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Question Number 13806    Answers: 0   Comments: 1

Prove that for −(π/2)<x<(π/2) , (1/1^3 )cos x−(1/3^3 )cos 3x+(1/5^3 )cos 5x−....to infinity =(π/8)((π^2 /4)−x^2 ) .

$${Prove}\:{that}\:{for}\:−\frac{\pi}{\mathrm{2}}<{x}<\frac{\pi}{\mathrm{2}}\:, \\ $$$$\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} }\mathrm{cos}\:{x}−\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }\mathrm{cos}\:\mathrm{3}{x}+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} }\mathrm{cos}\:\mathrm{5}{x}−....{to}\:{infinity} \\ $$$$\:\:=\frac{\pi}{\mathrm{8}}\left(\frac{\pi^{\mathrm{2}} }{\mathrm{4}}−{x}^{\mathrm{2}} \right)\:. \\ $$

Question Number 13778    Answers: 0   Comments: 0

Question Number 13756    Answers: 1   Comments: 5

Question Number 13751    Answers: 1   Comments: 4

(ds/dt)=v,(dv/dt)=a,(da/dt)=b,(db/dt)=e,(de/dt)=f (df/dt)=g,(dg/dt)=h,(dh/dt)=i,(di/dt)=j,(dj/dt)=k,..... now if we continue this process to infinity..and if v_0 ,v,a,b,e,f,g,h,i, j,................=1 .then calculate the formula of v and s ...

$$\frac{{ds}}{{dt}}={v},\frac{{dv}}{{dt}}={a},\frac{{da}}{{dt}}={b},\frac{{db}}{{dt}}={e},\frac{{de}}{{dt}}={f} \\ $$$$\frac{{df}}{{dt}}={g},\frac{{dg}}{{dt}}={h},\frac{{dh}}{{dt}}={i},\frac{{di}}{{dt}}={j},\frac{{dj}}{{dt}}={k},..... \\ $$$${now}\:{if}\:{we}\:{continue}\:{this}\:{process}\:{to} \\ $$$${infinity}..{and}\:{if}\:{v}_{\mathrm{0}} ,{v},{a},{b},{e},{f},{g},{h},{i}, \\ $$$${j},................=\mathrm{1}\:.{then}\:{calculate} \\ $$$${the}\:{formula}\:{of}\:{v}\:{and}\:{s}\:... \\ $$$$ \\ $$

Question Number 13745    Answers: 2   Comments: 1

The velocity of a particle moving in straight line is given by the graph shown here. Draw the acceleration position graph.

$$\mathrm{The}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{moving}\:\mathrm{in} \\ $$$$\mathrm{straight}\:\mathrm{line}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{shown} \\ $$$$\mathrm{here}.\:\mathrm{Draw}\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{position} \\ $$$$\mathrm{graph}. \\ $$

Question Number 13744    Answers: 1   Comments: 4

Solve the following 7^x ≡13(mod 18) Pl give complete process.

$${Solve}\:{the}\:{following} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{7}^{{x}} \equiv\mathrm{13}\left({mod}\:\mathrm{18}\right) \\ $$$${Pl}\:{give}\:{complete}\:{process}. \\ $$

Question Number 13738    Answers: 1   Comments: 1

P,Q,R,S are four locations on the same horizontal plane.Q is on a bearing of 041° from P and the distance is 40km. S is 28km from R on a bearing 074°, R is directly due north of P and the distance between Q and R is 38km. (a)the bearing of R from Q (b)the distance between Q and S (c)the distance between P and R

