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Question Number 107512 Answers: 0 Comments: 3

I seen that the function insert separator (right and left)don′t work,ask Titurkara fix this problem!

$$\mathrm{I}\:\mathrm{seen}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{insert}\:\mathrm{separator}\:\left(\mathrm{right}\:\mathrm{and}\:\mathrm{left}\right)\mathrm{don}'\mathrm{t}\: \\ $$$$\mathrm{work},\mathrm{ask}\:\mathrm{Titurkara}\:\mathrm{fix}\:\mathrm{this}\:\mathrm{problem}! \\ $$

Question Number 107516 Answers: 3 Comments: 0

⊚BeMath⊚ lim_(t→0) ((1−(√(cos 2t)))/(sin ((π/2)−t)−cos 2t)) ?

$$\:\:\:\:\circledcirc\mathcal{B}{e}\mathcal{M}{ath}\circledcirc \\ $$$$\:\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\sqrt{\mathrm{cos}\:\mathrm{2}{t}}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−{t}\right)−\mathrm{cos}\:\mathrm{2}{t}}\:? \\ $$

Question Number 107500 Answers: 1 Comments: 0

Given a ∈R−{±1} 1. Show that ∀x∈R 1−2acos(x)+a^2 >0 2. Show that; Π_(k=1) ^n (1−2acos(((2kπ)/n))+a^2 )=Π_(k=1) ^n (a−e^(2ikπ/n) )(a−e^(−2ikπ/n) ) 3. Deduce that; Π_(k=1) ^n (1−2acos(((2kπ)/n))+a^2 )=(a^n −1)^2 4. Using Reimann′s sum, calculate I=∫_0 ^(2π) ln(1−2acos(x)+a^2 )dx

$$\mathrm{Given}\:\mathrm{a}\:\in\mathbb{R}−\left\{\pm\mathrm{1}\right\} \\ $$$$\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{x}\in\mathbb{R}\:\mathrm{1}−\mathrm{2acos}\left(\mathrm{x}\right)+\mathrm{a}^{\mathrm{2}} >\mathrm{0} \\ $$$$\mathrm{2}.\:\mathrm{Show}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{1}−\mathrm{2acos}\left(\frac{\mathrm{2k}\pi}{\mathrm{n}}\right)+\mathrm{a}^{\mathrm{2}} \right)=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{a}−\mathrm{e}^{\mathrm{2ik}\pi/\mathrm{n}} \right)\left(\mathrm{a}−\mathrm{e}^{−\mathrm{2ik}\pi/\mathrm{n}} \right) \\ $$$$\mathrm{3}.\:\mathrm{Deduce}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{1}−\mathrm{2acos}\left(\frac{\mathrm{2k}\pi}{\mathrm{n}}\right)+\mathrm{a}^{\mathrm{2}} \right)=\left(\mathrm{a}^{\mathrm{n}} −\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\mathrm{4}.\:\:\mathrm{Using}\:\mathrm{Reimann}'\mathrm{s}\:\mathrm{sum},\:\mathrm{calculate} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{ln}\left(\mathrm{1}−\mathrm{2acos}\left(\mathrm{x}\right)+\mathrm{a}^{\mathrm{2}} \right)\mathrm{dx} \\ $$

Question Number 107498 Answers: 1 Comments: 0

Show that Π_(k=1) ^n (a−e^((2ikπ)/n) )(a−e^(−((2ikπ)/n)) )=(a^n −1)^2

$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{a}−\mathrm{e}^{\frac{\mathrm{2}{i}\mathrm{k}\pi}{\mathrm{n}}} \right)\left(\mathrm{a}−\mathrm{e}^{−\frac{\mathrm{2}{i}\mathrm{k}\pi}{\mathrm{n}}} \right)=\left(\mathrm{a}^{\mathrm{n}} −\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 107496 Answers: 1 Comments: 0

Question Number 107487 Answers: 2 Comments: 0

Question Number 107486 Answers: 3 Comments: 1

∦BeMath∦ (2/5)+(5/(25))+(8/(125))+((11)/(625))+((14)/(3125))+... = ?

$$\:\:\:\:\:\:\nparallel\mathcal{B}{e}\mathcal{M}{ath}\nparallel \\ $$$$\frac{\mathrm{2}}{\mathrm{5}}+\frac{\mathrm{5}}{\mathrm{25}}+\frac{\mathrm{8}}{\mathrm{125}}+\frac{\mathrm{11}}{\mathrm{625}}+\frac{\mathrm{14}}{\mathrm{3125}}+...\:=\:? \\ $$

