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AllQuestion and Answers: Page 1846

Question Number 20456    Answers: 0   Comments: 1

∫cot^4 xdx

$$\int\mathrm{cot}\:^{\mathrm{4}} {xdx} \\ $$

Question Number 20455    Answers: 0   Comments: 1

∫sec^6 xdx

$$\int{sec}^{\mathrm{6}} {xdx} \\ $$

Question Number 20454    Answers: 0   Comments: 1

∫sec^3 xdx

$$\int\mathrm{sec}\:^{\mathrm{3}} {xdx} \\ $$

Question Number 20453    Answers: 0   Comments: 0

∫(dx/(3+4sin x))

$$\int\frac{{dx}}{\mathrm{3}+\mathrm{4sin}\:{x}} \\ $$

Question Number 20452    Answers: 0   Comments: 0

∫(dx/(3+2sin x+cos x))

$$\int\frac{{dx}}{\mathrm{3}+\mathrm{2sin}\:{x}+\mathrm{cos}\:{x}} \\ $$

Question Number 20451    Answers: 1   Comments: 0

∫sin^3 xcos^4 xdx

$$\int{sin}^{\mathrm{3}} {x}\mathrm{cos}\:^{\mathrm{4}} {xdx} \\ $$

Question Number 20450    Answers: 1   Comments: 0

∫sin^4 xcos^3 xdx

$$\int\mathrm{sin}\:^{\mathrm{4}} {x}\mathrm{cos}\:^{\mathrm{3}} {xdx} \\ $$

Question Number 20551    Answers: 1   Comments: 8

Find the minimum value of ∣a + bω + cω^2 ∣, where a, b and c are all not equal integers and ω(≠1) is a cube root of unity.

$${Find}\:{the}\:{minimum}\:{value}\:{of} \\ $$$$\mid{a}\:+\:{b}\omega\:+\:{c}\omega^{\mathrm{2}} \mid,\:{where}\:{a},\:{b}\:{and}\:{c}\:{are}\:{all} \\ $$$${not}\:{equal}\:{integers}\:{and}\:\omega\left(\neq\mathrm{1}\right)\:{is}\:{a}\:{cube} \\ $$$${root}\:{of}\:{unity}. \\ $$

Question Number 20488    Answers: 1   Comments: 1

Question Number 20436    Answers: 0   Comments: 0

If A lies in the third quadrant and 3 tan A − 4 = 0, then 5 sin 2A + 3 sin A + 4cos A =

$$\mathrm{If}\:\:{A}\:\mathrm{lies}\:\mathrm{in}\:\mathrm{the}\:\mathrm{third}\:\mathrm{quadrant}\:\mathrm{and} \\ $$$$\mathrm{3}\:\mathrm{tan}\:{A}\:−\:\mathrm{4}\:=\:\mathrm{0},\:\mathrm{then}\: \\ $$$$\mathrm{5}\:\mathrm{sin}\:\mathrm{2}{A}\:+\:\mathrm{3}\:\mathrm{sin}\:{A}\:+\:\mathrm{4cos}\:{A}\:=\: \\ $$

Question Number 20435    Answers: 1   Comments: 0

If A lies in the third quadrant and 3 tan A − 4 = 0, then 5 sin 2A + 3 sin A + 4cos A =

$$\mathrm{If}\:\:{A}\:\mathrm{lies}\:\mathrm{in}\:\mathrm{the}\:\mathrm{third}\:\mathrm{quadrant}\:\mathrm{and} \\ $$$$\mathrm{3}\:\mathrm{tan}\:{A}\:−\:\mathrm{4}\:=\:\mathrm{0},\:\mathrm{then}\: \\ $$$$\mathrm{5}\:\mathrm{sin}\:\mathrm{2}{A}\:+\:\mathrm{3}\:\mathrm{sin}\:{A}\:+\:\mathrm{4cos}\:{A}\:=\: \\ $$

Question Number 20434    Answers: 2   Comments: 0

If cos x=tan y, cos y=tan z, cos z=tan x, then the value of sin x is

$$\mathrm{If}\:\:\mathrm{cos}\:{x}=\mathrm{tan}\:{y},\:\mathrm{cos}\:{y}=\mathrm{tan}\:{z},\:\mathrm{cos}\:{z}=\mathrm{tan}\:{x}, \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{sin}\:{x}\:\:\mathrm{is} \\ $$

