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In a quadrilateral ABCD, it is given that AB is parallel to CD and the diagonals AC and BD are perpendicular to each other. Show that (a) AD.BC ≥ AB.CD; (b) AD + BC ≥ AB + CD.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{quadrilateral}\:{ABCD},\:\mathrm{it}\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{that}\:{AB}\:\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:{CD}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{diagonals}\:{AC}\:\mathrm{and}\:{BD}\:\mathrm{are}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$$$\mathrm{Show}\:\mathrm{that} \\ $$$$\left(\mathrm{a}\right)\:{AD}.{BC}\:\geqslant\:{AB}.{CD}; \\ $$$$\left(\mathrm{b}\right)\:{AD}\:+\:{BC}\:\geqslant\:{AB}\:+\:{CD}. \\ $$

Question Number 22503 Answers: 1 Comments: 4

If (a + bx)^(−2) = (1/4) − 3x + ..., then (a, b) =

$$\mathrm{If}\:\left({a}\:+\:{bx}\right)^{−\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:\mathrm{3}{x}\:+\:...,\:\mathrm{then}\:\left({a},\:{b}\right)\:= \\ $$

Question Number 22499 Answers: 1 Comments: 1

A cylinder of weight 200 N is supported on a smooth horizontal plane by a light cord AC and pulled with force of 400 N. The normal reaction at B is equal to

$$\mathrm{A}\:\mathrm{cylinder}\:\mathrm{of}\:\mathrm{weight}\:\mathrm{200}\:\mathrm{N}\:\mathrm{is}\:\mathrm{supported} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{smooth}\:\mathrm{horizontal}\:\mathrm{plane}\:\mathrm{by}\:\mathrm{a}\:\mathrm{light} \\ $$$$\mathrm{cord}\:{AC}\:\mathrm{and}\:\mathrm{pulled}\:\mathrm{with}\:\mathrm{force}\:\mathrm{of}\:\mathrm{400}\:\mathrm{N}. \\ $$$$\mathrm{The}\:\mathrm{normal}\:\mathrm{reaction}\:\mathrm{at}\:{B}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 22492 Answers: 1 Comments: 1

A ball of mass 400 g travels horizontally along the ground and collides with a wall. The velocity-time graph below represents the motion of the ball for the first 1.2 seconds. The magnitude of average force between the ball and the wall is

$$\mathrm{A}\:\mathrm{ball}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{400}\:\mathrm{g}\:\mathrm{travels}\:\mathrm{horizontally} \\ $$$$\mathrm{along}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{and}\:\mathrm{collides}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{wall}.\:\mathrm{The}\:\mathrm{velocity}-\mathrm{time}\:\mathrm{graph}\:\mathrm{below} \\ $$$$\mathrm{represents}\:\mathrm{the}\:\mathrm{motion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{for}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{1}.\mathrm{2}\:\mathrm{seconds}. \\ $$$$\mathrm{The}\:\mathrm{magnitude}\:\mathrm{of}\:\mathrm{average}\:\mathrm{force}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{ball}\:\mathrm{and}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{is} \\ $$

Question Number 22491 Answers: 1 Comments: 0

The coefficient of x^r in the expansion of (1 − 2x)^(−1/2) is (1) (((2r)!)/((r!)^2 )) (2) (((2r)!)/(2^r (r!)^2 )) (3) (((2r)!)/((r!)^2 2^(2r) )) (4) (((2r)!)/(2^r (r + 1)!(r − 1)!))

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}\:−\:\mathrm{2}{x}\right)^{−\mathrm{1}/\mathrm{2}} \:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\left({r}!\right)^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\mathrm{2}^{{r}} \:\left({r}!\right)^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\left({r}!\right)^{\mathrm{2}} \:\mathrm{2}^{\mathrm{2}{r}} } \\ $$$$\left(\mathrm{4}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\mathrm{2}^{{r}} \:\left({r}\:+\:\mathrm{1}\right)!\left({r}\:−\:\mathrm{1}\right)!} \\ $$

Question Number 22487 Answers: 1 Comments: 1

Question Number 22483 Answers: 0 Comments: 0

Predict the density of Cs from the density of the following elements K 0.86 g/cm^3 Ca 1.548 g/cm^3 Sc 2.991 g/cm^3 Rb 1.532 g/cm^3 Sr 2.68 g/cm^3 Y 4.34 g/cm^3 Cs ? Ba 3.51 g/cm^3 La 6.16 g/cm^3

$$\mathrm{Predict}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\:\mathrm{Cs}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{density}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{elements} \\ $$$$\mathrm{K}\:\mathrm{0}.\mathrm{86}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \:\:\:\:\:\:\:\:\mathrm{Ca}\:\mathrm{1}.\mathrm{548}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \\ $$$$\mathrm{Sc}\:\mathrm{2}.\mathrm{991}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \:\:\:\:\:\mathrm{Rb}\:\mathrm{1}.\mathrm{532}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \\ $$$$\mathrm{Sr}\:\mathrm{2}.\mathrm{68}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \:\:\:\:\:\:\:\:\mathrm{Y}\:\mathrm{4}.\mathrm{34}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \\ $$$$\mathrm{Cs}\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Ba}\:\mathrm{3}.\mathrm{51}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{La}\:\mathrm{6}.\mathrm{16}\:\mathrm{g}/\mathrm{cm}^{\mathrm{3}} \\ $$

