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Question Number 18107    Answers: 0   Comments: 0

The first and second ionization potentials of helium atoms are 24.58 eV and 54.4 eV per mole respectively. Calculate the energy in kJ required to produce 1 mole of He^(2+) ions.

$$\mathrm{The}\:\mathrm{first}\:\mathrm{and}\:\mathrm{second}\:\mathrm{ionization} \\ $$$$\mathrm{potentials}\:\mathrm{of}\:\mathrm{helium}\:\mathrm{atoms}\:\mathrm{are}\:\mathrm{24}.\mathrm{58}\:\mathrm{eV} \\ $$$$\mathrm{and}\:\mathrm{54}.\mathrm{4}\:\mathrm{eV}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{respectively}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{in}\:\mathrm{kJ}\:\mathrm{required}\:\mathrm{to} \\ $$$$\mathrm{produce}\:\mathrm{1}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{He}^{\mathrm{2}+} \:\mathrm{ions}. \\ $$

Question Number 18106    Answers: 0   Comments: 0

The ionization potential of hydrogen is 13.60 eV/mole. Calculate the energy in kJ required to produce 0.1 mole of H^+ ions. Given, 1 eV = 96.49 kJ mol^(−1) )

$$\mathrm{The}\:\mathrm{ionization}\:\mathrm{potential}\:\mathrm{of}\:\mathrm{hydrogen}\:\mathrm{is} \\ $$$$\mathrm{13}.\mathrm{60}\:\mathrm{eV}/\mathrm{mole}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{in} \\ $$$$\mathrm{kJ}\:\mathrm{required}\:\mathrm{to}\:\mathrm{produce}\:\mathrm{0}.\mathrm{1}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{H}^{+} \\ $$$$\left.\mathrm{ions}.\:\mathrm{Given},\:\mathrm{1}\:\mathrm{eV}\:=\:\mathrm{96}.\mathrm{49}\:\mathrm{kJ}\:\mathrm{mol}^{−\mathrm{1}} \right) \\ $$

Question Number 18095    Answers: 1   Comments: 0

A boy travelling in an open car moving on a levelled road with constant speed tosses a ball vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy standing on the footpath. Give explanation to support your diagram.

$$\mathrm{A}\:\mathrm{boy}\:\mathrm{travelling}\:\mathrm{in}\:\mathrm{an}\:\mathrm{open}\:\mathrm{car}\:\mathrm{moving} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{levelled}\:\mathrm{road}\:\mathrm{with}\:\mathrm{constant}\:\mathrm{speed} \\ $$$$\mathrm{tosses}\:\mathrm{a}\:\mathrm{ball}\:\mathrm{vertically}\:\mathrm{up}\:\mathrm{in}\:\mathrm{the}\:\mathrm{air}\:\mathrm{and} \\ $$$$\mathrm{catches}\:\mathrm{it}\:\mathrm{back}.\:\mathrm{Sketch}\:\mathrm{the}\:\mathrm{motion}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{ball}\:\mathrm{as}\:\mathrm{observed}\:\mathrm{by}\:\mathrm{a}\:\mathrm{boy}\:\mathrm{standing} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{footpath}.\:\mathrm{Give}\:\mathrm{explanation}\:\mathrm{to} \\ $$$$\mathrm{support}\:\mathrm{your}\:\mathrm{diagram}. \\ $$

Question Number 18094    Answers: 0   Comments: 3

A^→ , B^(→) and C^(→) are three non-collinear, non co-planar vectors. What can you say about direction of A^(→) ×(B^(→) ×C^(→) )?

$$\overset{\rightarrow} {{A}},\:\overset{\rightarrow} {{B}}\:\mathrm{and}\:\overset{\rightarrow} {{C}}\:\mathrm{are}\:\mathrm{three}\:\mathrm{non}-\mathrm{collinear}, \\ $$$$\mathrm{non}\:\mathrm{co}-\mathrm{planar}\:\mathrm{vectors}.\:\mathrm{What}\:\mathrm{can}\:\mathrm{you} \\ $$$$\mathrm{say}\:\mathrm{about}\:\mathrm{direction}\:\mathrm{of}\:\overset{\rightarrow} {{A}}×\left(\overset{\rightarrow} {{B}}×\overset{\rightarrow} {{C}}\right)? \\ $$