Question Number 107484 Answers: 2 Comments: 0

∦BeMath∦ (√(x+(√(x+(√(x+(√(x+...)))))))) = (√(4(√(4(√(4(√(4...)))))))) x=?

$$\:\:\:\:\:\nparallel\mathcal{B}{e}\mathcal{M}{ath}\nparallel \\ $$$$\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+...}}}}\:=\:\sqrt{\mathrm{4}\sqrt{\mathrm{4}\sqrt{\mathrm{4}\sqrt{\mathrm{4}...}}}} \\ $$$${x}=?\: \\ $$

Question Number 107483 Answers: 2 Comments: 0

⋇JS⋇ ∣1+(1/x) ∣−∣x−3∣ > 2 find solution set. (A) 3−(√(10)) < x < 2−(√3) ; x≠0 (B) 3−(√(10)) < x < 3+(√(10)) ; x≠0 (C) 3−(√(10)) < x < 2+(√(10)) ; x≠0 (D) 2+(√(10)) < x < 3+(√(10)) ; x≠0 (E) none of these

$$\:\:\:\:\:\:\:\:\divideontimes\mathcal{JS}\divideontimes \\ $$$$\:\:\:\:\:\mid\mathrm{1}+\frac{\mathrm{1}}{{x}}\:\mid−\mid{x}−\mathrm{3}\mid\:>\:\mathrm{2}\: \\ $$$${find}\:{solution}\:{set}. \\ $$$$\left({A}\right)\:\mathrm{3}−\sqrt{\mathrm{10}}\:<\:{x}\:<\:\mathrm{2}−\sqrt{\mathrm{3}}\:;\:{x}\neq\mathrm{0} \\ $$$$\left({B}\right)\:\mathrm{3}−\sqrt{\mathrm{10}}\:<\:{x}\:<\:\mathrm{3}+\sqrt{\mathrm{10}}\:;\:{x}\neq\mathrm{0} \\ $$$$\left({C}\right)\:\mathrm{3}−\sqrt{\mathrm{10}}\:<\:{x}\:<\:\mathrm{2}+\sqrt{\mathrm{10}}\:;\:{x}\neq\mathrm{0} \\ $$$$\left({D}\right)\:\mathrm{2}+\sqrt{\mathrm{10}}\:<\:{x}\:<\:\mathrm{3}+\sqrt{\mathrm{10}}\:;\:{x}\neq\mathrm{0} \\ $$$$\left({E}\right)\:{none}\:{of}\:{these}\: \\ $$

Question Number 107481 Answers: 1 Comments: 0

If n∈Z^+ , show that Σ_(k=1) ^n ln^2 (1+(1/k))<1

$$\mathrm{If}\:\mathrm{n}\in\mathbb{Z}^{+} ,\:\mathrm{show}\:\mathrm{that}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{k}}\right)<\mathrm{1} \\ $$

Question Number 107471 Answers: 3 Comments: 0

sec x − cosec x=(√(35)) tan x+cot x=?

$$\mathrm{sec}\:\mathrm{x}\:−\:\mathrm{cosec}\:\mathrm{x}=\sqrt{\mathrm{35}} \\ $$$$\mathrm{tan}\:\mathrm{x}+\mathrm{cot}\:\mathrm{x}=? \\ $$

Question Number 107470 Answers: 3 Comments: 0

↺BeMath↻ (1)(1+tan 3°)(1+tan 4°)(1+tan 41°)(1+tan 42°)=? (2)f(x)=g(h(x)); h(x)=2x^2 −3x. If f ′(−1)=14 then g ′(5)=?

$$\:\:\:\:\:\:\:\:\circlearrowleft\mathcal{B}{e}\mathcal{M}{ath}\circlearrowright \\ $$$$\left(\mathrm{1}\right)\left(\mathrm{1}+\mathrm{tan}\:\mathrm{3}°\right)\left(\mathrm{1}+\mathrm{tan}\:\mathrm{4}°\right)\left(\mathrm{1}+\mathrm{tan}\:\mathrm{41}°\right)\left(\mathrm{1}+\mathrm{tan}\:\mathrm{42}°\right)=? \\ $$$$\left(\mathrm{2}\right){f}\left({x}\right)={g}\left({h}\left({x}\right)\right);\:{h}\left({x}\right)=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}. \\ $$$${If}\:{f}\:'\left(−\mathrm{1}\right)=\mathrm{14}\:{then}\:{g}\:'\left(\mathrm{5}\right)=? \\ $$

