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Question Number 24879 Answers: 0 Comments: 4

A particle of mass m is fixed to one end of a light rigid rod of length l and rotated in a vertical circular path about its other end. The minimum speed of the particle at its highest point must be

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{fixed}\:\mathrm{to}\:\mathrm{one}\:\mathrm{end} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{light}\:\mathrm{rigid}\:\mathrm{rod}\:\mathrm{of}\:\mathrm{length}\:{l}\:\mathrm{and}\:\mathrm{rotated} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{vertical}\:\mathrm{circular}\:\mathrm{path}\:\mathrm{about}\:\mathrm{its} \\ $$$$\mathrm{other}\:\mathrm{end}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{at}\:\mathrm{its}\:\mathrm{highest}\:\mathrm{point}\:\mathrm{must}\:\mathrm{be} \\ $$

Question Number 24877 Answers: 0 Comments: 7

Question Number 24866 Answers: 2 Comments: 0

3^x =2 2^y =4 (((3^x )^(4y−1) ×(3^(2x) )^(y+1) )/((3^(3y+2) )^x ))=

$$\mathrm{3}^{{x}} =\mathrm{2} \\ $$$$\mathrm{2}^{{y}} =\mathrm{4} \\ $$$$\frac{\left(\mathrm{3}^{{x}} \right)^{\mathrm{4}{y}−\mathrm{1}} ×\left(\mathrm{3}^{\mathrm{2}{x}} \right)^{{y}+\mathrm{1}} }{\left(\mathrm{3}^{\mathrm{3}{y}+\mathrm{2}} \right)^{{x}} }= \\ $$

Question Number 24865 Answers: 1 Comments: 0

((27^(x−1) +81^x )/3^(3x) )=(4/(27))

$$\frac{\mathrm{27}^{{x}−\mathrm{1}} +\mathrm{81}^{{x}} }{\mathrm{3}^{\mathrm{3}{x}} }=\frac{\mathrm{4}}{\mathrm{27}} \\ $$

Question Number 24864 Answers: 1 Comments: 0

Solve the following trigonometric limit: lim_(x → (π/4)) (5tg(x)) =

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{trigonometric}\:\mathrm{limit}: \\ $$$$ \\ $$$$\underset{\mathrm{x}\:\rightarrow\:\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\left(\mathrm{5tg}\left(\mathrm{x}\right)\right)\:=\: \\ $$

Question Number 24858 Answers: 2 Comments: 0

If three positive numbers a, b, c are in A.P. and (1/a^2 ), (1/b^2 ), (1/c^2 ) also in A.P., then (1) a = b = c (2) 2b = 3a + c (3) b^2 = ((ac)/8) (4) 2c = 2b + a

$$\mathrm{If}\:\mathrm{three}\:\mathrm{positive}\:\mathrm{numbers}\:{a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{in} \\ $$$$\mathrm{A}.\mathrm{P}.\:\mathrm{and}\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} },\:\frac{\mathrm{1}}{{b}^{\mathrm{2}} },\:\frac{\mathrm{1}}{{c}^{\mathrm{2}} }\:\mathrm{also}\:\mathrm{in}\:\mathrm{A}.\mathrm{P}.,\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{a}\:=\:{b}\:=\:{c} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2}{b}\:=\:\mathrm{3}{a}\:+\:{c} \\ $$$$\left(\mathrm{3}\right)\:{b}^{\mathrm{2}} \:=\:\frac{{ac}}{\mathrm{8}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}{c}\:=\:\mathrm{2}{b}\:+\:{a} \\ $$

Question Number 24852 Answers: 0 Comments: 1

please prove that∫_0 ^(π/2) log(sinx)dx=−(π/2)log2 or ∫_0 ^((π )/2) log(cosx)dx=−(π/2)log2

$$\mathrm{please}\:\mathrm{prove}\:\mathrm{that}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{log}\left(\mathrm{sinx}\right)\mathrm{dx}=−\frac{\pi}{\mathrm{2}}\mathrm{log2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{or} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi\:\:}{\mathrm{2}}} \mathrm{log}\left(\mathrm{cosx}\right)\mathrm{dx}=−\frac{\pi}{\mathrm{2}}\mathrm{log2} \\ $$

