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Question Number 17890    Answers: 0   Comments: 0

Question Number 17948    Answers: 1   Comments: 3

Question Number 17886    Answers: 0   Comments: 7

Question Number 17884    Answers: 1   Comments: 1

Question Number 17879    Answers: 0   Comments: 0

Question Number 17878    Answers: 0   Comments: 0

Question Number 17872    Answers: 2   Comments: 0

Solve : ∣x − 1∣ + ∣x∣ + ∣x + 1∣ = x + 2

$$\mathrm{Solve}\:: \\ $$$$\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\mid\:+\:\mid{x}\:+\:\mathrm{1}\mid\:=\:{x}\:+\:\mathrm{2} \\ $$

Question Number 17867    Answers: 1   Comments: 0

prove that (2+(√3))^(2n) +(2−(√3))^(2n) is an even integer and that (2+(√3))^(2n) −(2−(√3))^(2n) =w(√3) for some integers w,for all integer n≥1.

$${prove}\:{that}\:\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}} +\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}} {is}\:{an} \\ $$$${even}\:{integer}\:{and}\:{that}\:\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}} −\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}} ={w}\sqrt{\mathrm{3}} \\ $$$${for}\:{some}\:{integers}\:{w},{for}\:{all}\:{integer}\:{n}\geqslant\mathrm{1}. \\ $$$$ \\ $$

Question Number 17866    Answers: 1   Comments: 0

Prove that sin 5A = 5 cos^4 A sin A − 10 cos^2 A sin^3 A + sin^5 A

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{sin}\:\mathrm{5}{A}\:=\:\mathrm{5}\:\mathrm{cos}^{\mathrm{4}} \:{A}\:\mathrm{sin}\:{A}\:− \\ $$$$\mathrm{10}\:\mathrm{cos}^{\mathrm{2}} \:{A}\:\mathrm{sin}^{\mathrm{3}} \:{A}\:+\:\mathrm{sin}^{\mathrm{5}} \:{A} \\ $$

Question Number 17852    Answers: 0   Comments: 1

A particle moves in a straight line (x-axis) with the velocity shown in the figure. Knowing that x = −16 m at t = 0, draw the a-t and x-t curves for the interval 0 < t < 40 s and determine (i) Maximum value of the position coordinate of the particle. (ii) The values of t for which the particle is at a distance of 50 m from the origin.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\left({x}-\mathrm{axis}\right)\:\mathrm{with}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{figure}.\:\mathrm{Knowing}\:\mathrm{that}\:{x}\:=\:−\mathrm{16}\:\mathrm{m}\:\mathrm{at} \\ $$$${t}\:=\:\mathrm{0},\:\mathrm{draw}\:\mathrm{the}\:{a}-{t}\:\mathrm{and}\:{x}-{t}\:\mathrm{curves}\:\mathrm{for} \\ $$$$\mathrm{the}\:\mathrm{interval}\:\mathrm{0}\:<\:{t}\:<\:\mathrm{40}\:\mathrm{s}\:\mathrm{and}\:\mathrm{determine} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{position} \\ $$$$\mathrm{coordinate}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:{t}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{50}\:\mathrm{m}\:\mathrm{from} \\ $$$$\mathrm{the}\:\mathrm{origin}. \\ $$

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

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