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Question Number 113211 Answers: 3 Comments: 0

If α is a root of the equation 4x^2 +2x−1=0 . How do you prove the other root is 4α^3 −3α ?

$$\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{4x}^{\mathrm{2}} +\mathrm{2x}−\mathrm{1}=\mathrm{0}\:.\:\mathrm{How}\:\mathrm{do}\:\mathrm{you} \\ $$$$\mathrm{prove}\:\mathrm{the}\:\mathrm{other}\:\mathrm{root}\:\mathrm{is} \\ $$$$\mathrm{4}\alpha^{\mathrm{3}} −\mathrm{3}\alpha\:?\: \\ $$

Question Number 113203 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/(x^4 +2x^2 +3))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}} \\ $$

Question Number 113200 Answers: 2 Comments: 0

prove that 2tan^(−1) ((1/3))+tan^(−1) ((1/7))=(π/4)

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{2tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{7}}\right)=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 113199 Answers: 1 Comments: 8

Change the following decimal number into binary number: 73.108

$${Change}\:{the}\:{following}\:{decimal} \\ $$$${number}\:{into}\:{binary}\:{number}: \\ $$$$\mathrm{73}.\mathrm{108} \\ $$

Question Number 113198 Answers: 1 Comments: 0

.... calculus.... Evaluate ::: I :=∫_0 ^( 1) (1/( (√(x(x+1)(x+2)(x+3)+1))−3x))dx=??? M.N.july 1970#

$$\:\:\:\:\:\:\:\:\:....\:{calculus}.... \\ $$$$\:\:\:\:\:\:\mathscr{E}{valuate}\:::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{I}\::=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)+\mathrm{1}}−\mathrm{3}{x}}{dx}=??? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathscr{M}.\mathscr{N}.{july}\:\mathrm{1970}# \\ $$$$\:\: \\ $$

Question Number 113196 Answers: 1 Comments: 0

y = sinh^(−1) (sin x) , (dy/dx) =?

$$\:\mathrm{y}\:=\:\mathrm{sinh}\:^{−\mathrm{1}} \left(\mathrm{sin}\:\mathrm{x}\right)\:,\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=? \\ $$

Question Number 113191 Answers: 1 Comments: 0

Solve 2cos^2 (x/2)+3cos(x/2)+1=0

$$\mathrm{Solve}\:\mathrm{2cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{3cos}\frac{{x}}{\mathrm{2}}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 113190 Answers: 1 Comments: 3

∫_0 ^1 ((logx)/(x−1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{logx}}{{x}−\mathrm{1}}{dx} \\ $$

Question Number 113188 Answers: 1 Comments: 0

proporsed by m.n july 1790 ∫_0 ^π ln(1−(1/2)sinx)dx

$${proporsed}\:{by}\:{m}.{n}\:{july}\:\mathrm{1790} \\ $$$$\int_{\mathrm{0}} ^{\pi} \mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}{x}\right){dx} \\ $$

Question Number 113187 Answers: 1 Comments: 0

a,b,c ∈N such that ((a(√3) +b)/(b(√3)+c)) ∈ Q, show that ((a^2 +b^2 +c^2 )/(a+b+c)) ∈ Z

$$\mathrm{a},\mathrm{b},\mathrm{c}\:\in\mathbb{N}\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{a}\sqrt{\mathrm{3}}\:+\mathrm{b}}{\mathrm{b}\sqrt{\mathrm{3}}+\mathrm{c}}\:\in\:\mathrm{Q},\:\mathrm{show} \\ $$$$\mathrm{that}\:\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} }{\mathrm{a}+\mathrm{b}+\mathrm{c}}\:\in\:\mathbb{Z} \\ $$

Question Number 113185 Answers: 1 Comments: 0

Question Number 113160 Answers: 2 Comments: 4

Question Number 113159 Answers: 2 Comments: 0

∫_0 ^1 x^2 log(1−x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {log}\left(\mathrm{1}−{x}\right){dx} \\ $$

Question Number 113154 Answers: 1 Comments: 0

If ⌊x + (√5)⌋ = ⌊x⌋ + ⌊5⌋ then ⌊x⌋ − x would be greater than (a) (√5) − 2 (b) (√5) − 3 (c) (√5) (d) (√5) + 1 (e) (√5) − 1

$$\mathrm{If}\:\:\:\:\:\:\lfloor\mathrm{x}\:\:+\:\:\sqrt{\mathrm{5}}\rfloor\:\:\:=\:\:\:\lfloor\mathrm{x}\rfloor\:\:+\:\:\lfloor\mathrm{5}\rfloor \\ $$$$\mathrm{then}\:\:\:\:\:\lfloor\mathrm{x}\rfloor\:\:−\:\:\mathrm{x}\:\:\:\:\mathrm{would}\:\mathrm{be}\:\mathrm{greater}\:\mathrm{than} \\ $$$$\left(\mathrm{a}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\mathrm{2}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\mathrm{3}\:\:\:\:\:\:\left(\mathrm{c}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:+\:\:\mathrm{1}\:\:\:\:\:\:\:\left(\mathrm{e}\right)\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\mathrm{1} \\ $$

Question Number 113151 Answers: 1 Comments: 4

Prove ((sin2A+cos2A+1)/(sin2A+cos2A−1))=((tan(A+45°))/(tanA)) Hence, prove that tan15°=2−(√3)

$$\mathrm{Prove}\:\frac{\mathrm{sin2A}+\mathrm{cos2A}+\mathrm{1}}{\mathrm{sin2A}+\mathrm{cos2A}−\mathrm{1}}=\frac{\mathrm{tan}\left(\mathrm{A}+\mathrm{45}°\right)}{\mathrm{tanA}} \\ $$$$\mathrm{Hence},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{tan15}°=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$

