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Question Number 100792 Answers: 1 Comments: 0

Question Number 100789 Answers: 2 Comments: 0

Question Number 100785 Answers: 1 Comments: 0

Question Number 100771 Answers: 2 Comments: 1

sho that (0,(1/2)) is a point of symetry for the curve f(x) = x +(1/(1−e^x )) Please make a reference to a book i can understand centre of symmetry of rational functions and functions like this

$$\:\mathrm{sho}\:\mathrm{that}\:\:\left(\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{of}\:\mathrm{symetry}\:\mathrm{for}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\:{f}\left({x}\right)\:=\:{x}\:+\frac{\mathrm{1}}{\mathrm{1}−{e}^{{x}} } \\ $$$$\mathrm{Please}\:\mathrm{make}\:\mathrm{a}\:\mathrm{reference}\:\mathrm{to}\:\mathrm{a}\:\mathrm{book}\:\mathrm{i}\:\mathrm{can}\:\mathrm{understand} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{symmetry}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{functions}\:\mathrm{and}\:\mathrm{functions} \\ $$$$\mathrm{like}\:\mathrm{this} \\ $$

Question Number 100769 Answers: 0 Comments: 0

Consider the sequences (u_n ) and (v_n ) defined by { ((u_0 = 1)),((u_(n+1) = ((2u_n v_n )/(u_n + v_n )))) :} and { ((v_0 = 2)),((v_(n+1) = ((u_n + v_n )/2))) :} ∀ n∈ N (1) Show that (u_n ) and (v_n ) are strictly positive also Show that (u_n ) and (v_n ) are of opposite sense of variation. (2) let w_n = v_n −u_n show that 0 ≤ w_(n+1) ≤ (1/2)w_n (3) Prove by induction that 0 ≤ w_n ≤ (1/2^n )

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequences}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right)\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\begin{cases}{{u}_{\mathrm{0}} \:=\:\mathrm{1}}\\{{u}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}{u}_{{n}} {v}_{{n}} }{{u}_{{n}} \:+\:{v}_{{n}} }}\end{cases}\:\mathrm{and}\:\begin{cases}{{v}_{\mathrm{0}} \:=\:\mathrm{2}}\\{{v}_{{n}+\mathrm{1}} \:=\:\frac{{u}_{{n}} \:+\:{v}_{{n}} }{\mathrm{2}}}\end{cases}\:\:\forall\:{n}\in\:\mathbb{N} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Show}\:\mathrm{that}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right)\:\mathrm{are}\:\mathrm{strictly}\:\mathrm{positive}\:\mathrm{also} \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \:\right)\:\mathrm{are}\:\mathrm{of}\:\mathrm{opposite}\:\mathrm{sense}\:\mathrm{of}\:\mathrm{variation}. \\ $$$$\left(\mathrm{2}\right)\:\mathrm{let}\:{w}_{{n}} \:=\:{v}_{{n}} −{u}_{{n}} \:\:\mathrm{show}\:\mathrm{that}\:\:\mathrm{0}\:\leqslant\:{w}_{{n}+\mathrm{1}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}{w}_{{n}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Prove}\:\mathrm{by}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{0}\:\leqslant\:{w}_{{n}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} } \\ $$

Question Number 100767 Answers: 2 Comments: 0

sketch x^2 = y^3 and x^2 + y^2 = 16 and hence solve (x^2 −y^3 )(x^2 + y^2 −16) ≥0

$$\:\mathrm{sketch}\:\:{x}^{\mathrm{2}} \:=\:{y}^{\mathrm{3}} \:\mathrm{and}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:\mathrm{16} \\ $$$$\mathrm{and}\:\mathrm{hence}\:\mathrm{solve}\:\left({x}^{\mathrm{2}} −{y}^{\mathrm{3}} \right)\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} −\mathrm{16}\right)\:\geqslant\mathrm{0} \\ $$

Question Number 100766 Answers: 4 Comments: 0

Question Number 100746 Answers: 1 Comments: 0

∫_(−∞) ^∞ ((cos3x)/((1+x^2 )^2 ))dx

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

Question Number 100744 Answers: 0 Comments: 17

Happy tau day to all !!! 28 june

$$\mathrm{Happy}\:\boldsymbol{\mathrm{tau}}\:\boldsymbol{\mathrm{day}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{all}}\:!!! \\ $$$$ \\ $$$$\mathrm{28}\:{june}\: \\ $$

Question Number 100743 Answers: 1 Comments: 0

(√(((2+5)/(35))+6))+30×((15+90)/(255))

$$\sqrt{\frac{\mathrm{2}+\mathrm{5}}{\mathrm{35}}+\mathrm{6}}+\mathrm{30}×\frac{\mathrm{15}+\mathrm{90}}{\mathrm{255}} \\ $$

Question Number 100733 Answers: 0 Comments: 6

A positive integer such as 4334 is a palindrome if it reads the same forwards or backwards. What is the only prime palindrome with an even number of digits?

$$\mathrm{A}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{as}\:\mathrm{4334}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{palindrome}\:\mathrm{if}\:\mathrm{it}\:\mathrm{reads}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{forwards}\:\mathrm{or}\:\mathrm{backwards}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{only}\:\mathrm{prime}\:\mathrm{palindrome}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{even}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}?\: \\ $$

