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Question Number 17743 Answers: 2 Comments: 0

Question Number 17736 Answers: 0 Comments: 1

evaluate; ∫ln (sin 2x)dx

$${evaluate};\:\int\mathrm{ln}\:\left(\mathrm{sin}\:\mathrm{2}{x}\right){dx} \\ $$

Question Number 17735 Answers: 1 Comments: 0

Solve: (dy/dx) + ysec(x) = tan(x)

$$\mathrm{Solve}:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\mathrm{ysec}\left(\mathrm{x}\right)\:=\:\mathrm{tan}\left(\mathrm{x}\right) \\ $$

Question Number 17734 Answers: 0 Comments: 0

If x^2 + y^3 − 3xy = 0, Show that, (d^2 y/dx^2 ) = ((− 2xy)/(y^2 − x^2 ))

$$\mathrm{If}\:\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{3}} \:−\:\mathrm{3xy}\:=\:\mathrm{0}, \\ $$$$\mathrm{Show}\:\mathrm{that},\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:=\:\frac{−\:\mathrm{2xy}}{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 17729 Answers: 1 Comments: 3

A lotus plant in a pool of water is (1/2) cubit above water level. When propelled by air, the lotus sinks in the pool 2 cubits away from its position. Find the depth of water in the pool.

$$\mathrm{A}\:\mathrm{lotus}\:\mathrm{plant}\:\mathrm{in}\:\mathrm{a}\:\mathrm{pool}\:\mathrm{of}\:\mathrm{water}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{cubit}\:\mathrm{above}\:\mathrm{water}\:\mathrm{level}.\:\mathrm{When} \\ $$$$\mathrm{propelled}\:\mathrm{by}\:\mathrm{air},\:\mathrm{the}\:\mathrm{lotus}\:\mathrm{sinks}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{pool}\:\mathrm{2}\:\mathrm{cubits}\:\mathrm{away}\:\mathrm{from}\:\mathrm{its}\:\mathrm{position}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{water}\:\mathrm{in}\:\mathrm{the}\:\mathrm{pool}. \\ $$

Question Number 17704 Answers: 2 Comments: 1

A monkey climbs up a slippery pole for 3 seconds and subsequently slips for 3 seconds. Its velocity at time t is given by v (t) = 2t(3 − t) ; 0 < t < 3 and v (t) = − (t − 3)(6 − t) for 3 < t < 6 s in m/s. It repeats this cycle till it reaches the height of 20 m. At what time is its average velocity maximum?

$$\mathrm{A}\:\mathrm{monkey}\:\mathrm{climbs}\:\mathrm{up}\:\mathrm{a}\:\mathrm{slippery}\:\mathrm{pole}\:\mathrm{for} \\ $$$$\mathrm{3}\:\mathrm{seconds}\:\mathrm{and}\:\mathrm{subsequently}\:\mathrm{slips}\:\mathrm{for}\:\mathrm{3} \\ $$$$\mathrm{seconds}.\:\mathrm{Its}\:\mathrm{velocity}\:\mathrm{at}\:\mathrm{time}\:{t}\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{by}\:{v}\:\left({t}\right)\:=\:\mathrm{2}{t}\left(\mathrm{3}\:−\:{t}\right)\:;\:\mathrm{0}\:<\:{t}\:<\:\mathrm{3}\:\mathrm{and} \\ $$$${v}\:\left({t}\right)\:=\:−\:\left({t}\:−\:\mathrm{3}\right)\left(\mathrm{6}\:−\:{t}\right)\:\mathrm{for}\:\mathrm{3}\:<\:{t}\:<\:\mathrm{6}\:\mathrm{s} \\ $$$$\mathrm{in}\:\mathrm{m}/\mathrm{s}.\:\mathrm{It}\:\mathrm{repeats}\:\mathrm{this}\:\mathrm{cycle}\:\mathrm{till}\:\mathrm{it} \\ $$$$\mathrm{reaches}\:\mathrm{the}\:\mathrm{height}\:\mathrm{of}\:\mathrm{20}\:\mathrm{m}.\:\mathrm{At}\:\mathrm{what} \\ $$$$\mathrm{time}\:\mathrm{is}\:\mathrm{its}\:\mathrm{average}\:\mathrm{velocity}\:\mathrm{maximum}? \\ $$

Question Number 17692 Answers: 1 Comments: 0

Question Number 17689 Answers: 1 Comments: 0

A bird is tossing (flying to and fro) between two cars moving towards each other on a straight road. One car has a speed of 18 km/h while the other has the speed of 27 km/h. The bird starts moving from first car towards the other and is moving with the speed of 36 km/h and when the two cars were separated by 36 km. What is the total displacement of the bird?

