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Question Number 13904 Answers: 2 Comments: 0

Show that : sin(50) + sin(40) = (√2) cos(5)

$$\mathrm{Show}\:\mathrm{that}\:\::\:\:\mathrm{sin}\left(\mathrm{50}\right)\:+\:\mathrm{sin}\left(\mathrm{40}\right)\:=\:\sqrt{\mathrm{2}}\:\mathrm{cos}\left(\mathrm{5}\right) \\ $$

Question Number 13903 Answers: 2 Comments: 1

Find the values of x in the range 0° to 360° for which sin(3x)sin(x) = 2cos(2x) + 1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{range}\:\mathrm{0}°\:\mathrm{to}\:\mathrm{360}°\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\mathrm{sin}\left(\mathrm{3x}\right)\mathrm{sin}\left(\mathrm{x}\right)\:=\:\mathrm{2cos}\left(\mathrm{2x}\right)\:+\:\mathrm{1} \\ $$

Question Number 13893 Answers: 0 Comments: 3

Let n be an odd positive integer. On some field, n gunmen are placed such that all pairwise distances between them are different. At a signal, every gunman takes out his gun and shoots the closest gunman. Prove that: (a) at least one gunman survives; (b) no gunman is shot more than 5 times; (c) the trajectories of the bullets do not intersect.

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{positive}\:\mathrm{integer}.\:\mathrm{On} \\ $$$$\mathrm{some}\:\mathrm{field},\:{n}\:\mathrm{gunmen}\:\mathrm{are}\:\mathrm{placed}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{all}\:\mathrm{pairwise}\:\mathrm{distances}\:\mathrm{between} \\ $$$$\mathrm{them}\:\mathrm{are}\:\mathrm{different}.\:\mathrm{At}\:\mathrm{a}\:\mathrm{signal},\:\mathrm{every} \\ $$$$\mathrm{gunman}\:\mathrm{takes}\:\mathrm{out}\:\mathrm{his}\:\mathrm{gun}\:\mathrm{and}\:\mathrm{shoots} \\ $$$$\mathrm{the}\:\mathrm{closest}\:\mathrm{gunman}.\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{gunman}\:\mathrm{survives}; \\ $$$$\left(\mathrm{b}\right)\:\mathrm{no}\:\mathrm{gunman}\:\mathrm{is}\:\mathrm{shot}\:\mathrm{more}\:\mathrm{than}\:\mathrm{5} \\ $$$$\mathrm{times}; \\ $$$$\left(\mathrm{c}\right)\:\mathrm{the}\:\mathrm{trajectories}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bullets}\:\mathrm{do} \\ $$$$\mathrm{not}\:\mathrm{intersect}. \\ $$

Question Number 13891 Answers: 0 Comments: 1

Consider n red and n blue points in the plane, no three of them being collinear. Prove that one can connect each red point to a blue one with a segment such that no two segments intersect.

$$\mathrm{Consider}\:{n}\:\mathrm{red}\:\mathrm{and}\:{n}\:\mathrm{blue}\:\mathrm{points}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{plane},\:\mathrm{no}\:\mathrm{three}\:\mathrm{of}\:\mathrm{them}\:\mathrm{being}\:\mathrm{collinear}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{one}\:\mathrm{can}\:\mathrm{connect}\:\mathrm{each}\:\mathrm{red} \\ $$$$\mathrm{point}\:\mathrm{to}\:\mathrm{a}\:\mathrm{blue}\:\mathrm{one}\:\mathrm{with}\:\mathrm{a}\:\mathrm{segment} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{segments}\:\mathrm{intersect}. \\ $$

Question Number 13875 Answers: 0 Comments: 0

Question Number 13872 Answers: 1 Comments: 0

An alpha particle of mass 6.68 × 10^(−27) kg and charge q = +2, are accelerated from rest through the potential difference of 1kV. it then enters a magnetic field B = 0.2 T perpendicular to their direction of motion. Calculate the radius of their path.