Question Number 107461 Answers: 2 Comments: 1

Question Number 107454 Answers: 2 Comments: 0

Given the function f(x) = ((x + 3)/(x−2)) and g(x) = (1/2)xe^x (1) Find the centre of symmetry of f. (2) Define the monotony of g and if possible draw a variation table for g(x). (3) Sketch the function g(x) (4) determine if f and g intersect.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:=\:\frac{{x}\:+\:\mathrm{3}}{{x}−\mathrm{2}}\:\mathrm{and}\:\mathrm{g}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}{xe}^{{x}} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{symmetry}\:\mathrm{of}\:{f}. \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Define}\:\mathrm{the}\:\mathrm{monotony}\:\mathrm{of}\:\mathrm{g}\:\mathrm{and}\:\mathrm{if}\:\mathrm{possible}\:\mathrm{draw}\:\mathrm{a}\:\mathrm{variation} \\ $$$$\mathrm{table}\:\mathrm{for}\:\mathrm{g}\left({x}\right). \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Sketch}\:\mathrm{the}\:\mathrm{function}\:\mathrm{g}\left({x}\right) \\ $$$$\left(\mathrm{4}\right)\:\mathrm{determine}\:\mathrm{if}\:{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{intersect}. \\ $$

Question Number 107453 Answers: 0 Comments: 0

A man takes 1Hour15mn to travel 4.95km. Each 5 min he travels 10km in minus from the previous distance travelled in 5 min. We admit that this man start travelling at 6H00 am. What distance could he travel from 6H10 am to 6H15 am?

$${A}\:{man}\:{takes}\:\mathrm{1}{Hour}\mathrm{15}{mn}\:{to} \\ $$$${travel}\:\mathrm{4}.\mathrm{95}{km}.\:{Each}\:\mathrm{5}\:{min}\:{he} \\ $$$${travels}\:\mathrm{10}{km}\:{in}\:{minus}\:{from}\:{the} \\ $$$${previous}\:{distance}\:{travelled}\:{in}\:\mathrm{5} \\ $$$${min}.\:{We}\:{admit}\:{that}\:{this}\:{man}\: \\ $$$${start}\:{travelling}\:{at}\:\mathrm{6}{H}\mathrm{00}\:{am}. \\ $$$${What}\:{distance}\:{could}\:{he}\:{travel}\:{from} \\ $$$$\mathrm{6}{H}\mathrm{10}\:{am}\:{to}\:\mathrm{6}{H}\mathrm{15}\:{am}? \\ $$

Question Number 107452 Answers: 1 Comments: 0

Given a function f which is periodic of period 2 defined by f(x) = { ((3x^2 −4 , if 0 ≤ x < 3)),((x−3, if 3 ≤ x < 6)) :} (1) State in a similar manner f ′(x). (2) Check if f is continuous. (3) find f (7) and skech the curve y = f(x).

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\:{f}\:\mathrm{which}\:\mathrm{is}\:\mathrm{periodic}\:\mathrm{of}\:\mathrm{period}\:\mathrm{2}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:{f}\left({x}\right)\:=\:\begin{cases}{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}\:,\:\mathrm{if}\:\mathrm{0}\:\leqslant\:{x}\:<\:\mathrm{3}}\\{{x}−\mathrm{3},\:\mathrm{if}\:\:\mathrm{3}\:\leqslant\:{x}\:<\:\mathrm{6}}\end{cases} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{State}\:\mathrm{in}\:\mathrm{a}\:\mathrm{similar}\:\mathrm{manner}\:{f}\:'\left({x}\right). \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Check}\:\mathrm{if}\:{f}\:\mathrm{is}\:\mathrm{continuous}. \\ $$$$\left(\mathrm{3}\right)\:\mathrm{find}\:{f}\:\left(\mathrm{7}\right)\:\mathrm{and}\:\mathrm{skech}\:\mathrm{the}\:\mathrm{curve}\:{y}\:=\:{f}\left({x}\right). \\ $$

Question Number 107451 Answers: 1 Comments: 1

How many words can you form using the letters in UNUSUALLY such that no same letters are next to each other? [Answer: 10200]

$${How}\:{many}\:{words}\:{can}\:{you}\:{form}\:{using} \\ $$$${the}\:{letters}\:\:{in}\:\boldsymbol{{UNUSUALLY}} \\ $$$${such}\:{that}\:{no}\:{same}\:{letters}\:{are}\:\:{next} \\ $$$${to}\:{each}\:{other}? \\ $$$$ \\ $$$$\left[{Answer}:\:\mathrm{10200}\right] \\ $$

Question Number 107447 Answers: 0 Comments: 2