Question Number 18093    Answers: 1   Comments: 1

The equation sinx + sin2x + 2sinxsin2x = 2cosx + cos2x is satisfied by values of x for which (1) x = nπ + (−1)^n (π/6) , n ∈ I (2) x = 2nπ + ((2π)/3) , n ∈ I (3) x = 2nπ − ((2π)/3) , n ∈ I (4) x = 2nπ − (π/2) , n ∈ I

$$\mathrm{The}\:\mathrm{equation}\:\mathrm{sin}{x}\:+\:\mathrm{sin2}{x}\:+\:\mathrm{2sin}{x}\mathrm{sin2}{x} \\ $$$$=\:\mathrm{2cos}{x}\:+\:\mathrm{cos2}{x}\:\mathrm{is}\:\mathrm{satisfied}\:\mathrm{by}\:\mathrm{values} \\ $$$$\mathrm{of}\:{x}\:\mathrm{for}\:\mathrm{which} \\ $$$$\left(\mathrm{1}\right)\:{x}\:=\:{n}\pi\:+\:\left(−\mathrm{1}\right)^{{n}} \frac{\pi}{\mathrm{6}}\:,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{2}\right)\:{x}\:=\:\mathrm{2}{n}\pi\:+\:\frac{\mathrm{2}\pi}{\mathrm{3}}\:,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{3}\right)\:{x}\:=\:\mathrm{2}{n}\pi\:−\:\frac{\mathrm{2}\pi}{\mathrm{3}}\:,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{4}\right)\:{x}\:=\:\mathrm{2}{n}\pi\:−\:\frac{\pi}{\mathrm{2}}\:,\:{n}\:\in\:{I} \\ $$

Question Number 18092    Answers: 1   Comments: 0

A value of θ satisfying 4cos^2 θsinθ − 2sin^2 θ = 3sinθ is (1) ((9π)/(10)) (2) (π/(10)) (3) −((13π)/(10)) (4) −((17π)/(10))

$$\mathrm{A}\:\mathrm{value}\:\mathrm{of}\:\theta\:\mathrm{satisfying} \\ $$$$\mathrm{4cos}^{\mathrm{2}} \theta\mathrm{sin}\theta\:−\:\mathrm{2sin}^{\mathrm{2}} \theta\:=\:\mathrm{3sin}\theta\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{9}\pi}{\mathrm{10}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\pi}{\mathrm{10}} \\ $$$$\left(\mathrm{3}\right)\:−\frac{\mathrm{13}\pi}{\mathrm{10}} \\ $$$$\left(\mathrm{4}\right)\:−\frac{\mathrm{17}\pi}{\mathrm{10}} \\ $$

Question Number 18091    Answers: 1   Comments: 0

Which of the following statement(s) is/are correct? (1) cos(sin1) > sin(cos1) (2) cos(sin1.5) > sin(cos1.5) (3) cos(sin((7π)/(18))) > sin(cos((7π)/(18))) (4) cos(sin((5π)/(18))) > sin(cos((5π)/(18)))

$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statement}\left(\mathrm{s}\right) \\ $$$$\mathrm{is}/\mathrm{are}\:\mathrm{correct}? \\ $$$$\left(\mathrm{1}\right)\:\mathrm{cos}\left(\mathrm{sin1}\right)\:>\:\mathrm{sin}\left(\mathrm{cos1}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{cos}\left(\mathrm{sin1}.\mathrm{5}\right)\:>\:\mathrm{sin}\left(\mathrm{cos1}.\mathrm{5}\right) \\ $$$$\left(\mathrm{3}\right)\:\mathrm{cos}\left(\mathrm{sin}\frac{\mathrm{7}\pi}{\mathrm{18}}\right)\:>\:\mathrm{sin}\left(\mathrm{cos}\frac{\mathrm{7}\pi}{\mathrm{18}}\right) \\ $$$$\left(\mathrm{4}\right)\:\mathrm{cos}\left(\mathrm{sin}\frac{\mathrm{5}\pi}{\mathrm{18}}\right)\:>\:\mathrm{sin}\left(\mathrm{cos}\frac{\mathrm{5}\pi}{\mathrm{18}}\right) \\ $$