Question Number 20430    Answers: 2   Comments: 1

Let a, b and c be such that a + b + c = 0 and P = (a^2 /(2a^2 + bc)) + (b^2 /(2b^2 + ca)) + (c^2 /(2c^2 + ab)) is defined. What is the value of P?

$$\mathrm{Let}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0} \\ $$$$\mathrm{and} \\ $$$${P}\:=\:\frac{{a}^{\mathrm{2}} }{\mathrm{2}{a}^{\mathrm{2}} \:+\:{bc}}\:+\:\frac{{b}^{\mathrm{2}} }{\mathrm{2}{b}^{\mathrm{2}} \:+\:{ca}}\:+\:\frac{{c}^{\mathrm{2}} }{\mathrm{2}{c}^{\mathrm{2}} \:+\:{ab}} \\ $$$$\mathrm{is}\:\mathrm{defined}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{P}? \\ $$

Question Number 20425    Answers: 1   Comments: 1

On a frictionless horizontal surface, assumed to be the x-y plane, a small trolley A is moving along a straight line parallel to the y-axis (see figure) with a constant velocity of ((√3) − 1) m/s. At a particular instant, when the line OA makes an angle of 45° with the x-axis, a ball is thrown along the surface from the origin O. Its velocity makes an angle φ with x-axis and it hits the trolley. 1. The motion of the ball is observed from the frame of the trolley. Calculate the angle θ made by the velocity vector of the ball with the x-axis in this frame. 2. Find the speed of the ball with respect to the surface, if φ = ((4θ)/3)

$$\mathrm{On}\:\mathrm{a}\:\mathrm{frictionless}\:\mathrm{horizontal}\:\mathrm{surface}, \\ $$$$\mathrm{assumed}\:\mathrm{to}\:\mathrm{be}\:\mathrm{the}\:{x}-{y}\:\mathrm{plane},\:\mathrm{a}\:\mathrm{small} \\ $$$$\mathrm{trolley}\:{A}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{along}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{y}-\mathrm{axis}\:\left(\mathrm{see}\:\mathrm{figure}\right) \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{velocity}\:\mathrm{of}\:\left(\sqrt{\mathrm{3}}\:−\:\mathrm{1}\right)\:\mathrm{m}/\mathrm{s}. \\ $$$$\mathrm{At}\:\mathrm{a}\:\mathrm{particular}\:\mathrm{instant},\:\mathrm{when}\:\mathrm{the}\:\mathrm{line} \\ $$$${OA}\:\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{45}°\:\mathrm{with}\:\mathrm{the} \\ $$$${x}-\mathrm{axis},\:\mathrm{a}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{along}\:\mathrm{the} \\ $$$$\mathrm{surface}\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin}\:{O}.\:\mathrm{Its}\:\mathrm{velocity} \\ $$$$\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\phi\:\mathrm{with}\:{x}-\mathrm{axis}\:\mathrm{and}\:\mathrm{it} \\ $$$$\mathrm{hits}\:\mathrm{the}\:\mathrm{trolley}. \\ $$$$\mathrm{1}.\:\mathrm{The}\:\mathrm{motion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{observed}\:\mathrm{from} \\ $$$$\mathrm{the}\:\mathrm{frame}\:\mathrm{of}\:\mathrm{the}\:\mathrm{trolley}.\:\mathrm{Calculate}\:\mathrm{the} \\ $$$$\mathrm{angle}\:\theta\:\mathrm{made}\:\mathrm{by}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{vector}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{ball}\:\mathrm{with}\:\mathrm{the}\:{x}-\mathrm{axis}\:\mathrm{in}\:\mathrm{this}\:\mathrm{frame}. \\ $$$$\mathrm{2}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{the}\:\mathrm{surface},\:\mathrm{if}\:\phi\:=\:\frac{\mathrm{4}\theta}{\mathrm{3}} \\ $$

Question Number 20412    Answers: 0   Comments: 3

A block is placed on a rough horizontal surface. The minimum force required to slide the block is

$$\mathrm{A}\:\mathrm{block}\:\mathrm{is}\:\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{rough}\:\mathrm{horizontal} \\ $$$$\mathrm{surface}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{force}\:\mathrm{required} \\ $$$$\mathrm{to}\:\mathrm{slide}\:\mathrm{the}\:\mathrm{block}\:\mathrm{is} \\ $$