Question Number 24873 Answers: 0 Comments: 6

A rigid body is made of three identical thin rods, each of length L, fastened together in the form of letter H. The body is free to rotate about a horizontal axis that runs along the length of one of legs of H. The body is allowed to fall from rest from a position in which plane of H is horizontal. The angular speed of body when plane of H is vertical is

$$\mathrm{A}\:\mathrm{rigid}\:\mathrm{body}\:\mathrm{is}\:\mathrm{made}\:\mathrm{of}\:\mathrm{three}\:\mathrm{identical} \\ $$$$\mathrm{thin}\:\mathrm{rods},\:\mathrm{each}\:\mathrm{of}\:\mathrm{length}\:{L},\:\mathrm{fastened} \\ $$$$\mathrm{together}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{of}\:\mathrm{letter}\:{H}.\:\mathrm{The} \\ $$$$\mathrm{body}\:\mathrm{is}\:\mathrm{free}\:\mathrm{to}\:\mathrm{rotate}\:\mathrm{about}\:\mathrm{a}\:\mathrm{horizontal} \\ $$$$\mathrm{axis}\:\mathrm{that}\:\mathrm{runs}\:\mathrm{along}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{one}\:\mathrm{of} \\ $$$$\mathrm{legs}\:\mathrm{of}\:{H}.\:\mathrm{The}\:\mathrm{body}\:\mathrm{is}\:\mathrm{allowed}\:\mathrm{to}\:\mathrm{fall} \\ $$$$\mathrm{from}\:\mathrm{rest}\:\mathrm{from}\:\mathrm{a}\:\mathrm{position}\:\mathrm{in}\:\mathrm{which}\:\mathrm{plane} \\ $$$$\mathrm{of}\:{H}\:\mathrm{is}\:\mathrm{horizontal}.\:\mathrm{The}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\mathrm{of}\:\mathrm{body}\:\mathrm{when}\:\mathrm{plane}\:\mathrm{of}\:{H}\:\mathrm{is}\:\mathrm{vertical}\:\mathrm{is} \\ $$

Question Number 24911 Answers: 1 Comments: 1

find the derivative of ((((1−x)(√(3x−8)))/(sin^2 (1−5x))))^(1/4) plzzz help

$$\mathrm{find}\:\mathrm{the}\:\mathrm{derivative}\:\mathrm{of}\: \\ $$$$\sqrt[{\mathrm{4}}]{\frac{\left(\mathrm{1}−\mathrm{x}\right)\sqrt{\mathrm{3x}−\mathrm{8}}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{5x}\right)}} \\ $$$$\mathrm{plzzz}\:\mathrm{help}\: \\ $$

Question Number 24840 Answers: 1 Comments: 1

Question Number 24839 Answers: 2 Comments: 1

((5^(2x+1) +5^(2x) )/(5^(2x) +5^(2x−1) ))=(0/04)^(x−1)

$$\frac{\mathrm{5}^{\mathrm{2}{x}+\mathrm{1}} +\mathrm{5}^{\mathrm{2}{x}} }{\mathrm{5}^{\mathrm{2}{x}} +\mathrm{5}^{\mathrm{2}{x}−\mathrm{1}} }=\left(\mathrm{0}/\mathrm{04}\right)^{{x}−\mathrm{1}} \\ $$

Question Number 24834 Answers: 0 Comments: 1

how many thirds are there in 1/3?

$${how}\:{many}\:{thirds}\:{are}\:{there}\:{in}\:\mathrm{1}/\mathrm{3}? \\ $$

Question Number 24831 Answers: 1 Comments: 0

∫_1 ^2 ∫_1 ^2 ln(x+y)dx dy

$$ \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} \int_{\mathrm{1}} ^{\mathrm{2}} {ln}\left({x}+{y}\right){dx}\:{dy} \\ $$

Question Number 24828 Answers: 1 Comments: 3

∫(√(e^x +1))dx=?

$$\int\sqrt{{e}^{{x}} +\mathrm{1}}{dx}=? \\ $$

Question Number 24825 Answers: 1 Comments: 3

∫_0 ^( 2) ∫_x ^( 3x − x^2 ) 6x^2 − 2xy dy dx = ?

$$\int_{\mathrm{0}} ^{\:\mathrm{2}} \int_{\mathrm{x}} ^{\:\:\mathrm{3x}\:−\:\mathrm{x}^{\mathrm{2}} } \:\mathrm{6x}^{\mathrm{2}} \:−\:\mathrm{2xy}\:\mathrm{dy}\:\mathrm{dx}\:=\:? \\ $$

Question Number 24822 Answers: 1 Comments: 1

prove that 3^n −1 is a multiple of 2 by mathematical induction

$${prove}\:{that}\:\mathrm{3}^{{n}} −\mathrm{1}\:{is}\:{a}\:{multiple}\:{of}\:\mathrm{2} \\ $$$${by}\:{mathematical}\:{induction} \\ $$

