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Question Number 20511 Answers: 0 Comments: 0

Given a sphere of unit radius. Find the expression of a circular spot on the sphere′s surface given the latitude β and the longitude λ of its center and its angular radius r.

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{unit}\:\mathrm{radius}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{spot}\:\mathrm{on}\:\mathrm{the}\:\mathrm{sphere}'\mathrm{s}\:\mathrm{surface}\:\mathrm{given} \\ $$$$\mathrm{the}\:\mathrm{latitude}\:\beta\:\mathrm{and}\:\mathrm{the}\:\mathrm{longitude}\:\lambda \\ $$$$\mathrm{of}\:\mathrm{its}\:\mathrm{center}\:\mathrm{and}\:\mathrm{its}\:\mathrm{angular}\:\mathrm{radius}\:{r}. \\ $$

Question Number 20509 Answers: 0 Comments: 0

Simplify: cos^(−1) (((sin x + cos x)/(√2))), ((5π)/4) < x < ((9π)/4)

$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}}{\sqrt{\mathrm{2}}}\right),\:\frac{\mathrm{5}\pi}{\mathrm{4}}\:<\:{x}\:<\:\frac{\mathrm{9}\pi}{\mathrm{4}} \\ $$

Question Number 20506 Answers: 1 Comments: 0

Simplify: cos^(−1) (((sin x + cos x)/(√2))), (π/4) < x < ((5π)/4)

$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}}{\sqrt{\mathrm{2}}}\right),\:\frac{\pi}{\mathrm{4}}\:<\:{x}\:<\:\frac{\mathrm{5}\pi}{\mathrm{4}} \\ $$

Question Number 20505 Answers: 1 Comments: 0

Simplify: cos^(−1) ((3/5) cos x + (4/5) sin x), where −((3π)/4) ≤ x ≤ (π/4)

$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{3}}{\mathrm{5}}\:\mathrm{cos}\:{x}\:+\:\frac{\mathrm{4}}{\mathrm{5}}\:\mathrm{sin}\:{x}\right),\:{where} \\ $$$$−\frac{\mathrm{3}\pi}{\mathrm{4}}\:\leqslant\:{x}\:\leqslant\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 20501 Answers: 0 Comments: 5

The force acting on the block is given by F = 5 − 2t. The frictional force acting on the block at t = 2 s. (The block is at rest at t = 0)

$$\mathrm{The}\:\mathrm{force}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{the}\:\mathrm{block}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$$${F}\:=\:\mathrm{5}\:−\:\mathrm{2}{t}.\:\mathrm{The}\:\mathrm{frictional}\:\mathrm{force}\:\mathrm{acting} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{block}\:\mathrm{at}\:{t}\:=\:\mathrm{2}\:\mathrm{s}.\:\left(\mathrm{The}\:\mathrm{block}\:\mathrm{is}\:\mathrm{at}\right. \\ $$$$\left.\mathrm{rest}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\right) \\ $$

Question Number 20471 Answers: 1 Comments: 2

Find the surface area of a solid that is common part of two cylinders x^2 +y^2 =a^2 , y^2 +z^2 =a^2 . Compute the volume also.

$${Find}\:{the}\:{surface}\:{area}\:{of}\:{a}\:{solid} \\ $$$${that}\:{is}\:{common}\:{part}\:{of}\:{two} \\ $$$${cylinders}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={a}^{\mathrm{2}} ,\:{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={a}^{\mathrm{2}} . \\ $$$$\boldsymbol{{Compute}}\:\boldsymbol{{the}}\:\boldsymbol{{volume}}\:\boldsymbol{{also}}. \\ $$

Question Number 20468 Answers: 1 Comments: 0

∫(dx/(sin^4 x−cos^4 x))

$$\int\frac{{dx}}{\mathrm{sin}\:^{\mathrm{4}} {x}−\mathrm{cos}\:^{\mathrm{4}} {x}} \\ $$

Question Number 20467 Answers: 1 Comments: 0

∫(asin^2 x+bcos^2 x)dx

$$\int\left({a}\mathrm{sin}\:^{\mathrm{2}} {x}+{b}\mathrm{cos}\:^{\mathrm{2}} {x}\right){dx} \\ $$

Question Number 20466 Answers: 1 Comments: 0

∫((sin xcos xdx)/(sin^4 x+cos^4 x))

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

Question Number 20465 Answers: 1 Comments: 0

∫((cos 2xdx)/(sin^2 2x+8))

$$\int\frac{\mathrm{cos}\:\mathrm{2}{xdx}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{x}+\mathrm{8}} \\ $$

Question Number 20461 Answers: 0 Comments: 4

Question Number 20460 Answers: 0 Comments: 1

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 =

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)

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