Question Number 20411    Answers: 1   Comments: 0

A stone of weight W is thrown straight up from the ground with an initial speed u. if a drag force of constant magnitude f acts on the stone through out its flight, the speed of stone just before reaching the ground is

$$\mathrm{A}\:\mathrm{stone}\:\mathrm{of}\:\mathrm{weight}\:{W}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{straight} \\ $$$$\mathrm{up}\:\mathrm{from}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{with}\:\mathrm{an}\:\mathrm{initial} \\ $$$$\mathrm{speed}\:{u}.\:\mathrm{if}\:\mathrm{a}\:\mathrm{drag}\:\mathrm{force}\:\mathrm{of}\:\mathrm{constant} \\ $$$$\mathrm{magnitude}\:{f}\:\mathrm{acts}\:\mathrm{on}\:\mathrm{the}\:\mathrm{stone}\:\mathrm{through} \\ $$$$\mathrm{out}\:\mathrm{its}\:\mathrm{flight},\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{stone}\:\mathrm{just} \\ $$$$\mathrm{before}\:\mathrm{reaching}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{is} \\ $$

Question Number 20409    Answers: 1   Comments: 1

Calculate the force (F) required to cause the block of mass m_1 = 20 kg just to slide under the block of mass m_2 = 10 kg [coefficient of friction μ = 0.25 for all surfaces]

$${Calculate}\:{the}\:{force}\:\left({F}\right)\:{required}\:{to} \\ $$$${cause}\:{the}\:{block}\:{of}\:{mass}\:{m}_{\mathrm{1}} \:=\:\mathrm{20}\:{kg} \\ $$$${just}\:{to}\:{slide}\:{under}\:{the}\:{block}\:{of}\:{mass} \\ $$$${m}_{\mathrm{2}} \:=\:\mathrm{10}\:{kg}\:\left[{coefficient}\:{of}\:{friction}\:\mu\right. \\ $$$$\left.=\:\mathrm{0}.\mathrm{25}\:{for}\:{all}\:{surfaces}\right] \\ $$

Question Number 20404    Answers: 0   Comments: 0

∫_0 ^( π/4) tan(θ)ln(1−tan(θ))dθ=?

$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} {tan}\left(\theta\right){ln}\left(\mathrm{1}−{tan}\left(\theta\right)\right){d}\theta=? \\ $$

Question Number 20390    Answers: 1   Comments: 0

∫sin^5 xdx

$$\int\mathrm{sin}\:^{\mathrm{5}} {xdx} \\ $$

Question Number 20389    Answers: 1   Comments: 0

∫sin^4 xdx

$$\int\mathrm{sin}\:^{\mathrm{4}} {xdx} \\ $$

Question Number 20388    Answers: 1   Comments: 0

∫((2sin x+3cos x)/(7sin x−2cos x))

$$\int\frac{\mathrm{2sin}\:{x}+\mathrm{3cos}\:{x}}{\mathrm{7sin}\:{x}−\mathrm{2cos}\:{x}} \\ $$

Question Number 20387    Answers: 2   Comments: 0

∫sin pxcos qxdx

$$\int\mathrm{sin}\:{px}\mathrm{cos}\:{qxdx} \\ $$

Question Number 20386    Answers: 1   Comments: 0

∫(√(1−a^2 x^2 dx))

$$\int\sqrt{\mathrm{1}−{a}^{\mathrm{2}} {x}^{\mathrm{2}} {dx}} \\ $$

Question Number 20385    Answers: 1   Comments: 0

∫(√(16−9x^2 dx))

$$\int\sqrt{\mathrm{16}−\mathrm{9}{x}^{\mathrm{2}} {dx}} \\ $$

Question Number 20384    Answers: 1   Comments: 0

∫((x^3 dx)/((2+3x)^2 ))

$$\int\frac{{x}^{\mathrm{3}} {dx}}{\left(\mathrm{2}+\mathrm{3}{x}\right)^{\mathrm{2}} } \\ $$

Question Number 20383    Answers: 1   Comments: 0

∫(dx/(x(√(2+(x)^(1/3) ))))

$$\int\frac{{dx}}{{x}\sqrt{\mathrm{2}+\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }} \\ $$

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