Question Number 18077    Answers: 1   Comments: 0

A mango in a tree is located (30,40) from the point of projection of stone.Find the minimum speed and the angle of projevtion of the stone so as to hit the mango

$${A}\:{mango}\:{in}\:{a}\:{tree}\:{is}\:{located}\:\left(\mathrm{30},\mathrm{40}\right)\:{from}\:{the} \\ $$$${point}\:{of}\:{projection}\:{of}\:{stone}.{Find}\:{the}\: \\ $$$${minimum}\:{speed}\:{and}\:{the}\:{angle}\:{of}\:{projevtion} \\ $$$${of}\:{the}\:{stone}\:{so}\:{as}\:{to}\:{hit}\:{the}\:{mango} \\ $$

Question Number 18069    Answers: 0   Comments: 1

ai) If θ is the angle in the fourth quadrant satisfying the equation : cot^2 θ = 4 find the value of the function: f(θ) = (1/(√5)) (secθ − cosecθ) aii) Prove that: (√((1 + cosθ)/(1 − cosθ))) = cosecθ + cotθ, if cosθ ≠ 1 (b) Let R be a positive real number and let α satisfy the inequality 0 < α < 360. express the function 2sinθ + cosθ in the form Rsin(θ + α). Hence, find the value of θ between 0 and 360 which satisfy the equation. 3cosθ + 6sinθ = 1

$$\left.\mathrm{ai}\right)\:\:\mathrm{If}\:\theta\:\mathrm{is}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{in}\:\mathrm{the}\:\mathrm{fourth}\:\mathrm{quadrant}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equation}\::\:\mathrm{cot}^{\mathrm{2}} \theta\:=\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}:\:\:\mathrm{f}\left(\theta\right)\:=\:\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}\:\left(\mathrm{sec}\theta\:−\:\mathrm{cosec}\theta\right) \\ $$$$\left.\mathrm{aii}\right)\:\:\mathrm{Prove}\:\mathrm{that}:\:\:\:\sqrt{\frac{\mathrm{1}\:+\:\mathrm{cos}\theta}{\mathrm{1}\:−\:\mathrm{cos}\theta}}\:\:=\:\:\mathrm{cosec}\theta\:+\:\mathrm{cot}\theta,\:\:\:\:\:\:\:\:\mathrm{if}\:\:\mathrm{cos}\theta\:\neq\:\mathrm{1} \\ $$$$\left(\mathrm{b}\right)\:\:\:\mathrm{Let}\:\:\mathrm{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\:\mathrm{and}\:\mathrm{let}\:\alpha\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{inequality}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:<\:\alpha\:<\:\mathrm{360}.\:\mathrm{express}\:\mathrm{the}\:\mathrm{function}\:\:\mathrm{2sin}\theta\:+\:\mathrm{cos}\theta\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\:\mathrm{Rsin}\left(\theta\:+\:\alpha\right). \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Hence},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\theta\:\mathrm{between}\:\mathrm{0}\:\mathrm{and}\:\mathrm{360}\:\mathrm{which}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3cos}\theta\:+\:\mathrm{6sin}\theta\:=\:\mathrm{1} \\ $$