$$\mathrm{A}\:\mathrm{bird}\:\mathrm{is}\:\mathrm{tossing}\:\left(\mathrm{flying}\:\mathrm{to}\:\mathrm{and}\:\mathrm{fro}\right) \\ $$$$\mathrm{between}\:\mathrm{two}\:\mathrm{cars}\:\mathrm{moving}\:\mathrm{towards} \\ $$$$\mathrm{each}\:\mathrm{other}\:\mathrm{on}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{road}.\:\mathrm{One}\:\mathrm{car} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{18}\:\mathrm{km}/\mathrm{h}\:\mathrm{while}\:\mathrm{the}\:\mathrm{other} \\ $$$$\mathrm{has}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{27}\:\mathrm{km}/\mathrm{h}.\:\mathrm{The}\:\mathrm{bird} \\ $$$$\mathrm{starts}\:\mathrm{moving}\:\mathrm{from}\:\mathrm{first}\:\mathrm{car}\:\mathrm{towards} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{and}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{the}\:\mathrm{speed} \\ $$$$\mathrm{of}\:\mathrm{36}\:\mathrm{km}/\mathrm{h}\:\mathrm{and}\:\mathrm{when}\:\mathrm{the}\:\mathrm{two}\:\mathrm{cars}\:\mathrm{were} \\ $$$$\mathrm{separated}\:\mathrm{by}\:\mathrm{36}\:\mathrm{km}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{total} \\ $$$$\mathrm{displacement}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bird}? \\ $$

Question Number 17676 Answers: 2 Comments: 0

Question Number 17675 Answers: 1 Comments: 0

Question Number 17653 Answers: 0 Comments: 6

A line segment moves in the plane with its end points on the coordinate axes so that the sum of the length of its intersect on the coordinate axes is a constant C . Find the locus of the mid points of this segment . Ans. is 8(∣x∣^3 +∣y∣^3 )=C . Λ means power . pls. solve it.

$$\mathrm{A}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{with}\:\mathrm{its}\:\mathrm{end}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{coordinate} \\ $$$$\mathrm{axes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{length} \\ $$$$\mathrm{of}\:\mathrm{its}\:\mathrm{intersect}\:\mathrm{on}\:\mathrm{the}\:\mathrm{coordinate}\: \\ $$$$\mathrm{axes}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{C}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{mid}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{this}\:\mathrm{segment}\:. \\ $$$$\mathrm{Ans}.\:\mathrm{is}\:\:\:\mathrm{8}\left(\mid\mathrm{x}\mid^{\mathrm{3}} +\mid\mathrm{y}\mid^{\mathrm{3}} \right)=\mathrm{C}\:. \\ $$$$\Lambda\:\:\mathrm{means}\:\mathrm{power}\:.\:\mathrm{pls}.\:\mathrm{solve}\:\mathrm{it}. \\ $$

Question Number 17638 Answers: 0 Comments: 4

please help me with this confusing question x^(2x/y) ×y^(y/x) =4......(1) (xy)^(xy+yx) =16.....(2) solve for x and y

$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{with}\:\mathrm{this} \\ $$$$\mathrm{confusing}\:\mathrm{question} \\ $$$$ \\ $$$$\mathrm{x}^{\mathrm{2x}/\mathrm{y}} ×\mathrm{y}^{\mathrm{y}/\mathrm{x}} =\mathrm{4}......\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\left(\mathrm{xy}\right)^{\mathrm{xy}+\mathrm{yx}} =\mathrm{16}.....\left(\mathrm{2}\right) \\ $$$$ \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

Question Number 17634 Answers: 1 Comments: 0

y = x! , Find y′

$$\mathrm{y}\:=\:\mathrm{x}!\:\:,\:\:\:\:\:\mathrm{Find}\:\:\:\mathrm{y}' \\ $$

Question Number 17625 Answers: 1 Comments: 1

Find the fourier series of : f(x) = x, from 0 < x < π

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{fourier}\:\mathrm{series}\:\mathrm{of}\::\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{x},\:\:\mathrm{from}\:\:\:\mathrm{0}\:<\:\mathrm{x}\:<\:\pi \\ $$

Question Number 17617 Answers: 0 Comments: 5

A string is stretched and fastened to two points l apart. Motion is started by displacing the string into the form y = (lx − x^2 ) from which it is release at time t = 0. Find the displacement of any point on the spring at a distance x from one end at time t.

