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Question Number 17935 Answers: 0 Comments: 0

Ball A is dropped from the top of a building. At the same instant ball B is thrown vertically upwards from the ground. When the ball collide, they are moving in opposite directions and the speed of A(u) is twice the speed of B. The relative velocity of the ball just before collision and relative acceleration between them is (only their magnitudes) (A) 0 and 0 (B) ((3u)/2) and 0 (C) ((3u)/2) and 2g (D) ((3u)/2) and g

$$\mathrm{Ball}\:{A}\:\mathrm{is}\:\mathrm{dropped}\:\mathrm{from}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{a}\:\mathrm{building}. \\ $$$$\mathrm{At}\:\mathrm{the}\:\mathrm{same}\:\mathrm{instant}\:\mathrm{ball}\:{B}\:\mathrm{is}\:\mathrm{thrown} \\ $$$$\mathrm{vertically}\:\mathrm{upwards}\:\mathrm{from}\:\mathrm{the}\:\mathrm{ground}. \\ $$$$\mathrm{When}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{collide},\:\mathrm{they}\:\mathrm{are}\:\mathrm{moving}\:\:\mathrm{in} \\ $$$$\mathrm{opposite}\:\mathrm{directions}\:\mathrm{and}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:{A}\left({u}\right) \\ $$$$\mathrm{is}\:\mathrm{twice}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:{B}.\:\mathrm{The}\:\mathrm{relative}\: \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{just}\:\mathrm{before}\:\mathrm{collision} \\ $$$$\mathrm{and}\:\mathrm{relative}\:\mathrm{acceleration}\:\mathrm{between}\:\mathrm{them} \\ $$$$\mathrm{is}\:\left(\mathrm{only}\:\mathrm{their}\:\mathrm{magnitudes}\right) \\ $$$$\left(\mathrm{A}\right)\:\mathrm{0}\:\mathrm{and}\:\mathrm{0}\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\frac{\mathrm{3}{u}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{0} \\ $$$$\left(\mathrm{C}\right)\:\frac{\mathrm{3}{u}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{2}{g}\:\:\:\left(\mathrm{D}\right)\:\frac{\mathrm{3}{u}}{\mathrm{2}}\:\mathrm{and}\:{g} \\ $$

Question Number 17835 Answers: 2 Comments: 1

A rocket is moving in a gravity free space with a constant acceleration of 2 m/s^2 along + x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in +x direction with a speed of 0.3 m/s relative to the rocket. At the same time, another ball is thrown in −x direction with a speed of 0.2 m/s from its right end relative to the rocket. The time in seconds when the two balls hit each other is

$$\mathrm{A}\:\mathrm{rocket}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{in}\:\mathrm{a}\:\mathrm{gravity}\:\mathrm{free} \\ $$$$\mathrm{space}\:\mathrm{with}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{acceleration}\:\mathrm{of} \\ $$$$\mathrm{2}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \:\mathrm{along}\:+\:\mathrm{x}\:\mathrm{direction}\:\left(\mathrm{see}\:\mathrm{figure}\right). \\ $$$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{a}\:\mathrm{chamber}\:\mathrm{inside}\:\mathrm{the} \\ $$$$\mathrm{rocket}\:\mathrm{is}\:\mathrm{4}\:\mathrm{m}.\:\mathrm{A}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{left}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{chamber}\:\mathrm{in}\:+{x}\:\mathrm{direction} \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{0}.\mathrm{3}\:\mathrm{m}/\mathrm{s}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{rocket}.\:\mathrm{At}\:\mathrm{the}\:\mathrm{same}\:\mathrm{time},\:\mathrm{another}\:\mathrm{ball} \\ $$$$\mathrm{is}\:\mathrm{thrown}\:\mathrm{in}\:−{x}\:\mathrm{direction}\:\mathrm{with}\:\mathrm{a}\:\mathrm{speed} \\ $$$$\mathrm{of}\:\mathrm{0}.\mathrm{2}\:\mathrm{m}/\mathrm{s}\:\mathrm{from}\:\mathrm{its}\:\mathrm{right}\:\mathrm{end}\:\mathrm{relative}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{rocket}.\:\mathrm{The}\:\mathrm{time}\:\mathrm{in}\:\mathrm{seconds}\:\mathrm{when} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{balls}\:\mathrm{hit}\:\mathrm{each}\:\mathrm{other}\:\mathrm{is} \\ $$