Question Number 18066    Answers: 0   Comments: 1

(a) Evaluate the integral of the function: y(x) = ((3x + 1)/(2x^2 − 2x + 3)) (b) Find the constant A, B, C in the identity: ((3x^2 − ax)/((x − 2a)(x^2 + a^2 ))) ≡ (A/((x − 2a))) + ((Bx + Ca)/((x^2 + a^2 ))) where a is a constant, hence prove that. ∫_0 ^( 2) ((3x^2 − ax)/((x − 2a)(x^2 + a^2 ))) dx = (π/4) − (3/2) ln(2)

$$\left(\mathrm{a}\right)\:\:\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}:\:\:\mathrm{y}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{3x}\:+\:\mathrm{1}}{\mathrm{2x}^{\mathrm{2}} \:−\:\mathrm{2x}\:+\:\mathrm{3}} \\ $$$$\left(\mathrm{b}\right)\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{A},\:\mathrm{B},\:\mathrm{C}\:\mathrm{in}\:\mathrm{the}\:\mathrm{identity}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{3x}^{\mathrm{2}} \:−\:\mathrm{ax}}{\left(\mathrm{x}\:−\:\mathrm{2a}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{a}^{\mathrm{2}} \right)}\:\equiv\:\frac{\mathrm{A}}{\left(\mathrm{x}\:−\:\mathrm{2a}\right)}\:+\:\frac{\mathrm{Bx}\:+\:\mathrm{Ca}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{a}^{\mathrm{2}} \right)} \\ $$$$\mathrm{where}\:\:\mathrm{a}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant},\:\:\mathrm{hence}\:\mathrm{prove}\:\mathrm{that}.\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\:\frac{\mathrm{3x}^{\mathrm{2}} \:−\:\mathrm{ax}}{\left(\mathrm{x}\:−\:\mathrm{2a}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{a}^{\mathrm{2}} \right)}\:\mathrm{dx}\:=\:\frac{\pi}{\mathrm{4}}\:−\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{ln}\left(\mathrm{2}\right) \\ $$

Question Number 18063    Answers: 1   Comments: 3

The angles A, B, C of a triangle ABC satisfy 4cosAcosB + sin2A + sin2B + sin2C = 4. Then which of the following statements is/are correct? (1) The triangle ABC is right angled (2) The triangle ABC is isosceles (3) The triangle ABC is neither isosceles nor right angled (4) The triangle ABC is equilateral

$$\mathrm{The}\:\mathrm{angles}\:{A},\:{B},\:{C}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:{ABC} \\ $$$$\mathrm{satisfy}\:\mathrm{4cos}{A}\mathrm{cos}{B}\:+\:\mathrm{sin2}{A}\:+\:\mathrm{sin2}{B}\:+ \\ $$$$\mathrm{sin2}{C}\:=\:\mathrm{4}.\:\mathrm{Then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{statements}\:\mathrm{is}/\mathrm{are}\:\mathrm{correct}? \\ $$$$\left(\mathrm{1}\right)\:\mathrm{The}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{right}\:\mathrm{angled} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{The}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{isosceles} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{The}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{neither} \\ $$$$\mathrm{isosceles}\:\mathrm{nor}\:\mathrm{right}\:\mathrm{angled} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{The}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{equilateral} \\ $$

Question Number 18062    Answers: 1   Comments: 0

If 0 < α, β < π and they satisfy cos α + cos β − cos (α + β) = (3/2) (1) α = β (2) α + β = ((2π)/3) (3) α = 2β (4) β = 2α

$$\mathrm{If}\:\mathrm{0}\:<\:\alpha,\:\beta\:<\:\pi\:\mathrm{and}\:\mathrm{they}\:\mathrm{satisfy} \\ $$$$\mathrm{cos}\:\alpha\:+\:\mathrm{cos}\:\beta\:−\:\mathrm{cos}\:\left(\alpha\:+\:\beta\right)\:=\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left(\mathrm{1}\right)\:\alpha\:=\:\beta \\ $$$$\left(\mathrm{2}\right)\:\alpha\:+\:\beta\:=\:\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$$\left(\mathrm{3}\right)\:\alpha\:=\:\mathrm{2}\beta \\ $$$$\left(\mathrm{4}\right)\:\beta\:=\:\mathrm{2}\alpha \\ $$

