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AllQuestion and Answers: Page 170

Question Number 206425    Answers: 1   Comments: 0

If cos𝛂 = (3/5) (0<𝛂<(𝛑/2)) Find: ((tan^2 (45Β° + (𝛂/2)))/3) = ?

$$\mathrm{If}\:\:\:\mathrm{cos}\boldsymbol{\alpha}\:=\:\frac{\mathrm{3}}{\mathrm{5}}\:\:\:\left(\mathrm{0}<\boldsymbol{\alpha}<\frac{\boldsymbol{\pi}}{\mathrm{2}}\right) \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{tan}^{\mathrm{2}} \:\left(\mathrm{45}Β°\:+\:\frac{\boldsymbol{\alpha}}{\mathrm{2}}\right)}{\mathrm{3}}\:=\:? \\ $$

Question Number 206421    Answers: 1   Comments: 0

If tanpΞΈ = ptanΞΈ then prove that ((sin^2 pΞΈ)/(sin^2 ΞΈ)) = (p^2 /(1 + (p^2 βˆ’ 1)sin^2 ΞΈ)) .

$$\mathrm{If}\:\mathrm{tan}{p}\theta\:=\:{p}\mathrm{tan}\theta\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{sin}^{\mathrm{2}} {p}\theta}{\mathrm{sin}^{\mathrm{2}} \theta}\:=\:\frac{{p}^{\mathrm{2}} }{\mathrm{1}\:+\:\left({p}^{\mathrm{2}} \:βˆ’\:\mathrm{1}\right)\mathrm{sin}^{\mathrm{2}} \theta}\:.\: \\ $$

Question Number 206399    Answers: 2   Comments: 1

Question Number 206396    Answers: 3   Comments: 0

Question Number 206394    Answers: 0   Comments: 1

Question Number 206393    Answers: 1   Comments: 0

find S=1+Ξ£_β„“ (((βˆ’)^β„“ )/β„“)((1/β„“)βˆ’(1/(β„“+1))) , β„“βˆˆ[1,∞) 1+Ξ£_β„“ (((βˆ’)^β„“ )/β„“)((1/β„“)βˆ’(1/(β„“+1))) 1βˆ’(1βˆ’(1/2))+(1/2)((1/2)βˆ’(1/3))βˆ’(1/3)((1/3)βˆ’(1/4))+(1/4)((1/4)βˆ’(1/5))βˆ’......

$$\mathrm{find}\:\mathrm{S}=\mathrm{1}+\underset{\ell} {\sum}\:\frac{\left(βˆ’\right)^{\ell} }{\ell}\left(\frac{\mathrm{1}}{\ell}βˆ’\frac{\mathrm{1}}{\ell+\mathrm{1}}\right)\:,\:\ell\in\left[\mathrm{1},\infty\right) \\ $$$$\mathrm{1}+\underset{\ell} {\sum}\:\frac{\left(βˆ’\right)^{\ell} }{\ell}\left(\frac{\mathrm{1}}{\ell}βˆ’\frac{\mathrm{1}}{\ell+\mathrm{1}}\right) \\ $$$$\mathrm{1}βˆ’\left(\mathrm{1}βˆ’\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}βˆ’\frac{\mathrm{1}}{\mathrm{3}}\right)βˆ’\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{3}}βˆ’\frac{\mathrm{1}}{\mathrm{4}}\right)+\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{4}}βˆ’\frac{\mathrm{1}}{\mathrm{5}}\right)βˆ’...... \\ $$

Question Number 206391    Answers: 2   Comments: 0

Find: ∫_(βˆ’3) ^( βˆ’2) (∣x∣ + ∣x βˆ’ 4∣) dx = ?

$$\mathrm{Find}: \\ $$$$\int_{βˆ’\mathrm{3}} ^{\:βˆ’\mathrm{2}} \:\left(\mid\mathrm{x}\mid\:+\:\mid\mathrm{x}\:βˆ’\:\mathrm{4}\mid\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 206365    Answers: 2   Comments: 4

Number series: a_3 = 2a + b βˆ’ 6 a_9 = a + b + 5 a_(15) = 3a + b βˆ’ 7 Find: a = ?

$$\mathrm{Number}\:\mathrm{series}: \\ $$$$\mathrm{a}_{\mathrm{3}} \:=\:\mathrm{2a}\:+\:\mathrm{b}\:βˆ’\:\mathrm{6} \\ $$$$\mathrm{a}_{\mathrm{9}} \:=\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{5} \\ $$$$\mathrm{a}_{\mathrm{15}} \:=\:\mathrm{3a}\:+\:\mathrm{b}\:βˆ’\:\mathrm{7} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}\:=\:? \\ $$