$$\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\:\mathrm{are}\:\mathrm{four}\:\mathrm{locations}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{horizontal}\:\mathrm{plane}.\mathrm{Q}\:\mathrm{is}\:\mathrm{on}\:\mathrm{a}\: \\ $$$$\mathrm{bearing}\:\mathrm{of}\:\mathrm{041}°\:\mathrm{from}\:\mathrm{P}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{distance}\:\mathrm{is}\:\mathrm{40km}. \\ $$$$\mathrm{S}\:\mathrm{is}\:\mathrm{28km}\:\mathrm{from}\:\mathrm{R}\:\mathrm{on}\:\mathrm{a}\:\mathrm{bearing}\:\mathrm{074}°, \\ $$$$\mathrm{R}\:\mathrm{is}\:\mathrm{directly}\:\mathrm{due}\:\mathrm{north}\:\mathrm{of}\:\mathrm{P}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{Q}\:\mathrm{and}\:\mathrm{R}\:\mathrm{is} \\ $$$$\mathrm{38km}. \\ $$$$\left(\mathrm{a}\right)\mathrm{the}\:\mathrm{bearing}\:\mathrm{of}\:\mathrm{R}\:\mathrm{from}\:\mathrm{Q} \\ $$$$\left(\mathrm{b}\right)\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{Q}\:\mathrm{and}\:\mathrm{S} \\ $$$$\left(\mathrm{c}\right)\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{P}\:\mathrm{and}\:\mathrm{R} \\ $$

Question Number 13737    Answers: 1   Comments: 1

A platform and a building are on the same horizontal plane.The angle of depression of the bottom C of the building from the top A of the platform is 39°.The angle of elevation of the top D of the building from the top of the platform is 56°.Given that the distance between the foot of the platform and that of the building is 10m,calculate the height of the building to the nearest whole number.

$$\mathrm{A}\:\mathrm{platform}\:\mathrm{and}\:\mathrm{a}\:\mathrm{building}\:\mathrm{are}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{horizontal}\:\mathrm{plane}.\mathrm{The} \\ $$$$\mathrm{angle}\:\mathrm{of}\:\mathrm{depression}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bottom} \\ $$$$\mathrm{C}\:\mathrm{of}\:\mathrm{the}\:\mathrm{building}\:\mathrm{from}\:\mathrm{the}\:\mathrm{top}\:\mathrm{A}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{platform}\:\mathrm{is}\:\mathrm{39}°.\mathrm{The}\:\mathrm{angle}\:\mathrm{of}\: \\ $$$$\mathrm{elevation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{top}\:\mathrm{D}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{building}\:\mathrm{from}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{platform}\:\mathrm{is}\:\mathrm{56}°.\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{platform}\:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\:\mathrm{building} \\ $$$$\mathrm{is}\:\mathrm{10m},\mathrm{calculate}\:\mathrm{the}\:\mathrm{height}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{building}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{whole} \\ $$$$\mathrm{number}. \\ $$

Question Number 13735    Answers: 2   Comments: 6

Question Number 13731    Answers: 0   Comments: 1

Prove that if p>q>0 and x≥0 (1/p)((x^p /(p+1))−1)≥(1/q)((x^q /(q+1))−1).

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{p}>{q}>\mathrm{0}\:\mathrm{and}\:{x}\geqslant\mathrm{0} \\ $$$$\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right).\: \\ $$

Question Number 13733    Answers: 0   Comments: 4

Question continuing from mrW1 post on p^2 mod n≡1. Find a number n such that for all m<n such that HCF(m,n)=1 m^2 mod n =1 e.g. for 12 possible value of m are 1,5,7,11.

$$\mathrm{Question}\:\mathrm{continuing}\:\mathrm{from} \\ $$$$\mathrm{mrW1}\:\mathrm{post}\:\mathrm{on}\:{p}^{\mathrm{2}} \:\mathrm{mod}\:\mathrm{n}\equiv\mathrm{1}. \\ $$$$\mathrm{Find}\:\mathrm{a}\:\mathrm{number}\:{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{all}\:{m}<{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{HCF}\left({m},{n}\right)=\mathrm{1} \\ $$$${m}^{\mathrm{2}} \:\mathrm{mod}\:{n}\:=\mathrm{1} \\ $$$${e}.{g}.\:\mathrm{for}\:\mathrm{12}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{m} \\ $$$$\mathrm{are}\:\mathrm{1},\mathrm{5},\mathrm{7},\mathrm{11}. \\ $$

Question Number 13728    Answers: 1   Comments: 0

Prove that n!>((n/3))^n

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{n}!>\left(\frac{{n}}{\mathrm{3}}\right)^{{n}} \\ $$