$$\mathrm{An}\:\mathrm{alpha}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:\:\mathrm{6}.\mathrm{68}\:×\:\mathrm{10}^{−\mathrm{27}} \:\mathrm{kg}\:\mathrm{and}\:\mathrm{charge}\:\:\mathrm{q}\:=\:+\mathrm{2},\:\mathrm{are}\: \\ $$$$\mathrm{accelerated}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{through}\:\mathrm{the}\:\mathrm{potential}\:\mathrm{difference}\:\mathrm{of}\:\:\mathrm{1kV}.\:\mathrm{it}\:\mathrm{then}\:\mathrm{enters} \\ $$$$\mathrm{a}\:\mathrm{magnetic}\:\mathrm{field}\:\mathrm{B}\:=\:\mathrm{0}.\mathrm{2}\:\mathrm{T}\:\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{their}\:\mathrm{direction}\:\mathrm{of}\:\mathrm{motion}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{their}\:\mathrm{path}. \\ $$

Question Number 13871 Answers: 0 Comments: 0

why the function,sin(x) is a power series??

$${why}\:{the}\:{function},{sin}\left({x}\right)\:{is}\:{a}\:{power} \\ $$$${series}?? \\ $$

Question Number 13842 Answers: 3 Comments: 0

Find the number of solutions of the equation sin 5x cos 3x = sin 6x cos 2x, x ∈ [0, π]

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{sin}\:\mathrm{5}{x}\:\mathrm{cos}\:\mathrm{3}{x}\:=\:\mathrm{sin}\:\mathrm{6}{x}\:\mathrm{cos}\:\mathrm{2}{x}, \\ $$$${x}\:\in\:\left[\mathrm{0},\:\pi\right] \\ $$

Question Number 13840 Answers: 1 Comments: 0

Solve: (a) tanx + secx = 2cosx (b) sinθ + tanθ − sin2θ = 0

$$\mathrm{Solve}: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{tan}{x}\:+\:\mathrm{sec}{x}\:=\:\mathrm{2cos}{x} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{sin}\theta\:+\:\mathrm{tan}\theta\:−\:\mathrm{sin2}\theta\:=\:\mathrm{0} \\ $$

Question Number 13849 Answers: 0 Comments: 2

Given below is a graph between speed and time for a particle. Is the particle undergoing positive displacement or negative displacement?

$$\mathrm{Given}\:\mathrm{below}\:\mathrm{is}\:\mathrm{a}\:\mathrm{graph}\:\mathrm{between}\:\mathrm{speed} \\ $$$$\mathrm{and}\:\mathrm{time}\:\mathrm{for}\:\mathrm{a}\:\mathrm{particle}.\:\mathrm{Is}\:\mathrm{the}\:\mathrm{particle} \\ $$$$\mathrm{undergoing}\:\mathrm{positive}\:\mathrm{displacement}\:\mathrm{or} \\ $$$$\mathrm{negative}\:\mathrm{displacement}? \\ $$

Question Number 13835 Answers: 1 Comments: 1

Question Number 13830 Answers: 1 Comments: 0

Question Number 13831 Answers: 1 Comments: 0

Question Number 13832 Answers: 1 Comments: 0

Question Number 13812 Answers: 1 Comments: 0

The domain of f(x) = (√(((4/3))^(4−x) −(((27)/(64)))^(x−5) )); is?

$$\mathrm{The}\:\mathrm{domain}\:\mathrm{of}\:{f}\left({x}\right)\:= \\ $$$$\sqrt{\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{4}−{x}} −\left(\frac{\mathrm{27}}{\mathrm{64}}\right)^{{x}−\mathrm{5}} };\:\mathrm{is}? \\ $$

Question Number 13811 Answers: 1 Comments: 1

The domain of f(x) = (√(2 − 2^x − 2^(2x) )) is?

$$\mathrm{The}\:\mathrm{domain}\:\mathrm{of}\:{f}\left({x}\right)\:=\:\sqrt{\mathrm{2}\:−\:\mathrm{2}^{{x}} \:−\:\mathrm{2}^{\mathrm{2}{x}} }\:\mathrm{is}? \\ $$

Question Number 13808 Answers: 1 Comments: 0

If f(x) defined over the domain [0, 1] then domain of function f(10^x ) is?

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{defined}\:\mathrm{over}\:\mathrm{the}\:\mathrm{domain}\:\left[\mathrm{0},\:\mathrm{1}\right] \\ $$$$\mathrm{then}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{function}\:{f}\left(\mathrm{10}^{{x}} \right)\:\mathrm{is}? \\ $$

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.

Question Number 13735 Answers: 2 Comments: 6

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