Question Number 206364    Answers: 2   Comments: 0

Question Number 206363    Answers: 0   Comments: 2

If 0<a<1 Compare: (1/(aβˆ’1)) , (a/(aβˆ’1)) , (1/(1βˆ’a)) , (a/(1βˆ’a)) , (a/(2a))

$$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}<\mathrm{1} \\ $$$$\mathrm{Compare}: \\ $$$$\frac{\mathrm{1}}{\mathrm{a}βˆ’\mathrm{1}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{a}βˆ’\mathrm{1}}\:\:,\:\:\frac{\mathrm{1}}{\mathrm{1}βˆ’\mathrm{a}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{1}βˆ’\mathrm{a}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{2a}} \\ $$

Question Number 206357    Answers: 2   Comments: 0

Find: 1 + cos444Β° βˆ’ cos84Β° + cot45Β° = ?

$$\mathrm{Find}: \\ $$$$\mathrm{1}\:+\:\mathrm{cos444}Β°\:βˆ’\:\mathrm{cos84}Β°\:+\:\mathrm{cot45}Β°\:=\:? \\ $$

Question Number 206355    Answers: 2   Comments: 0

if the sum of three positive real numbers is equal to their product, prove that at least one of the numbers is larger than 1.7.

$${if}\:{the}\:{sum}\:{of}\:{three}\:{positive}\:{real}\: \\ $$$${numbers}\:{is}\:{equal}\:{to}\:{their}\:{product}, \\ $$$${prove}\:{that}\:{at}\:{least}\:{one}\:{of}\:{the}\: \\ $$$${numbers}\:{is}\:{larger}\:{than}\:\mathrm{1}.\mathrm{7}. \\ $$

Question Number 206353    Answers: 0   Comments: 6

Question Number 206351    Answers: 0   Comments: 1

expression of the sequence (a_n ) defined by { ((a_0 >0 , a_1 >0)),((a_(n+2) =((2(βˆ’1)^n )/(n+2))βˆ’((2(βˆ’1)^n (2n+3))/(n+2))a_(n+1) +((n+1)/(n+2))a_n )) :}

$${expression}\:{of}\:{the}\:{sequence}\:\left({a}_{{n}} \right)\:{defined} \\ $$$${by}\: \\ $$$$\begin{cases}{{a}_{\mathrm{0}} >\mathrm{0}\:,\:{a}_{\mathrm{1}} >\mathrm{0}}\\{{a}_{{n}+\mathrm{2}} =\frac{\mathrm{2}\left(βˆ’\mathrm{1}\right)^{{n}} }{{n}+\mathrm{2}}βˆ’\frac{\mathrm{2}\left(βˆ’\mathrm{1}\right)^{{n}} \left(\mathrm{2}{n}+\mathrm{3}\right)}{{n}+\mathrm{2}}{a}_{{n}+\mathrm{1}} +\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}{a}_{{n}} }\end{cases} \\ $$

Question Number 206340    Answers: 2   Comments: 0

∫_0 ^( 1) (( ln(1βˆ’x )ln(1+x ))/x)dx = Ξ£_(n=1) ^∞ Ξ©_n find : Ξ£_(n=1) ^∞ n Ξ©_n = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\:{ln}\left(\mathrm{1}βˆ’{x}\:\right){ln}\left(\mathrm{1}+{x}\:\right)}{{x}}{dx}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\Omega_{{n}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{find}\::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:{n}\:\Omega_{{n}} \:=\:? \\ $$

Question Number 206339    Answers: 1   Comments: 0

E βŠ† Y βŠ† ( X , d )∣_(metric space) prove E is open in Y if and only if βˆƒ G (open set ) in X such that E = G ∩ Y .... (mathematical analysis (I))

$$ \\ $$$$\:\:\:\:\:{E}\:\subseteq\:{Y}\:\subseteq\:\left(\:{X}\:,\:{d}\:\right)\mid_{{metric}\:{space}} \\ $$$$\:\:\:\:{prove}\:\:{E}\:{is}\:{open}\:{in}\:{Y}\:{if}\:{and}\:\:{only}\:{if} \\ $$$$\:\:\:\:\:\:\:\exists\:{G}\:\left({open}\:{set}\:\right)\:{in}\:{X}\:\:{such}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:{E}\:=\:{G}\:\cap\:{Y}\:\:\:....\:\left({mathematical}\:{analysis}\:\left({I}\right)\right) \\ $$

Question Number 206338    Answers: 1   Comments: 0

Question Number 206367    Answers: 4   Comments: 2

Find: (1/6) + (1/(24)) + (1/(60)) + ... + ... (1/(720)) = ?