$$\mathrm{A}\:\mathrm{string}\:\mathrm{is}\:\mathrm{stretched}\:\mathrm{and}\:\mathrm{fastened}\:\mathrm{to}\:\mathrm{two}\:\mathrm{points}\:\:\mathrm{l}\:\:\mathrm{apart}.\:\mathrm{Motion}\:\mathrm{is}\:\mathrm{started} \\ $$$$\mathrm{by}\:\mathrm{displacing}\:\mathrm{the}\:\mathrm{string}\:\mathrm{into}\:\mathrm{the}\:\mathrm{form}\:\:\mathrm{y}\:=\:\left(\mathrm{lx}\:−\:\mathrm{x}^{\mathrm{2}} \right)\:\mathrm{from}\:\mathrm{which}\:\mathrm{it}\:\mathrm{is}\:\mathrm{release} \\ $$$$\mathrm{at}\:\mathrm{time}\:\mathrm{t}\:=\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{displacement}\:\mathrm{of}\:\mathrm{any}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{spring}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance} \\ $$$$\mathrm{x}\:\mathrm{from}\:\mathrm{one}\:\mathrm{end}\:\mathrm{at}\:\mathrm{time}\:\:\mathrm{t}.\: \\ $$

Question Number 17614 Answers: 0 Comments: 3

The triangle ABC has CA = CB. P is a point on the circumcircle between A and B (and on the opposite side of the line AB to C). D is the foot of the perpendicular from C to PB. Show that PA + PB = 2∙PD.

$$\mathrm{The}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{has}\:\mathrm{CA}\:=\:\mathrm{CB}.\:\mathrm{P}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{between}\:\mathrm{A} \\ $$$$\mathrm{and}\:\mathrm{B}\:\left(\mathrm{and}\:\mathrm{on}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\right. \\ $$$$\left.\mathrm{line}\:\mathrm{AB}\:\mathrm{to}\:\mathrm{C}\right).\:\mathrm{D}\:\mathrm{is}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{perpendicular}\:\mathrm{from}\:\mathrm{C}\:\mathrm{to}\:\mathrm{PB}.\:\mathrm{Show}\:\mathrm{that} \\ $$$$\mathrm{PA}\:+\:\mathrm{PB}\:=\:\mathrm{2}\centerdot\mathrm{PD}. \\ $$

Question Number 17612 Answers: 1 Comments: 2

The accompanying diagram is a road- plan of a city. All the roads go east- west or north-south, with the exception of one shown. Due to repairs one road is impassable at the point X, Of all the possible routes from P to Q, there are several shortest routes. How many such shortest routes are there?

$$\mathrm{The}\:\mathrm{accompanying}\:\mathrm{diagram}\:\mathrm{is}\:\mathrm{a}\:\mathrm{road}- \\ $$$$\mathrm{plan}\:\mathrm{of}\:\mathrm{a}\:\mathrm{city}.\:\mathrm{All}\:\mathrm{the}\:\mathrm{roads}\:\mathrm{go}\:\mathrm{east}- \\ $$$$\mathrm{west}\:\mathrm{or}\:\mathrm{north}-\mathrm{south},\:\mathrm{with}\:\mathrm{the} \\ $$$$\mathrm{exception}\:\mathrm{of}\:\mathrm{one}\:\mathrm{shown}.\:\mathrm{Due}\:\mathrm{to}\:\mathrm{repairs} \\ $$$$\mathrm{one}\:\mathrm{road}\:\mathrm{is}\:\mathrm{impassable}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{X}, \\ $$$$\mathrm{Of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{routes}\:\mathrm{from}\:\mathrm{P}\:\mathrm{to}\:\mathrm{Q}, \\ $$$$\mathrm{there}\:\mathrm{are}\:\mathrm{several}\:\mathrm{shortest}\:\mathrm{routes}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{such}\:\mathrm{shortest}\:\mathrm{routes}\:\mathrm{are}\:\mathrm{there}? \\ $$

Question Number 17610 Answers: 1 Comments: 1

A train, after travelling 70 km from a station A towards a station B, develops a fault in the engine at C, and covers the remaining journey to B at (3/4) of its earlier speed and arrives at B 1 hour and 20 minutes late. If the fault had developed 35 km further on at D, it would have arrived 20 minutes sooner. Find the speed of the train and the distance from A to B.