Question Number 24821 Answers: 1 Comments: 0

Draw the graph of the function ((f/g))(x) if f,g:R→R are given by f(x)=2x−1,g(x)=x+1.Find the domain and the range of ((f/g))(x)

$${Draw}\:{the}\:{graph}\:{of}\:{the}\:{function} \\ $$$$\left(\frac{{f}}{{g}}\right)\left({x}\right)\:{if}\:{f},{g}:\mathbb{R}\rightarrow\mathbb{R}\:{are}\:{given}\:{by} \\ $$$${f}\left({x}\right)=\mathrm{2}{x}−\mathrm{1},{g}\left({x}\right)={x}+\mathrm{1}.{Find}\:{the} \\ $$$${domain}\:{and}\:{the}\:{range}\:{of}\:\left(\frac{{f}}{{g}}\right)\left({x}\right) \\ $$$$ \\ $$

Question Number 24817 Answers: 1 Comments: 0

Question Number 24814 Answers: 0 Comments: 3

Question Number 24813 Answers: 0 Comments: 1

please find value of x 2x+2=0

$$\mathrm{please}\:\mathrm{find}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$$$\mathrm{2x}+\mathrm{2}=\mathrm{0} \\ $$

Question Number 24800 Answers: 1 Comments: 0

A closed box measures externally 9dm long, 6dm broad, 4(1/2)dm high, and is made of wood 2(1/2)cm thick. Find the cost of lining it on the inside with metal at 6 paise per sq.m.

$$\mathrm{A}\:\mathrm{closed}\:\mathrm{box}\:\mathrm{measures}\:\mathrm{externally}\: \\ $$$$\mathrm{9dm}\:\mathrm{long},\:\mathrm{6dm}\:\mathrm{broad},\:\mathrm{4}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{dm}\:\mathrm{high},\:\mathrm{and} \\ $$$$\mathrm{is}\:\mathrm{made}\:\mathrm{of}\:\mathrm{wood}\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cm}\:\mathrm{thick}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{cost}\:\mathrm{of}\:\mathrm{lining}\:\mathrm{it}\:\mathrm{on}\:\mathrm{the}\:\mathrm{inside}\:\mathrm{with}\:\mathrm{metal} \\ $$$$\mathrm{at}\:\mathrm{6}\:\mathrm{paise}\:\mathrm{per}\:\mathrm{sq}.\mathrm{m}. \\ $$

Question Number 24786 Answers: 1 Comments: 1

A particle of mass m is moving in yz-plane with a uniform velocity v with its trajectory running parallel to +ve y- axis and intersecting z-axis at z = a. The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is : (1) mvae_x ^∧ (2) 2mvae_x ^∧ (3) ymve_x ^∧ (4) 2ymve_x ^∧

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{in}\:{yz}-\mathrm{plane} \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{uniform}\:\mathrm{velocity}\:{v}\:\mathrm{with}\:\mathrm{its} \\ $$$$\mathrm{trajectory}\:\mathrm{running}\:\mathrm{parallel}\:\mathrm{to}\:+\mathrm{ve}\:{y}- \\ $$$$\mathrm{axis}\:\mathrm{and}\:\mathrm{intersecting}\:{z}-\mathrm{axis}\:\mathrm{at}\:{z}\:=\:{a}. \\ $$$$\mathrm{The}\:\mathrm{change}\:\mathrm{in}\:\mathrm{its}\:\mathrm{angular}\:\mathrm{momentum} \\ $$$$\mathrm{about}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{as}\:\mathrm{it}\:\mathrm{bounces}\:\mathrm{elastically} \\ $$$$\mathrm{from}\:\mathrm{a}\:\mathrm{wall}\:\mathrm{at}\:{y}\:=\:\mathrm{constant}\:\mathrm{is}\:: \\ $$$$\left(\mathrm{1}\right)\:{mva}\overset{\wedge} {{e}}_{{x}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2}{mva}\overset{\wedge} {{e}}_{{x}} \\ $$$$\left(\mathrm{3}\right)\:{ymv}\overset{\wedge} {{e}}_{{x}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}{ymv}\overset{\wedge} {{e}}_{{x}} \\ $$

Question Number 24778 Answers: 0 Comments: 4

Show that the shortest distance between two opposite edges a,d of a tetrahedron is 6V/adsin 𝛉, where θ is the angle between the edges and V is the volume of the tetrahedron.

