that ∫_0 ^π (dx/((a+bcos x)^2 )) = ((πa)/((a^2 −b^2 )^(3/2) )) ; a^2 >b^2 and ∫_0 ^π ((cos x dx)/((a+bcos x)^2 )) = ((−πb)/((a^2 −b^2 )^(3/2) )) ; a^2 >b^2

$$\:\:\:{evaluate} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{{a}+{b}\mathrm{cos}{x}\:}\:\:\:\:\:\:,\:\:\:{a}\:>\:\mathrm{0} \\ $$$$\:\:\:{and}\:{deduce}\:{that} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:\:=\:\:\:\frac{\pi{a}}{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:;\:\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \\ $$$${and}\:\:\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{cos}\:{x}\:{dx}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:\:=\:\frac{−\pi{b}}{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:;\:\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \\ $$

Question Number 188508 Answers: 2 Comments: 0

Question Number 188507 Answers: 0 Comments: 0

Question Number 188551 Answers: 1 Comments: 0

you randomly select a 5 digit number. what′s the probability that this number has exactly 3 different digits?

$${you}\:{randomly}\:{select}\:{a}\:\mathrm{5}\:{digit}\:{number}. \\ $$$${what}'{s}\:{the}\:{probability}\:{that}\:{this}\:{number} \\ $$$${has}\:{exactly}\:\mathrm{3}\:{different}\:{digits}? \\ $$

Question Number 188493 Answers: 0 Comments: 0

Question Number 188492 Answers: 2 Comments: 2

Question Number 188482 Answers: 1 Comments: 0

512x^(1−x^(−3) ) =−1 find volue of Σ_(n=1) ^∞ (x^2 )^n =?

$$\mathrm{512}{x}^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\mathrm{1} \\ $$$${find}\:\:{volue}\:\:{of}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left({x}^{\mathrm{2}} \right)^{{n}} =? \\ $$

Question Number 188475 Answers: 1 Comments: 0

sin ((π/2)(4x+(√x) ))cos (π(x+7(√x)))=1 x=?

$$\:\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}\left(\mathrm{4x}+\sqrt{\mathrm{x}}\:\right)\right)\mathrm{cos}\:\left(\pi\left(\mathrm{x}+\mathrm{7}\sqrt{\mathrm{x}}\right)\right)=\mathrm{1} \\ $$$$\:\mathrm{x}=? \\ $$

Question Number 188470 Answers: 0 Comments: 0

∫_0 ^(π/4) arctan((√((1−tan^2 x)/2)))dx = ?

$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}{arctan}\left(\sqrt{\frac{\mathrm{1}−{tan}^{\mathrm{2}} {x}}{\mathrm{2}}}\right){dx}\:=\:? \\ $$

Question Number 188456 Answers: 0 Comments: 0

Question Number 188455 Answers: 1 Comments: 0

Question Number 188451 Answers: 3 Comments: 0

Question Number 188449 Answers: 1 Comments: 0

calculate lim_( x→ (π/4)) ( tan (x ))^( tan(2x )) = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{calculate} \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\mathrm{lim}_{\:\:{x}\rightarrow\:\frac{\pi}{\mathrm{4}}} \:\:\left(\:\:\mathrm{tan}\:\left({x}\:\right)\right)^{\:\mathrm{tan}\left(\mathrm{2}{x}\:\right)} \:\:=\:?\:\:\: \\ $$$$\:\: \\ $$

Question Number 188444 Answers: 1 Comments: 0

Question Number 188443 Answers: 1 Comments: 0

Question Number 188441 Answers: 1 Comments: 0

Question Number 188442 Answers: 1 Comments: 0

Solve by computer programing (if possible) d<a<b & c < a, b>2c a^2 +b^2 = 5c^2 +2d^2 (a, b, c, d ∈ N) c^2 +d^2 = a^2 .........(i) (2c)^2 +d^2 = b^2 .........(ii) (a, b, c, d) = ?

$$ \\ $$$${Solve}\:{by}\:{computer}\:{programing} \\ $$$$\left({if}\:{possible}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{d}<{a}<{b}\:\&\:{c}\:<\:{a},\:{b}>\mathrm{2}{c} \\ $$$$\cancel{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\:\mathrm{5}{c}^{\mathrm{2}} +\mathrm{2}{d}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\left({a},\:{b},\:{c},\:{d}\:\:\in\:\mathrm{N}\right) \\ $$$${c}^{\mathrm{2}} +{d}^{\mathrm{2}} \:=\:{a}^{\mathrm{2}} \:\:\:\:\:\:\:\:.........\left({i}\right) \\ $$$$\left(\mathrm{2}{c}\right)^{\mathrm{2}} +{d}^{\mathrm{2}} \:=\:{b}^{\mathrm{2}} \:\:.........\left({ii}\right) \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a},\:{b},\:{c},\:{d}\right)\:=\:? \\ $$

Question Number 188430 Answers: 0 Comments: 0

2⌊ x ⌋ − ⌊ −x ⌋ =4 −−−− if x∈Z ⇒ 2x +x = 4 ⇒ x=(4/3) ,impossible if x∉ Z ⇒^(⌊−x⌋=−⌊x⌋−1) 2⌊x⌋+⌊x⌋=3 ⇒ ⌊ x ⌋= 1 ⇒ 1≤ x < 2 ⇒^(x≠1) x∈ (1 , 2) ✓

$$ \\ $$$$\:\:\:\:\mathrm{2}\lfloor\:{x}\:\rfloor\:−\:\lfloor\:−{x}\:\rfloor\:=\mathrm{4} \\ $$$$\:\:\:−−−− \\ $$$$\:\:{if}\:\:{x}\in\mathbb{Z}\:\Rightarrow\:\:\mathrm{2}{x}\:+{x}\:=\:\mathrm{4}\:\Rightarrow\:{x}=\frac{\mathrm{4}}{\mathrm{3}}\:\:,{impossible} \\ $$$$\:\:{if}\:{x}\notin\:\mathbb{Z}\:\overset{\lfloor−{x}\rfloor=−\lfloor{x}\rfloor−\mathrm{1}} {\Rightarrow}\mathrm{2}\lfloor{x}\rfloor+\lfloor{x}\rfloor=\mathrm{3} \\ $$$$\:\:\:\:\:\Rightarrow\:\lfloor\:{x}\:\rfloor=\:\mathrm{1}\:\Rightarrow\:\:\mathrm{1}\leqslant\:{x}\:<\:\mathrm{2}\:\:\:\:\overset{{x}\neq\mathrm{1}} {\Rightarrow}\:{x}\in\:\left(\mathrm{1}\:,\:\mathrm{2}\right)\:\:\:\checkmark \\ $$$$ \\ $$

Question Number 188418 Answers: 2 Comments: 0

Question Number 188417 Answers: 1 Comments: 1

Question Number 188416 Answers: 1 Comments: 0

Question Number 188407 Answers: 3 Comments: 0

xf(x) = f(x + 2) f(2) = 2 f(8) = ?

$${xf}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{2}\right) \\ $$$${f}\left(\mathrm{2}\right)\:=\:\mathrm{2} \\ $$$${f}\left(\mathrm{8}\right)\:=\:?\: \\ $$

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