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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

Question Number 17530    Answers: 0   Comments: 7

Two masses 5 kg and M are hanging with the help of light rope and pulley as shown below. If the system is in equilibrium then M =

$$\mathrm{Two}\:\mathrm{masses}\:\mathrm{5}\:\mathrm{kg}\:\mathrm{and}\:{M}\:\mathrm{are}\:\mathrm{hanging} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{help}\:\mathrm{of}\:\mathrm{light}\:\mathrm{rope}\:\mathrm{and}\:\mathrm{pulley} \\ $$$$\mathrm{as}\:\mathrm{shown}\:\mathrm{below}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}\:\mathrm{then}\:{M}\:= \\ $$

Question Number 17525    Answers: 0   Comments: 1

Evaluate ∫_(−1) ^1 (1−x^2 )^(n/2) dx for n ∈ Z∩[0;∞) (i.e. 0, 1, 2, ...) and: a) n ≡ 0(mod 2) b) n ≡ 1(mod 2)

$$\mathrm{Evaluate}\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} {dx}\:\:\mathrm{for}\: \\ $$$${n}\:\in\:\mathbb{Z}\cap\left[\mathrm{0};\infty\right)\:\left(\mathrm{i}.\mathrm{e}.\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:...\right)\:\mathrm{and}: \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\:\:{n}\:\equiv\:\mathrm{0}\left({mod}\:\mathrm{2}\right) \\ $$$$\left.\boldsymbol{{b}}\right)\:{n}\:\equiv\:\mathrm{1}\left({mod}\:\mathrm{2}\right) \\ $$

Question Number 17524    Answers: 1   Comments: 0

The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY be a diameter of Ω which is also tangent to ω. Assume PY > PX. Let PY intersect ω at Z. If YZ = 2PZ, what is the magnitude of ∠PYX in degrees?

$$\mathrm{The}\:\mathrm{circle}\:\omega\:\mathrm{touches}\:\mathrm{the}\:\mathrm{circle}\:\Omega \\ $$$$\mathrm{internally}\:\mathrm{at}\:{P}.\:\mathrm{The}\:\mathrm{centre}\:{O}\:\mathrm{of}\:\Omega\:\mathrm{is} \\ $$$$\mathrm{outside}\:\omega.\:\mathrm{Let}\:{XY}\:\mathrm{be}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{of}\:\Omega \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{also}\:\mathrm{tangent}\:\mathrm{to}\:\omega.\:\mathrm{Assume} \\ $$$${PY}\:>\:{PX}.\:\mathrm{Let}\:{PY}\:\mathrm{intersect}\:\omega\:\mathrm{at}\:{Z}.\:\mathrm{If} \\ $$$${YZ}\:=\:\mathrm{2}{PZ},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{magnitude}\:\mathrm{of} \\ $$$$\angle{PYX}\:\mathrm{in}\:\mathrm{degrees}? \\ $$

Question Number 17522    Answers: 0   Comments: 1

∫(dx/(sin^5 x+cos^5 x))

$$\int\frac{{dx}}{\mathrm{sin}\:^{\mathrm{5}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}} \\ $$

Question Number 17520    Answers: 1   Comments: 1

Find the coordinate of the point in RΛ3 which is the reflection the point (1,2,3) with respect to plane X+Y+Z=1 .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coordinate}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{in} \\ $$$$\mathrm{R}\Lambda\mathrm{3}\:\mathrm{which}\:\mathrm{is}\:\mathrm{the}\:\mathrm{reflection}\:\mathrm{the}\:\mathrm{point} \\ $$$$\left(\mathrm{1},\mathrm{2},\mathrm{3}\right)\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{plane}\: \\ $$$$\mathrm{X}+\mathrm{Y}+\mathrm{Z}=\mathrm{1}\:. \\ $$

Question Number 17514    Answers: 1   Comments: 2

S(n)=∫_0 ^1 x^(2n) sin(2nπx)dx

$${S}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}{n}} {sin}\left(\mathrm{2}{n}\pi{x}\right){dx} \\ $$

Question Number 17512    Answers: 1   Comments: 0

∫ ((e^(−t) ln(1 + e^(−t) ))/(1 + e^(−t) )) dt

$$\int\:\frac{\mathrm{e}^{−\mathrm{t}} \:\mathrm{ln}\left(\mathrm{1}\:+\:\mathrm{e}^{−\mathrm{t}} \right)}{\mathrm{1}\:+\:\mathrm{e}^{−\mathrm{t}} }\:\mathrm{dt} \\ $$

Question Number 17506    Answers: 1   Comments: 0

∫ tan^(−1) ((√((x + 1)/(x − 1)))) dx

$$\int\:\mathrm{tan}^{−\mathrm{1}} \left(\sqrt{\frac{\mathrm{x}\:+\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{1}}}\right)\:\mathrm{dx} \\ $$

Question Number 17497    Answers: 1   Comments: 0

Evaluate: sin(9)°

$$\mathrm{Evaluate}:\:\:\:\mathrm{sin}\left(\mathrm{9}\right)° \\ $$

Question Number 17481    Answers: 0   Comments: 0

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