Question Number 113368 Answers: 1 Comments: 0

There are 4 identical mathematics books, 3 identical physics books, 2 identical chemistry books. in how many ways can you compile the 9 books such that same books are not mutually adjacent.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{4}\:\mathrm{identical}\:\mathrm{mathematics} \\ $$$$\mathrm{books},\:\mathrm{3}\:\mathrm{identical}\:\mathrm{physics}\:\mathrm{books},\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{chemistry}\:\mathrm{books}. \\ $$$$\mathrm{in}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\:\mathrm{can}\:\mathrm{you}\:\mathrm{compile} \\ $$$$\mathrm{the}\:\mathrm{9}\:\mathrm{books}\:\mathrm{such}\:\mathrm{that}\:\mathrm{same}\:\mathrm{books}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{mutually}\:\mathrm{adjacent}. \\ $$

Question Number 113134 Answers: 3 Comments: 2

any one can explain me how to change decimal number to biner number. i′m forgot. example (315)_(10) = (...)_2 thank you

$$\mathrm{any}\:\mathrm{one}\:\mathrm{can}\:\mathrm{explain}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to} \\ $$$$\mathrm{change}\:\mathrm{decimal}\:\mathrm{number}\:\mathrm{to}\: \\ $$$$\mathrm{biner}\:\mathrm{number}.\:\mathrm{i}'\mathrm{m}\:\mathrm{forgot}. \\ $$$$\mathrm{example}\:\left(\mathrm{315}\right)_{\mathrm{10}} \:=\:\left(...\right)_{\mathrm{2}} \\ $$$$\mathrm{thank}\:\mathrm{you} \\ $$

Question Number 113114 Answers: 1 Comments: 1

Question Number 113113 Answers: 0 Comments: 0

suppose that the quantity demanded Q_d =13−6p+2(dp/dt)+(dp^2 /dt^2 ) and quantity supplied Q_s =−3+2p where p is the price find the equilibrium price for market clearance

$${suppose}\:{that}\:{the}\:{quantity}\:{demanded}\:{Q}_{{d}} =\mathrm{13}−\mathrm{6}{p}+\mathrm{2}\frac{{dp}}{{dt}}+\frac{{dp}^{\mathrm{2}} }{{dt}^{\mathrm{2}} }\:{and}\:{quantity}\:{supplied}\:{Q}_{{s}} =−\mathrm{3}+\mathrm{2}{p}\:{where}\:{p}\:{is}\:{the}\:{price}\:{find}\:{the}\:{equilibrium}\:{price}\:{for}\:{market}\:{clearance} \\ $$

Question Number 113111 Answers: 1 Comments: 0

find the area bounded by the curve y^2 =x^3 and the lines x=0 y=1 and y=2

$${find}\:{the}\:{area}\:{bounded}\:{by}\:{the}\:{curve}\:{y}^{\mathrm{2}} ={x}^{\mathrm{3}} \:{and}\:{the}\:{lines}\:{x}=\mathrm{0}\:{y}=\mathrm{1}\:{and}\:{y}=\mathrm{2} \\ $$

Question Number 113110 Answers: 2 Comments: 2

∫_0 ^1 (√(x(x−1)dx))

$$\overset{\mathrm{1}} {\int}_{\mathrm{0}} \sqrt{{x}\left({x}−\mathrm{1}\right){dx}} \\ $$

Question Number 113109 Answers: 1 Comments: 2

Question Number 113107 Answers: 1 Comments: 0

Question Number 113101 Answers: 0 Comments: 2

Prove that GCD ((a,b),b)=(a,b)

$${Prove}\:{that}\:{GCD}\:\left(\left({a},{b}\right),{b}\right)=\left({a},{b}\right) \\ $$

Question Number 113097 Answers: 1 Comments: 0

If 4 women earn as much as 9 boys; 4 men as much as 15 boys; 27 girls as much as 20 women, how many girls will earn the same amount as 24 men?

$$\mathrm{If}\:\mathrm{4}\:\mathrm{women}\:\mathrm{earn}\:\mathrm{as}\:\mathrm{much}\:\mathrm{as}\:\mathrm{9}\:\mathrm{boys};\:\mathrm{4}\:\mathrm{men}\:\mathrm{as}\:\mathrm{much}\:\mathrm{as} \\ $$$$\mathrm{15}\:\mathrm{boys};\:\mathrm{27}\:\mathrm{girls}\:\mathrm{as}\:\mathrm{much}\:\mathrm{as}\:\mathrm{20}\:\mathrm{women},\:\mathrm{how}\:\mathrm{many}\:\mathrm{girls} \\ $$$$\mathrm{will}\:\mathrm{earn}\:\mathrm{the}\:\mathrm{same}\:\mathrm{amount}\:\mathrm{as}\:\mathrm{24}\:\mathrm{men}? \\ $$

Question Number 113092 Answers: 2 Comments: 0

What is the area of a tringle where the sides of triangle are 91 cm, 98 cm, and 105 cm

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{tringle}\:\mathrm{where}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{are}\:\mathrm{91}\:\mathrm{cm},\:\mathrm{98}\:\mathrm{cm},\:\mathrm{and}\:\mathrm{105}\:\mathrm{cm} \\ $$

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