Question Number 17830 Answers: 2 Comments: 0

Two candles of the same height are lighted together. First one gets burnt up completely in 3 hours while the second in 4 hours. At what point of time, the length of second candle will be double the length of the first candle?

$$\mathrm{Two}\:\mathrm{candles}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{height}\:\mathrm{are} \\ $$$$\mathrm{lighted}\:\mathrm{together}.\:\mathrm{First}\:\mathrm{one}\:\mathrm{gets}\:\mathrm{burnt}\:\mathrm{up} \\ $$$$\mathrm{completely}\:\mathrm{in}\:\mathrm{3}\:\mathrm{hours}\:\mathrm{while}\:\mathrm{the}\:\mathrm{second} \\ $$$$\mathrm{in}\:\mathrm{4}\:\mathrm{hours}.\:\mathrm{At}\:\mathrm{what}\:\mathrm{point}\:\mathrm{of}\:\mathrm{time},\:\mathrm{the} \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{second}\:\mathrm{candle}\:\mathrm{will}\:\mathrm{be}\:\mathrm{double} \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{candle}? \\ $$

Question Number 17828 Answers: 1 Comments: 0

Question Number 17815 Answers: 1 Comments: 4

For x ∈ (0, π), the equation sin x + 2 sin 2x − sin 3x = 3 has (1) Infinitely many solutions (2) Three solutions (3) One solution (4) No solution

$$\mathrm{For}\:{x}\:\in\:\left(\mathrm{0},\:\pi\right),\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{sin}\:{x}\:+\:\mathrm{2}\:\mathrm{sin}\:\mathrm{2}{x}\:−\:\mathrm{sin}\:\mathrm{3}{x}\:=\:\mathrm{3}\:\mathrm{has} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Infinitely}\:\mathrm{many}\:\mathrm{solutions} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Three}\:\mathrm{solutions} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{One}\:\mathrm{solution} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{No}\:\mathrm{solution} \\ $$

Question Number 17805 Answers: 0 Comments: 0

If θ=t^n e^(−r^(2/(ut)) ) ,find the value of n which will make (1/r^2 ) (∂/∂r)(r^2 (∂θ/(∂r))) equal to (∂θ/∂t)

$${If}\:\theta={t}^{{n}} {e}^{−{r}^{\frac{\mathrm{2}}{{ut}}} } ,{find}\:{the}\:{value}\:{of} \\ $$$${n}\:{which}\:{will}\:{make}\:\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\:\frac{\partial}{\partial{r}}\left({r}^{\mathrm{2}} \frac{\partial\theta}{\left.\partial{r}\right)}\right. \\ $$$${equal}\:{to}\:\frac{\partial\theta}{\partial{t}} \\ $$

Question Number 17782 Answers: 0 Comments: 1

let a,b,c,x,y and z be complex numbers such that : a=((b+c)/(x−2)), b=((c+a)/(y−2)), c=((a+b)/(z−2)) if xy+yz+zx=1000 and x+y+z=2016, find the value of xyz

$${let}\:{a},{b},{c},{x},{y}\:{and}\:{z}\:{be}\:{complex}\:{numbers} \\ $$$${such}\:{that}\:: \\ $$$${a}=\frac{{b}+{c}}{{x}−\mathrm{2}},\:{b}=\frac{{c}+{a}}{{y}−\mathrm{2}},\:{c}=\frac{{a}+{b}}{{z}−\mathrm{2}} \\ $$$${if}\:{xy}+{yz}+{zx}=\mathrm{1000}\:{and}\:{x}+{y}+{z}=\mathrm{2016}, \\ $$$${find}\:{the}\:{value}\:{of}\:{xyz} \\ $$