Question Number 13727    Answers: 1   Comments: 1

A light wire which obeys hooke′s law hangs vertically on a fixed support. The wire has an unstretched lenght of 15cm. The lenght of the wire however increase to 20cm when a load of 0.5kg is attached to it lower end . What is the tension in the wire when the load is at rest ?. If the load is pulled down until the lenght of the wire is 22cm. What is the new tension in the wire (g = 9.8 m/s).

$$\mathrm{A}\:\mathrm{light}\:\mathrm{wire}\:\mathrm{which}\:\mathrm{obeys}\:\mathrm{hooke}'\mathrm{s}\:\mathrm{law}\:\mathrm{hangs}\:\mathrm{vertically}\:\mathrm{on}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{support}. \\ $$$$\mathrm{The}\:\mathrm{wire}\:\mathrm{has}\:\mathrm{an}\:\mathrm{unstretched}\:\mathrm{lenght}\:\mathrm{of}\:\mathrm{15cm}.\:\:\mathrm{The}\:\mathrm{lenght}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{however} \\ $$$$\mathrm{increase}\:\mathrm{to}\:\mathrm{20cm}\:\mathrm{when}\:\mathrm{a}\:\mathrm{load}\:\mathrm{of}\:\mathrm{0}.\mathrm{5kg}\:\mathrm{is}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{it}\:\mathrm{lower}\:\mathrm{end}\:.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{tension}\:\mathrm{in}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{when}\:\mathrm{the}\:\mathrm{load}\:\mathrm{is}\:\mathrm{at}\:\mathrm{rest}\:?.\: \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{load}\:\mathrm{is}\:\mathrm{pulled}\:\mathrm{down}\:\mathrm{until}\:\mathrm{the}\:\mathrm{lenght}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{is}\:\mathrm{22cm}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{new} \\ $$$$\mathrm{tension}\:\mathrm{in}\:\mathrm{the}\:\mathrm{wire}\:\left(\mathrm{g}\:=\:\mathrm{9}.\mathrm{8}\:\mathrm{m}/\mathrm{s}\right). \\ $$

Question Number 13725    Answers: 0   Comments: 2

(1/7)=.142857^(−) (1/7) is a recurring decimal of period 6. What will be the period of (1/7^(20) )?

$$\frac{\mathrm{1}}{\mathrm{7}}=.\overline {\mathrm{142857}} \\ $$$$\frac{\mathrm{1}}{\mathrm{7}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{recurring}\:\mathrm{decimal}\:\mathrm{of}\:\mathrm{period}\:\mathrm{6}. \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{period}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{7}^{\mathrm{20}} }? \\ $$

Question Number 13724    Answers: 2   Comments: 3

Expansion of 1000! has 249, 0′s at the end Find the first non−zero digit from right. 1000!=......d000...00 What is the value of d?

$$\mathrm{Expansion}\:\mathrm{of}\:\mathrm{1000}!\:\mathrm{has}\:\mathrm{249},\:\mathrm{0}'{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero}\:\mathrm{digit}\:\mathrm{from} \\ $$$$\mathrm{right}. \\ $$$$\mathrm{1000}!=......{d}\mathrm{000}...\mathrm{00} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{d}? \\ $$

Question Number 13721    Answers: 0   Comments: 0

what is NBS?

$$\mathrm{what}\:\mathrm{is}\:\mathrm{NBS}? \\ $$

Question Number 13706    Answers: 1   Comments: 0

The volume of a right circular cone is 5 litres . Calculate the volumes of the two parts into which the cone is divided by a plane parallel to the base , One third of the way down from the vertex to the base. Give your answer to the nearest ml.

$$\mathrm{The}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{5}\:\mathrm{litres}\:.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{volumes}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{parts}\:\mathrm{into}\:\mathrm{which}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{base}\:,\:\mathrm{One}\:\mathrm{third} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{way}\:\mathrm{down}\:\mathrm{from}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{to}\:\mathrm{the}\:\mathrm{base}.\:\mathrm{Give}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest} \\ $$$$\mathrm{ml}. \\ $$

Question Number 13705    Answers: 1   Comments: 3

Assuming no air resistance, and angle of projection α=(π/4) , find the ratio of the length of trajectory L of a projectile motion (by the time it hits the ground) to its horizontal range R on ground. (L/R)=?