Question Number 100732 Answers: 1 Comments: 5

(1) If ((x+yi)/(1+i)) = (7/(7+i)) where x and y are real , what is the value of x+y (2)What are all values of x which satisfy x^2 −cos x+1 = 0 (3)What are all values of x between 0^o and 360^o which satisfy (5+2(√6))^(sin x) + (5−2(√6))^(sin x) = 2(√3)

$$\left(\mathrm{1}\right)\:\mathrm{If}\:\frac{{x}+{yi}}{\mathrm{1}+{i}}\:=\:\frac{\mathrm{7}}{\mathrm{7}+{i}}\:\mathrm{where}\:{x}\:\mathrm{and}\:\mathrm{y}\: \\ $$$$\mathrm{are}\:\mathrm{real}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}+{y}\: \\ $$$$\left(\mathrm{2}\right)\mathrm{What}\:\mathrm{are}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{which} \\ $$$$\mathrm{satisfy}\:{x}^{\mathrm{2}} −\mathrm{cos}\:{x}+\mathrm{1}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{3}\right){W}\mathrm{hat}\:\mathrm{are}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{between}\: \\ $$$$\mathrm{0}^{\mathrm{o}} \:\mathrm{and}\:\mathrm{360}^{\mathrm{o}} \:\mathrm{which}\:\mathrm{satisfy}\: \\ $$$$\left(\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{sin}\:\mathrm{x}} \:+\:\left(\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{sin}\:\mathrm{x}} \:=\:\mathrm{2}\sqrt{\mathrm{3}} \\ $$

Question Number 100730 Answers: 0 Comments: 1

((2z^2 )/(z^2 +∣z+1∣)) < 1

$$\frac{\mathrm{2}{z}^{\mathrm{2}} }{{z}^{\mathrm{2}} +\mid{z}+\mathrm{1}\mid}\:<\:\mathrm{1}\: \\ $$

Question Number 100720 Answers: 1 Comments: 3

Question Number 100703 Answers: 0 Comments: 2

Question Number 100696 Answers: 0 Comments: 4

1. 25≠66 = True or False? 2. 44=44 =True or false? 3. 39>169=True or False? 4. 15<61 = True or false? Make sure you have to answer correctly

$$\mathrm{1}.\:\:\:\:\mathrm{25}\neq\mathrm{66}\:=\:\mathrm{True}\:\mathrm{or}\:\mathrm{False}? \\ $$$$\mathrm{2}.\:\:\:\:\mathrm{44}=\mathrm{44}\:=\mathrm{True}\:\mathrm{or}\:\mathrm{false}? \\ $$$$\mathrm{3}.\:\:\:\:\:\mathrm{39}>\mathrm{169}=\mathrm{True}\:\mathrm{or}\:\mathrm{False}? \\ $$$$\mathrm{4}.\:\:\:\:\mathrm{15}<\mathrm{61}\:=\:\mathrm{True}\:\mathrm{or}\:\mathrm{false}? \\ $$$$\: \\ $$$$\mathrm{Make}\:\mathrm{sure}\:\mathrm{you}\:\mathrm{have}\:\mathrm{to} \\ $$$$\mathrm{answer}\:\mathrm{correctly} \\ $$

Question Number 100695 Answers: 2 Comments: 3

Question Number 100908 Answers: 2 Comments: 2

what the value of angle formed by a long needle and short needle on analog clock that shows at 15.50 ? (A) 175^o (B) 174^o (C) 173^o (D) 172^o (E) 170^o

$$\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{angle} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{a}\:\mathrm{long}\:\mathrm{needle}\:\mathrm{and}\: \\ $$$$\mathrm{short}\:\mathrm{needle}\:\mathrm{on}\:\mathrm{analog}\:\mathrm{clock}\: \\ $$$$\mathrm{that}\:\mathrm{shows}\:\mathrm{at}\:\mathrm{15}.\mathrm{50}\:? \\ $$$$\left(\mathrm{A}\right)\:\mathrm{175}^{\mathrm{o}} \:\:\:\left(\mathrm{B}\right)\:\mathrm{174}^{\mathrm{o}} \:\:\:\left(\mathrm{C}\right)\:\mathrm{173}^{\mathrm{o}} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{172}^{\mathrm{o}} \:\:\:\:\left(\mathrm{E}\right)\:\mathrm{170}^{\mathrm{o}} \\ $$

Question Number 100690 Answers: 0 Comments: 1

(√(((15929)/(30.25569))+15^5 ))+(√(30.509))

$$\sqrt{\frac{\mathrm{15929}}{\mathrm{30}.\mathrm{25569}}+\mathrm{15}^{\mathrm{5}} }+\sqrt{\mathrm{30}.\mathrm{509}} \\ $$

Question Number 100684 Answers: 1 Comments: 0

(x^2 +xy) (dy/dx) = xy + y^2

$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{xy}\right)\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{xy}\:+\:\mathrm{y}^{\mathrm{2}} \\ $$

Question Number 100677 Answers: 0 Comments: 0

for m,n positive integers m > n prove that lcd(m,n) + lcd(m+1,n+1) > ((2mn)/(√(m−n)))

$$\mathrm{for}\:\mathrm{m},\mathrm{n}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{m}\:>\:\mathrm{n}\: \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{lcd}\left(\mathrm{m},\mathrm{n}\right)\:+\:\mathrm{lcd}\left(\mathrm{m}+\mathrm{1},\mathrm{n}+\mathrm{1}\right)\:>\:\frac{\mathrm{2mn}}{\sqrt{\mathrm{m}−\mathrm{n}}} \\ $$

Question Number 100675 Answers: 1 Comments: 1

If log _(2x) ((1/(18))) = log _(18) ((1/(3y))) = log _(3y) ((1/(2x))) find 3x−2y