Question Number 22481 Answers: 1 Comments: 1

Question Number 22479 Answers: 0 Comments: 2

A ladder of mass m is leaning against a wall. It is in static equilibrium making an angle θ with the horizontal floor. The coefficient of friction between the wall and the ladder is μ_1 and that between the floor and the ladder is μ_2 . The normal reaction of the wall on the ladder is N_1 and that of the floor is N_2 . If the ladder is about to slip, then (1) μ_1 = 0, μ_2 ≠ 0 and N_2 tan θ = mg/2 (2) μ_1 ≠ 0, μ_2 = 0 and N_1 tan θ = mg/2 (3) μ_1 ≠ 0, μ_2 ≠ 0 and N_2 = ((mg)/(1 + μ_1 μ_2 )) (4) μ_1 = 0, μ_2 ≠ 0 and N_1 tan θ = ((mg)/2)

$$\mathrm{A}\:\mathrm{ladder}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{leaning}\:\mathrm{against}\:\mathrm{a} \\ $$$$\mathrm{wall}.\:\mathrm{It}\:\mathrm{is}\:\mathrm{in}\:\mathrm{static}\:\mathrm{equilibrium}\:\mathrm{making} \\ $$$$\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{floor}. \\ $$$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{wall}\:\mathrm{and}\:\mathrm{the}\:\mathrm{ladder}\:\mathrm{is}\:\mu_{\mathrm{1}} \:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{between}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{and}\:\mathrm{the}\:\mathrm{ladder}\:\mathrm{is}\:\mu_{\mathrm{2}} . \\ $$$$\mathrm{The}\:\mathrm{normal}\:\mathrm{reaction}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{ladder}\:\mathrm{is}\:\mathrm{N}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{is}\:\mathrm{N}_{\mathrm{2}} . \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{ladder}\:\mathrm{is}\:\mathrm{about}\:\mathrm{to}\:\mathrm{slip},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0},\:\mu_{\mathrm{2}} \:\neq\:\mathrm{0}\:\mathrm{and}\:{N}_{\mathrm{2}} \:\mathrm{tan}\:\theta\:=\:{mg}/\mathrm{2} \\ $$$$\left(\mathrm{2}\right)\:\mu_{\mathrm{1}} \:\neq\:\mathrm{0},\:\mu_{\mathrm{2}} \:=\:\mathrm{0}\:\mathrm{and}\:{N}_{\mathrm{1}} \:\mathrm{tan}\:\theta\:=\:{mg}/\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\mu_{\mathrm{1}} \:\neq\:\mathrm{0},\:\mu_{\mathrm{2}} \:\neq\:\mathrm{0}\:\mathrm{and}\:{N}_{\mathrm{2}} \:=\:\frac{{mg}}{\mathrm{1}\:+\:\mu_{\mathrm{1}} \mu_{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0},\:\mu_{\mathrm{2}} \:\neq\:\mathrm{0}\:\mathrm{and}\:{N}_{\mathrm{1}} \:\mathrm{tan}\:\theta\:=\:\frac{{mg}}{\mathrm{2}} \\ $$

Question Number 22478 Answers: 0 Comments: 1

{ ((x+y^2 +z^3 =3)),((y+z^2 +x^3 =3)),((z+x^2 +z^3 =3)) :} trouver les solutions positives

$$\begin{cases}{{x}+{y}^{\mathrm{2}} +{z}^{\mathrm{3}} =\mathrm{3}}\\{{y}+{z}^{\mathrm{2}} +{x}^{\mathrm{3}} =\mathrm{3}}\\{{z}+{x}^{\mathrm{2}} +{z}^{\mathrm{3}} =\mathrm{3}}\end{cases} \\ $$$${trouver}\:{les}\:{solutions}\:{positives} \\ $$$$ \\ $$

Question Number 22696 Answers: 1 Comments: 5

A force F^→ = 2xj^∧ newton acts in a region where a particle moves anticlockwise in a square loop of 2 m in x-y plane. Calculate the total amount of work done. Is this force a conservative force or a non-conservative force?

$$\mathrm{A}\:\mathrm{force}\:\overset{\rightarrow} {{F}}\:=\:\mathrm{2}{x}\overset{\wedge} {{j}}\:\mathrm{newton}\:\mathrm{acts}\:\mathrm{in}\:\mathrm{a}\:\mathrm{region} \\ $$$$\mathrm{where}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{anticlockwise} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{square}\:\mathrm{loop}\:\mathrm{of}\:\mathrm{2}\:\mathrm{m}\:\mathrm{in}\:{x}-{y}\:\mathrm{plane}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{total}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{work} \\ $$$$\mathrm{done}.\:\mathrm{Is}\:\mathrm{this}\:\mathrm{force}\:\mathrm{a}\:\mathrm{conservative}\:\mathrm{force} \\ $$$$\mathrm{or}\:\mathrm{a}\:\mathrm{non}-\mathrm{conservative}\:\mathrm{force}? \\ $$

Question Number 22475 Answers: 0 Comments: 1

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