Question Number 18501    Answers: 0   Comments: 2

In a triangle ABC (1) sinA.sinB.sinC = (Δ/(2R^2 )) (2) sinA.sinB.sinC = (r/(2R))(sinA + sinB + sinC) (3) acosA + bcosB + ccosC = ((abc)/(2R^2 )) (4) sinA.sinB.sinC = (R/(2r))(sinA + sinB + sinC)

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{ABC} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{sin}{A}.\mathrm{sin}{B}.\mathrm{sin}{C}\:=\:\frac{\Delta}{\mathrm{2}{R}^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\mathrm{sin}{A}.\mathrm{sin}{B}.\mathrm{sin}{C}\:=\:\frac{{r}}{\mathrm{2}{R}}\left(\mathrm{sin}{A}\:+\:\mathrm{sin}{B}\:+\:\mathrm{sin}{C}\right) \\ $$$$\left(\mathrm{3}\right)\:{a}\mathrm{cos}{A}\:+\:{b}\mathrm{cos}{B}\:+\:{c}\mathrm{cos}{C}\:=\:\frac{{abc}}{\mathrm{2}{R}^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\mathrm{sin}{A}.\mathrm{sin}{B}.\mathrm{sin}{C}\:=\:\frac{{R}}{\mathrm{2}{r}}\left(\mathrm{sin}{A}\:+\:\mathrm{sin}{B}\:+\:\mathrm{sin}{C}\right) \\ $$

Question Number 18053    Answers: 0   Comments: 5

Question Number 18046    Answers: 1   Comments: 0

Question Number 18036    Answers: 0   Comments: 4

solve: 4cos(x) + 2sin(x) = 2 + (√3)

$$\mathrm{solve}: \\ $$$$ \\ $$$$\mathrm{4cos}\left(\mathrm{x}\right)\:+\:\mathrm{2sin}\left(\mathrm{x}\right)\:=\:\mathrm{2}\:+\:\sqrt{\mathrm{3}} \\ $$

Question Number 18034    Answers: 2   Comments: 0

solve for a 5log _4 a+48log _a 4=(a/8)

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{a} \\ $$$$ \\ $$$$\mathrm{5log}\:_{\mathrm{4}} \mathrm{a}+\mathrm{48log}\:_{\mathrm{a}} \mathrm{4}=\frac{\mathrm{a}}{\mathrm{8}} \\ $$

Question Number 18031    Answers: 1   Comments: 0

What is the solution set of ((x + 2)/(x + 1)) = 1

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\:\mathrm{of}\:\:\:\frac{\mathrm{x}\:+\:\mathrm{2}}{\mathrm{x}\:+\:\mathrm{1}}\:=\:\mathrm{1} \\ $$

Question Number 18022    Answers: 1   Comments: 0

The first term of an A.P is log^a and second term is log^b .show that the sum of first n terms in (1/2)log[(b^(n(n−1)) /a^(n(−3)) )]

$${The}\:{first}\:{term}\:{of}\:{an}\:{A}.{P}\:\:{is}\:{log}^{{a}} \:{and}\:{second}\:{term}\:{is}\: \\ $$$${log}^{{b}} .{show}\:{that}\:{the}\:{sum}\:{of}\:{first}\:{n}\:{terms}\:{in}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{log}\left[\frac{{b}^{{n}\left({n}−\mathrm{1}\right)} }{{a}^{{n}\left(−\mathrm{3}\right)} }\right] \\ $$

Question Number 18024    Answers: 2   Comments: 0

Question Number 18015    Answers: 2   Comments: 1

Just for fun Prove that there are no real numbers A and B that satisfy sinA=(2/(sinB))

$${Just}\:{for}\:{fun} \\ $$$${Prove}\:{that}\:{there}\:{are}\:{no}\:{real}\:{numbers}\:{A}\:{and}\:{B}\:{that} \\ $$$${satisfy}\: \\ $$$${sinA}=\frac{\mathrm{2}}{{sinB}} \\ $$

