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Question Number 20908 Answers: 1 Comments: 0

integrate with respect to x ∫(((2x+1)/(x^2 +4x+8)))dx

$${integrate}\:{with}\:{respect}\:{to}\:{x}\: \\ $$$$\int\left(\frac{\mathrm{2}{x}+\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{8}}\right){dx} \\ $$

Question Number 20907 Answers: 1 Comments: 0

How many zeroes (0) there in 1×2×3×4..........99×100

$${How}\:{many}\:{zeroes}\:\left(\mathrm{0}\right)\:{there}\:{in}\:\mathrm{1}×\mathrm{2}×\mathrm{3}×\mathrm{4}..........\mathrm{99}×\mathrm{100}\: \\ $$

Question Number 20905 Answers: 1 Comments: 1

∫e^x^2 dx

$$\int{e}^{{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 20903 Answers: 1 Comments: 0

Question Number 20891 Answers: 0 Comments: 11

The Figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed ω and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ω/2. The ring and disc are separated by frictionless ball bearing. The system is in the x-z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle 30° with the horizontal. Then with respect to the horizontal surface (a) The point O has a linear velocity 3Rωi^∧ (b) The point P has a linear velocity ((11)/4)Rωi^∧ + ((√3)/4)Rωk^∧

$$\mathrm{The}\:\mathrm{Figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{system}\:\mathrm{consisting} \\ $$$$\mathrm{of}\:\left({i}\right)\:\mathrm{a}\:\mathrm{ring}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{radius}\:\mathrm{3}{R}\:\mathrm{rolling} \\ $$$$\mathrm{clockwise}\:\mathrm{without}\:\mathrm{slipping}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{with}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\omega\:\mathrm{and}\:\left({ii}\right)\:\mathrm{an}\:\mathrm{inner}\:\mathrm{disc}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{2}{R} \\ $$$$\mathrm{rotating}\:\mathrm{anti}-\mathrm{clockwise}\:\mathrm{with}\:\mathrm{angular} \\ $$$$\mathrm{speed}\:\omega/\mathrm{2}.\:\mathrm{The}\:\mathrm{ring}\:\mathrm{and}\:\mathrm{disc}\:\mathrm{are} \\ $$$$\mathrm{separated}\:\mathrm{by}\:\mathrm{frictionless}\:\mathrm{ball}\:\mathrm{bearing}. \\ $$$$\mathrm{The}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:{x}-{z}\:\mathrm{plane}.\:\mathrm{The} \\ $$$$\mathrm{point}\:{P}\:\mathrm{on}\:\mathrm{the}\:\mathrm{inner}\:\mathrm{disc}\:\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance} \\ $$$${R}\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin},\:\mathrm{where}\:{OP}\:\mathrm{makes}\:\mathrm{an} \\ $$$$\mathrm{angle}\:\mathrm{30}°\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{Then} \\ $$$$\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{surface} \\ $$$$\left({a}\right)\:\mathrm{The}\:\mathrm{point}\:{O}\:\mathrm{has}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{velocity} \\ $$$$\mathrm{3}{R}\omega\overset{\wedge} {{i}} \\ $$$$\left({b}\right)\:\mathrm{The}\:\mathrm{point}\:{P}\:\mathrm{has}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{velocity} \\ $$$$\frac{\mathrm{11}}{\mathrm{4}}{R}\omega\overset{\wedge} {{i}}\:+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}{R}\omega\overset{\wedge} {{k}} \\ $$

Question Number 20886 Answers: 1 Comments: 0

if sin x=msin y so proof that tan (1/2)(x−y)=((m−1)/(m+1))tan (1/2)(x+y)

$${if}\:\mathrm{sin}\:{x}={m}\mathrm{sin}\:{y} \\ $$$${so}\:{proof}\:{that} \\ $$$$\mathrm{tan}\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}−{y}\right)=\frac{{m}−\mathrm{1}}{{m}+\mathrm{1}}\mathrm{tan}\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}\right) \\ $$

Question Number 20885 Answers: 0 Comments: 0

if (θ−ϕ)subtle and sin θ+sin ϕ= (cos ϕ−cos θ)(√3) so proof sin 3θ+sin 3ϕ=0

$${if}\:\left(\theta−\varphi\right){subtle}\:{and}\:\:\:\mathrm{sin}\:\theta+\mathrm{sin}\:\varphi= \\ $$$$\left(\mathrm{cos}\:\varphi−\mathrm{cos}\:\theta\right)\sqrt{\mathrm{3}} \\ $$$${so}\:{proof}\:\mathrm{sin}\:\mathrm{3}\theta+\mathrm{sin}\:\mathrm{3}\varphi=\mathrm{0} \\ $$

Question Number 20884 Answers: 1 Comments: 0

2cos (π/3)cos ((9π)/(13))+cos ((3π)/(13))+cos ((5π)/(13))=0

$$\mathrm{2cos}\:\frac{\pi}{\mathrm{3}}\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}}=\mathrm{0} \\ $$

Question Number 20882 Answers: 1 Comments: 0

Question Number 20881 Answers: 2 Comments: 0

Question Number 20876 Answers: 0 Comments: 1

Question Number 20873 Answers: 1 Comments: 0

∫_1 ^5 (e^x /x^2 ) dx

$$\int_{\mathrm{1}} ^{\mathrm{5}} \frac{{e}^{{x}} }{{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 20872 Answers: 1 Comments: 0

if y=[xtan^(−1) x]−[(1/2)ln(1+x^2 )] show that (1+x^2 )y^(′′) =1

$${if}\:\:{y}=\left[{xtan}^{−\mathrm{1}} {x}\right]−\left[\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right] \\ $$$${show}\:{that}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{''} =\mathrm{1} \\ $$