$${Assuming}\:{no}\:{air}\:{resistance}, \\ $$$${and}\:{angle}\:{of}\:{projection}\:\alpha=\frac{\pi}{\mathrm{4}}\:, \\ $$$${find}\:{the}\:{ratio}\:{of}\:{the}\:{length}\:{of} \\ $$$${trajectory}\:\boldsymbol{{L}}\:{of}\:{a}\:{projectile}\:{motion}\: \\ $$$$\left({by}\:{the}\:{time}\:{it}\:{hits}\:{the}\:{ground}\right) \\ $$$${to}\:{its}\:{horizontal}\:{range}\:\boldsymbol{{R}}\:{on}\: \\ $$$${ground}.\:\:\:\:\:\:\frac{\boldsymbol{{L}}}{\boldsymbol{{R}}}=? \\ $$

Question Number 13695    Answers: 1   Comments: 2

Volume of a bubble is 3 times larger when it reaches the surface from the bottom of the lake. What is the depth of the lake? (A) 10 m (D) 40 m (B) 20 m (E) 50 m (C) 30 m

$$\mathrm{Volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{bubble}\:\mathrm{is}\:\mathrm{3}\:\mathrm{times}\:\mathrm{larger} \\ $$$$\mathrm{when}\:\mathrm{it}\:\mathrm{reaches}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{from}\:\mathrm{the}\:\mathrm{bottom} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{lake}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lake}? \\ $$$$ \\ $$$$\left(\mathrm{A}\right)\:\mathrm{10}\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{40}\:\mathrm{m} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{20}\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{E}\right)\:\mathrm{50}\:\mathrm{m} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{30}\:\mathrm{m} \\ $$

Question Number 13688    Answers: 1   Comments: 0

Question Number 13687    Answers: 0   Comments: 4

Question Number 13681    Answers: 1   Comments: 3

Question Number 13658    Answers: 1   Comments: 0

Prove that cos^2 x + cos^2 3x + cos^2 5x + ... to n terms = (1/2)[n + ((sin4nx)/(2sin2x))]

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{cos}^{\mathrm{2}} {x}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{3}{x}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{5}{x}\:+\:...\:\mathrm{to}\:{n}\:\mathrm{terms} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left[{n}\:+\:\frac{\mathrm{sin4}{nx}}{\mathrm{2sin2}{x}}\right] \\ $$

Question Number 13654    Answers: 1   Comments: 0

The sum of the series sinθ + sin(((n − 4)/(n − 2)))θ + sin(((n − 6)/(n − 2)))θ + ... n terms is equal to (1) sin(((nθ)/(2 − n))) (2) cos(((2nθ)/(2 − n))) (3) tannθ (4) cotnθ

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\mathrm{sin}\theta\:+\:\mathrm{sin}\left(\frac{{n}\:−\:\mathrm{4}}{{n}\:−\:\mathrm{2}}\right)\theta\:+\:\mathrm{sin}\left(\frac{{n}\:−\:\mathrm{6}}{{n}\:−\:\mathrm{2}}\right)\theta\:+\:...\:{n}\:\mathrm{terms} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{sin}\left(\frac{{n}\theta}{\mathrm{2}\:−\:{n}}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{cos}\left(\frac{\mathrm{2}{n}\theta}{\mathrm{2}\:−\:{n}}\right) \\ $$$$\left(\mathrm{3}\right)\:\mathrm{tan}{n}\theta \\ $$$$\left(\mathrm{4}\right)\:\mathrm{cot}{n}\theta \\ $$

Question Number 13649    Answers: 1   Comments: 0

2xyy′+(x−1)y^2 =x^2 e^x

$$\mathrm{2}{xyy}'+\left({x}−\mathrm{1}\right){y}^{\mathrm{2}} ={x}^{\mathrm{2}} {e}^{{x}} \\ $$

Question Number 13647    Answers: 0   Comments: 6

x^2 +y^2 =5.....(1) 3x^2 +xy+y^2 =1.....(2) please help find x and y

$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{5}.....\left(\mathrm{1}\right) \\ $$$$\mathrm{3x}^{\mathrm{2}} +\mathrm{xy}+\mathrm{y}^{\mathrm{2}} =\mathrm{1}.....\left(\mathrm{2}\right) \\ $$$$ \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{find}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

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