Question Number 23721    Answers: 1   Comments: 0

If symbols have their usual meaning then (1/r^2 ) + (1/r_1 ^2 ) + (1/r_2 ^2 ) + (1/r_3 ^2 ) = (1) ((a^2 + b^2 + c^2 )/s^2 ) (2) (Δ/(a^2 + b^2 + c^2 )) (3) ((a^2 + b^2 + c^2 )/Δ^2 ) (4) ((a + b + c)/Δ^2 )

$$\mathrm{If}\:\mathrm{symbols}\:\mathrm{have}\:\mathrm{their}\:\mathrm{usual}\:\mathrm{meaning} \\ $$$$\mathrm{then}\:\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{r}_{\mathrm{1}} ^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{r}_{\mathrm{2}} ^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{r}_{\mathrm{3}} ^{\mathrm{2}} }\:= \\ $$$$\left(\mathrm{1}\right)\:\frac{{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} }{{s}^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{\Delta}{{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:\frac{{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} }{\Delta^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\frac{{a}\:+\:{b}\:+\:{c}}{\Delta^{\mathrm{2}} } \\ $$

Question Number 23722    Answers: 0   Comments: 4

A block of mass m is pulled on the smooth horizontal floor using two methods I and II. The ratio of acceleration (a_I /a_(II) ) is

$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{pulled}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{smooth}\:\mathrm{horizontal}\:\mathrm{floor}\:\mathrm{using}\:\mathrm{two} \\ $$$$\mathrm{methods}\:\mathrm{I}\:\mathrm{and}\:\mathrm{II}.\:\mathrm{The}\:\mathrm{ratio}\:\mathrm{of} \\ $$$$\mathrm{acceleration}\:\frac{{a}_{{I}} }{{a}_{{II}} }\:\mathrm{is} \\ $$

Question Number 18004    Answers: 1   Comments: 1

Question Number 18003    Answers: 1   Comments: 0

The value of cosA∙cos2A∙cos2^2 A ..... cos(2^(n − 1) A), where A ∈ R may be (1) 1 (2) 2 (3) −1 (4) ((sin 2^n A)/(2^n sin A))

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{cos}{A}\centerdot\mathrm{cos2}{A}\centerdot\mathrm{cos2}^{\mathrm{2}} {A}\:.....\:\mathrm{cos}\left(\mathrm{2}^{{n}\:−\:\mathrm{1}} {A}\right), \\ $$$$\mathrm{where}\:{A}\:\in\:{R}\:\mathrm{may}\:\mathrm{be} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:−\mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{sin}\:\mathrm{2}^{{n}} \:{A}}{\mathrm{2}^{{n}} \:\mathrm{sin}\:{A}} \\ $$

Question Number 17999    Answers: 1   Comments: 0

A large number of bullets are fired in all direction with same speed u. The maximum area on the ground covered by these bullets will be (1) π.(u^2 /g) (2) π.(u^4 /g^2 ) (3) (π/4).(u^4 /g^2 ) (4) (π/2).(u^4 /g^2 )

$$\mathrm{A}\:\mathrm{large}\:\mathrm{number}\:\mathrm{of}\:\mathrm{bullets}\:\mathrm{are}\:\mathrm{fired}\:\mathrm{in} \\ $$$$\mathrm{all}\:\mathrm{direction}\:\mathrm{with}\:\mathrm{same}\:\mathrm{speed}\:{u}.\:\mathrm{The} \\ $$$$\mathrm{maximum}\:\mathrm{area}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{covered} \\ $$$$\mathrm{by}\:\mathrm{these}\:\mathrm{bullets}\:\mathrm{will}\:\mathrm{be} \\ $$$$\left(\mathrm{1}\right)\:\pi.\frac{{u}^{\mathrm{2}} }{{g}} \\ $$$$\left(\mathrm{2}\right)\:\pi.\frac{{u}^{\mathrm{4}} }{{g}^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:\frac{\pi}{\mathrm{4}}.\frac{{u}^{\mathrm{4}} }{{g}^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\frac{\pi}{\mathrm{2}}.\frac{{u}^{\mathrm{4}} }{{g}^{\mathrm{2}} } \\ $$

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