$${Show}\:{that}\:{the}\:{shortest}\:{distance} \\ $$$${between}\:{two}\:{opposite}\:{edges}\:\boldsymbol{{a}},\boldsymbol{{d}}\: \\ $$$${of}\:{a}\:{tetrahedron}\:{is}\:\mathrm{6}\boldsymbol{{V}}/\boldsymbol{{ad}}\mathrm{sin}\:\boldsymbol{\theta}, \\ $$$${where}\:\theta\:{is}\:{the}\:{angle}\:{between}\:{the} \\ $$$${edges}\:{and}\:{V}\:{is}\:{the}\:{volume}\:{of}\:{the} \\ $$$${tetrahedron}. \\ $$

Question Number 24772 Answers: 0 Comments: 13

The ratio of acceleration of points A, B and C is [assume all surfaces are smooth, pulley and strings are light]

$$\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{points}\:{A}, \\ $$$${B}\:\mathrm{and}\:{C}\:\mathrm{is}\:\left[\mathrm{assume}\:\mathrm{all}\:\mathrm{surfaces}\:\mathrm{are}\right. \\ $$$$\left.\mathrm{smooth},\:\mathrm{pulley}\:\mathrm{and}\:\mathrm{strings}\:\mathrm{are}\:\mathrm{light}\right] \\ $$

Question Number 24764 Answers: 2 Comments: 0

Given that the function f:R→R is defined by f(x)=x^n .For what values of n,if any,is fof=f.f? For each of these values of n find fof.

$${Given}\:{that}\:{the}\:{function}\:{f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${is}\:{defined}\:{by}\:{f}\left({x}\right)={x}^{{n}} .{For}\:{what} \\ $$$${values}\:{of}\:{n},{if}\:{any},{is}\:{fof}={f}.{f}? \\ $$$${For}\:{each}\:{of}\:{these}\:{values}\:{of}\:{n}\:{find} \\ $$$${fof}. \\ $$

Question Number 24755 Answers: 0 Comments: 0

Given f(x) =Σ_(x=1) ^n tan((x/2^r )).sec((x/2^(r−1) )) where r and n εN g(x) =lim_(n→∝) ((ln(f(x)+tan(x/2^n )) −(f(x)+tan(x/2^n )).[sin(tan(x/2)))/(1+(f(x) + tan(x/2^n ))^n )) = k for x =(π/4) and the domain of g(x) is (0 ,(π/2)) where [.] denotes the g.i.f Find the value of k, if possible so that g(x) is continuous at x =(π/4) .

$${Given} \\ $$$${f}\left({x}\right)\:=\underset{{x}=\mathrm{1}} {\overset{{n}} {\sum}}{tan}\left(\frac{{x}}{\mathrm{2}^{{r}} }\right).{sec}\left(\frac{{x}}{\mathrm{2}^{{r}−\mathrm{1}} }\right)\: \\ $$$$\:\:\:\:\:\:\:\:\:{where}\:{r}\:{and}\:{n}\:\varepsilon{N} \\ $$$${g}\left({x}\right)\:=\underset{{n}\rightarrow\propto} {\mathrm{li}{m}}\:\:\frac{{ln}\left({f}\left({x}\right)+{tan}\frac{{x}}{\mathrm{2}^{{n}} }\right)\:−\left({f}\left({x}\right)+{tan}\frac{{x}}{\mathrm{2}^{{n}} }\right).\left[{sin}\left({tan}\frac{{x}}{\mathrm{2}}\right)\right.}{\mathrm{1}+\left({f}\left({x}\right)\:\:+\:\:{tan}\frac{{x}}{\mathrm{2}^{{n}} }\right)^{{n}} }\:\:=\:{k} \\ $$$${for}\:{x}\:=\frac{\pi}{\mathrm{4}}\:\:{and}\:{the}\:{domain}\:{of} \\ $$$${g}\left({x}\right)\:{is}\:\left(\mathrm{0}\:,\frac{\pi}{\mathrm{2}}\right) \\ $$$${where}\:\left[.\right]\:{denotes}\:{the}\:{g}.{i}.\mathrm{f} \\ $$$${Find}\:{the}\:{value}\:{of}\:{k},\:{if}\:{possible}\: \\ $$$${so}\:{that}\:{g}\left({x}\right)\:{is}\:{continuous}\:{at}\: \\ $$$${x}\:=\frac{\pi}{\mathrm{4}}\:. \\ $$$$ \\ $$$$ \\ $$

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