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{6}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{24}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{60}}\:\:+\:...\:+\:...\:\:\frac{\mathrm{1}}{\mathrm{720}}\:=\:? \\ $$

Question Number 206332    Answers: 1   Comments: 0

If log(a + b + c) = loga + logb + logc then prove that log(((2a)/(1 βˆ’ a^2 )) + ((2b)/(1 βˆ’ b^2 )) + ((2c)/(1 βˆ’ c^2 ))) = log(((2a)/(1 βˆ’ a^2 ))) + log(((2b)/(1 βˆ’ b^2 ))) + log(((2c)/(1 βˆ’ c^2 ))).

$$\mathrm{If}\:\mathrm{log}\left({a}\:+\:{b}\:+\:{c}\right)\:=\:\mathrm{log}{a}\:+\:\mathrm{log}{b}\:+\:\mathrm{log}{c}\: \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}\:βˆ’\:{a}^{\mathrm{2}} }\:+\:\frac{\mathrm{2}{b}}{\mathrm{1}\:βˆ’\:{b}^{\mathrm{2}} }\:+\:\frac{\mathrm{2}{c}}{\mathrm{1}\:βˆ’\:{c}^{\mathrm{2}} }\right)\:=\: \\ $$$$\mathrm{log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}\:βˆ’\:{a}^{\mathrm{2}} }\right)\:+\:\mathrm{log}\left(\frac{\mathrm{2}{b}}{\mathrm{1}\:βˆ’\:{b}^{\mathrm{2}} }\right)\:+\:\mathrm{log}\left(\frac{\mathrm{2}{c}}{\mathrm{1}\:βˆ’\:{c}^{\mathrm{2}} }\right). \\ $$

Question Number 206328    Answers: 2   Comments: 0

f(x)= ⌊ (( 1+ x +x^2 )/(x+ x^( 2) )) βŒ‹ is given. β‡’ { (( D_f = ? (domain ))),(( R_( f) = ?( range))) :}

$$ \\ $$$$\:\:\:\:\:\:\:{f}\left({x}\right)=\:\lfloor\:\frac{\:\mathrm{1}+\:{x}\:+{x}^{\mathrm{2}} }{{x}+\:{x}^{\:\mathrm{2}} }\:\rfloor\:{is}\:{given}. \\ $$$$\:\:\:\:\:\:\:\Rightarrow\begin{cases}{\:\:{D}_{{f}} \:=\:?\:\left({domain}\:\right)}\\{\:\:\:{R}_{\:{f}} \:=\:?\left(\:{range}\right)}\end{cases} \\ $$$$ \\ $$

Question Number 206323    Answers: 0   Comments: 0

Question Number 206322    Answers: 1   Comments: 1

If abc = 1 then prove that (1/(1 + a + b^(βˆ’1) )) + (1/(1 + b + c^(βˆ’1) )) + (1/(1 + c + a^(βˆ’1) )) = 1

$$\mathrm{If}\:{abc}\:=\:\mathrm{1}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{a}\:+\:{b}^{βˆ’\mathrm{1}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{b}\:+\:{c}^{βˆ’\mathrm{1}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{c}\:+\:{a}^{βˆ’\mathrm{1}} }\:=\:\mathrm{1} \\ $$

Question Number 206321    Answers: 2   Comments: 0

Question Number 206319    Answers: 0   Comments: 0

Question Number 206318    Answers: 0   Comments: 0

∫_0 ^1 ∫_0 ^1 ((xln(1+y)ln(1βˆ’x))/(1+x^2 y))dxdy

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{xln}\left(\mathrm{1}+{y}\right){ln}\left(\mathrm{1}βˆ’{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} {y}}{dxdy} \\ $$

Question Number 206362    Answers: 2   Comments: 0

sin(Ο€/7) Γ— sin((2Ο€)/7) Γ— sin((3Ο€)/7) = ?

$$\mathrm{sin}\frac{\pi}{\mathrm{7}}\:Γ—\:\mathrm{sin}\frac{\mathrm{2}\pi}{\mathrm{7}}\:Γ—\:\mathrm{sin}\frac{\mathrm{3}\pi}{\mathrm{7}}\:=\:? \\ $$

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