Question Number 17779 Answers: 1 Comments: 0

show that {log_a ab}{log_b ab}=logab_a +logab_b

$${show}\:{that}\:\left\{{lo}\underset{{a}} {{g}ab}\right\}\left\{{lo}\underset{{b}} {{g}ab}\right\}={loga}\underset{{a}} {{b}}+{loga}\underset{{b}} {{b}} \\ $$

Question Number 17775 Answers: 2 Comments: 0

Σ_(m=1) ^(10) (((sin 2πm)/(11)) − i((cos 2πm)/(11))) =????? Solve..it

$$\underset{{m}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left(\frac{\mathrm{sin}\:\mathrm{2}\pi{m}}{\mathrm{11}}\:−\:{i}\frac{\mathrm{cos}\:\mathrm{2}\pi{m}}{\mathrm{11}}\right)\:=????? \\ $$$${Solve}..{it} \\ $$$$ \\ $$

Question Number 17771 Answers: 1 Comments: 1

Question Number 17770 Answers: 1 Comments: 0

Find the number of ways the digits 0,1,2 and 3 can be permuted to give rise to a number greater than 2000.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{the} \\ $$$$\mathrm{digits}\:\mathrm{0},\mathrm{1},\mathrm{2}\:\mathrm{and}\:\mathrm{3}\:\mathrm{can}\:\mathrm{be}\:\mathrm{permuted} \\ $$$$\mathrm{to}\:\mathrm{give}\:\mathrm{rise}\:\mathrm{to}\:\mathrm{a}\:\mathrm{number}\:\mathrm{greater} \\ $$$$\mathrm{than}\:\mathrm{2000}. \\ $$

Question Number 17767 Answers: 1 Comments: 0

Σ_(n = 1) ^(30) (n^2 + 1) = (A) Σ_(n = 1) ^(15) (2n^2 + 30n + 224) (B) Σ_(n = 1) ^(15) (2n^2 + 30n + 225) (C) Σ_(n = 1) ^(15) (2n^2 + 30n + 226) (D) Σ_(n = 1) ^(15) (2n^2 + 30n + 227) (E) Σ_(n = 1) ^(15) (2n^2 + 30n + 228)

$$\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{30}} {\sum}}\left({n}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\: \\ $$$$\left(\mathrm{A}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{224}\right) \\ $$$$\left(\mathrm{B}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{225}\right) \\ $$$$\left(\mathrm{C}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{226}\right) \\ $$$$\left(\mathrm{D}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{227}\right) \\ $$$$\left(\mathrm{E}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{228}\right) \\ $$

Question Number 17759 Answers: 1 Comments: 0

2^(4(y^2 −2)) =y^((y^2 −8)) please help me find y... pls!

$$\mathrm{2}^{\mathrm{4}\left(\mathrm{y}^{\mathrm{2}} −\mathrm{2}\right)} =\mathrm{y}^{\left(\mathrm{y}^{\mathrm{2}} −\mathrm{8}\right)} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{find}\:\mathrm{y}...\:\mathrm{pls}! \\ $$

Question Number 17748 Answers: 0 Comments: 1

f p^2 =qr, prove thathat log_r ^p +log_(q=) ^p 2log_q ^p log_r ^p

$${f}\:{p}^{\mathrm{2}} ={qr},\:\:{prove}\:{thathat}\:{log}_{{r}} ^{{p}} +{log}_{{q}=} ^{{p}} \\ $$$$\mathrm{2}{log}_{{q}} ^{{p}} {log}_{{r}} ^{{p}} \\ $$

Question Number 17742 Answers: 1 Comments: 1

x^3 = 3^x , find x.

$$\mathrm{x}^{\mathrm{3}} \:=\:\mathrm{3}^{\mathrm{x}} ,\:\:\:\mathrm{find}\:\mathrm{x}. \\ $$

Question Number 17740 Answers: 0 Comments: 1

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

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