Question and Answers Forum

AllQuestion and Answers: Page 1040

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

Question Number 107446 Answers: 0 Comments: 3

Question Number 107444 Answers: 0 Comments: 0

Question Number 107445 Answers: 0 Comments: 0

Question Number 107438 Answers: 1 Comments: 0

Question Number 107435 Answers: 0 Comments: 0

Question Number 107434 Answers: 0 Comments: 1

Question Number 107433 Answers: 1 Comments: 0

∫x^x dx=?

$$\int{x}^{{x}} {dx}=? \\ $$

Question Number 107432 Answers: 0 Comments: 0

Question Number 107428 Answers: 1 Comments: 1

The polynomial P(x)=x^3 +ax^2 −4x+b, where a and b are constants. Given that x−2 is a factor of P(x) and that a remainder of 6 is obtained when P(x) is divided by (x+1), find the values of a and b.

$$\mathrm{The}\:\mathrm{polynomial}\:{P}\left({x}\right)=\mathrm{x}^{\mathrm{3}} +\mathrm{ax}^{\mathrm{2}} −\mathrm{4x}+\mathrm{b}, \\ $$$$\mathrm{where}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{constants}.\:\mathrm{Given}\:\mathrm{that} \\ $$$$\mathrm{x}−\mathrm{2}\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:{P}\left({x}\right)\:\mathrm{and}\:\mathrm{that}\:\mathrm{a}\:\mathrm{remainder} \\ $$$$\mathrm{of}\:\mathrm{6}\:\mathrm{is}\:\mathrm{obtained}\:\mathrm{when}\:{P}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left(\mathrm{x}+\mathrm{1}\right),\:\mathrm{find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}. \\ $$

Question Number 107420 Answers: 1 Comments: 1