$$\mathrm{A}\:\mathrm{train},\:\mathrm{after}\:\mathrm{travelling}\:\mathrm{70}\:\mathrm{km}\:\mathrm{from}\:\mathrm{a} \\ $$$$\mathrm{station}\:\mathrm{A}\:\mathrm{towards}\:\mathrm{a}\:\mathrm{station}\:\mathrm{B},\:\mathrm{develops} \\ $$$$\mathrm{a}\:\mathrm{fault}\:\mathrm{in}\:\mathrm{the}\:\mathrm{engine}\:\mathrm{at}\:\mathrm{C},\:\mathrm{and}\:\mathrm{covers} \\ $$$$\mathrm{the}\:\mathrm{remaining}\:\mathrm{journey}\:\mathrm{to}\:\mathrm{B}\:\mathrm{at}\:\frac{\mathrm{3}}{\mathrm{4}}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{earlier}\:\mathrm{speed}\:\mathrm{and}\:\mathrm{arrives}\:\mathrm{at}\:\mathrm{B}\:\mathrm{1}\:\mathrm{hour} \\ $$$$\mathrm{and}\:\mathrm{20}\:\mathrm{minutes}\:\mathrm{late}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{fault}\:\mathrm{had} \\ $$$$\mathrm{developed}\:\mathrm{35}\:\mathrm{km}\:\mathrm{further}\:\mathrm{on}\:\mathrm{at}\:\mathrm{D},\:\mathrm{it} \\ $$$$\mathrm{would}\:\mathrm{have}\:\mathrm{arrived}\:\mathrm{20}\:\mathrm{minutes}\:\mathrm{sooner}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{train}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{A}\:\mathrm{to}\:\mathrm{B}. \\ $$

Question Number 17604 Answers: 2 Comments: 0

∫_( 0) ^( n) x^2 (n − x)^p dx for p > 0

$$\int_{\:\:\mathrm{0}} ^{\:\:\mathrm{n}} \:\mathrm{x}^{\mathrm{2}} \left(\mathrm{n}\:−\:\mathrm{x}\right)^{\mathrm{p}} \:\mathrm{dx}\:\:\:\:\:\:\:\mathrm{for}\:\:\:\mathrm{p}\:>\:\mathrm{0} \\ $$

Question Number 17599 Answers: 1 Comments: 0

a particle starts with an initial speed u,it moves in a straight line with an accleration which varies as the square of the time the particle has been in motion. Find the speed at any time t,and the distance travelled.

$$\mathrm{a}\:\mathrm{particle}\:\mathrm{starts}\:\mathrm{with}\:\mathrm{an}\:\mathrm{initial} \\ $$$$\mathrm{speed}\:\mathrm{u},\mathrm{it}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{with}\:\mathrm{an}\:\mathrm{accleration}\:\mathrm{which} \\ $$$$\mathrm{varies}\:\mathrm{as}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{the}\:\mathrm{time} \\ $$$$\mathrm{the}\:\mathrm{particle}\:\mathrm{has}\:\mathrm{been}\:\mathrm{in}\:\mathrm{motion}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{at}\:\mathrm{any}\:\mathrm{time}\:\mathrm{t},\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{distance}\:\mathrm{travelled}. \\ $$

Question Number 17595 Answers: 1 Comments: 1

Let a,b,c be the posetive integer such that b/a is an also integer if a,b,c are in GP and AM of a,b,c is(b+2) then find the value of (a^2 +a−14)/(a+1)

$${Let}\:{a},{b},{c}\:{be}\:{the}\:{posetive}\:{integer}\:{such}\:{that}\: \\ $$$${b}/{a}\:{is}\:{an}\:{also}\:{integer}\:{if}\:{a},{b},{c}\:{are}\:{in}\:{GP}\:{and}\: \\ $$$${AM}\:{of}\:{a},{b},{c}\:{is}\left({b}+\mathrm{2}\right)\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$$\left({a}^{\mathrm{2}} +{a}−\mathrm{14}\right)/\left({a}+\mathrm{1}\right) \\ $$

Question Number 17600 Answers: 1 Comments: 0

Question Number 17603 Answers: 2 Comments: 0

∫_( 0) ^( a/2) x^2 (a^2 − x^2 )^(3/2) dx

$$\int_{\:\:\mathrm{0}} ^{\:\:\mathrm{a}/\mathrm{2}} \:\mathrm{x}^{\mathrm{2}} \left(\mathrm{a}^{\mathrm{2}} \:−\:\mathrm{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} \:\mathrm{dx} \\ $$

Question Number 17580 Answers: 0 Comments: 8

Question Number 17576 Answers: 1 Comments: 0

solve the equation x^y =y^x .......(1) x^2 =y^3 .......(2)

$$\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$ \\ $$$$\mathrm{x}^{\mathrm{y}} =\mathrm{y}^{\mathrm{x}} .......\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{x}^{\mathrm{2}} =\mathrm{y}^{\mathrm{3}} .......\left(\mathrm{2}\right) \\ $$

Question Number 17575 Answers: 0 Comments: 3

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