Question Number 20871 Answers: 1 Comments: 0

Question Number 20870 Answers: 0 Comments: 0

Question Number 20869 Answers: 0 Comments: 0

Question Number 20868 Answers: 0 Comments: 1

C×(A+B)=C×A+C×B prove it.

$$\boldsymbol{{C}}×\left(\boldsymbol{{A}}+\boldsymbol{{B}}\right)=\boldsymbol{{C}}×\boldsymbol{{A}}+\boldsymbol{{C}}×\boldsymbol{{B}}\: \\ $$$${prove}\:{it}. \\ $$

Question Number 20857 Answers: 0 Comments: 0

∫ (√(sin x)) + (√(cos x)) dx

$$\int\:\sqrt{\mathrm{sin}\:{x}}\:+\:\sqrt{\mathrm{cos}\:{x}}\:{dx} \\ $$

Question Number 20853 Answers: 0 Comments: 0

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Question Number 20851 Answers: 2 Comments: 0

if (x+y)^(m+n) =x^m y^n ,show that (dy/dx)=(y/x)

$${if}\:\left({x}+{y}\right)^{{m}+{n}} ={x}^{{m}} {y}^{{n}} ,{show}\:{that}\:\frac{{dy}}{{dx}}=\frac{{y}}{{x}} \\ $$

Question Number 20850 Answers: 1 Comments: 0

given that y=log(√((1+cos^2 x)/(1−e^(2x) ))),find (dy/dx)

$${given}\:{that}\:{y}={log}\sqrt{\frac{\mathrm{1}+{cos}^{\mathrm{2}} {x}}{\mathrm{1}−{e}^{\mathrm{2}{x}} }},{find} \\ $$$$\frac{{dy}}{{dx}} \\ $$

Question Number 20849 Answers: 1 Comments: 0

The time period,T of pendulum of length l is given by T=2Π(√(l/g))where l amd g are constant.Find the approximate percentage increase in time T, when the length of pendulum increases by 4%

$${The}\:{time}\:{period},{T}\:\:{of}\:{pendulum}\:{of} \\ $$$${length}\:{l}\:{is}\:{given}\:{by}\:{T}=\mathrm{2}\Pi\sqrt{\frac{{l}}{{g}}}{where} \\ $$$$\:{l}\:{amd}\:{g}\:{are}\:{constant}.{Find}\:{the}\: \\ $$$${approximate}\:{percentage}\:{increase}\: \\ $$$${in}\:{time}\:{T},\:{when}\:{the}\:{length}\:{of}\: \\ $$$${pendulum}\:{increases}\:{by}\:\mathrm{4\%} \\ $$

Question Number 20847 Answers: 1 Comments: 0

the rectangle is known to be twice as long as its wide.if the width is measured as 20 ±0.2cm. find the area in the form of (A±b)

$${the}\:{rectangle}\:{is}\:{known}\:{to}\:{be}\:{twice} \\ $$$${as}\:{long}\:{as}\:{its}\:{wide}.{if}\:{the}\:{width}\:{is} \\ $$$${measured}\:{as}\:\mathrm{20}\:\pm\mathrm{0}.\mathrm{2}{cm}. \\ $$$${find}\:{the}\:{area}\:{in}\:{the}\:{form}\:{of}\:\left({A}\pm{b}\right) \\ $$$$ \\ $$

Question Number 20867 Answers: 1 Comments: 0

The number of irrational roots of the equation (x − 1)(x − 2)(3x − 2)(3x + 1) = 21 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{irrational}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\left({x}\:−\:\mathrm{1}\right)\left({x}\:−\:\mathrm{2}\right)\left(\mathrm{3}{x}\:−\:\mathrm{2}\right)\left(\mathrm{3}{x}\:+\:\mathrm{1}\right)\:=\:\mathrm{21}\:\mathrm{is} \\ $$

Question Number 20842 Answers: 1 Comments: 0

Acceleration of a particle which is at rest at x = 0 is a^→ = (4 − 2x) i^∧ . Select the correct alternative(s). (a) Maximum speed of the particle is 4 units (b) Particle further comes to rest at x = 4 (c) Particle oscillates about x = 2 (d) Particle will continuously accelerate along the x-axis.

$$\mathrm{Acceleration}\:\mathrm{of}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{which}\:\mathrm{is}\:\mathrm{at} \\ $$$$\mathrm{rest}\:\mathrm{at}\:{x}\:=\:\mathrm{0}\:\mathrm{is}\:\overset{\rightarrow} {{a}}\:=\:\left(\mathrm{4}\:−\:\mathrm{2}{x}\right)\:\overset{\wedge} {{i}}.\:\mathrm{Select} \\ $$$$\mathrm{the}\:\mathrm{correct}\:\mathrm{alternative}\left(\mathrm{s}\right). \\ $$$$\left({a}\right)\:\mathrm{Maximum}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{is} \\ $$$$\mathrm{4}\:\mathrm{units} \\ $$$$\left({b}\right)\:\mathrm{Particle}\:\mathrm{further}\:\mathrm{comes}\:\mathrm{to}\:\mathrm{rest}\:\mathrm{at} \\ $$$${x}\:=\:\mathrm{4} \\ $$$$\left({c}\right)\:\mathrm{Particle}\:\mathrm{oscillates}\:\mathrm{about}\:{x}\:=\:\mathrm{2} \\ $$$$\left({d}\right)\:\mathrm{Particle}\:\mathrm{will}\:\mathrm{continuously}\:\mathrm{accelerate} \\ $$$$\mathrm{along}\:\mathrm{the}\:{x}-\mathrm{axis}. \\ $$

Question Number 20836 Answers: 0 Comments: 0

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