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Question Number 18364 Answers: 0 Comments: 0

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Question Number 18469 Answers: 1 Comments: 0

Find Z_x and Z_y for each of the functions below (a) Z = 8x^2 y + 14xy^2 + 5y^2 x^3 (b) Z = 4x^3 y^2 + 2x^2 y^3 − 7xy^5

$$\mathrm{Find}\:\mathrm{Z}_{\mathrm{x}} \:\mathrm{and}\:\mathrm{Z}_{\mathrm{y}} \:\mathrm{for}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{functions}\:\mathrm{below} \\ $$$$\left(\mathrm{a}\right)\:\:\mathrm{Z}\:=\:\mathrm{8x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{14xy}^{\mathrm{2}} \:+\:\mathrm{5y}^{\mathrm{2}} \mathrm{x}^{\mathrm{3}} \\ $$$$\left(\mathrm{b}\right)\:\:\mathrm{Z}\:=\:\mathrm{4x}^{\mathrm{3}} \mathrm{y}^{\mathrm{2}} \:+\:\mathrm{2x}^{\mathrm{2}} \mathrm{y}^{\mathrm{3}} \:−\:\mathrm{7xy}^{\mathrm{5}} \\ $$

Question Number 18327 Answers: 1 Comments: 1

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Σ((cos 2rθ)/(sin^2 2rθ−sin^2 θ))

$$\Sigma\frac{\mathrm{cos}\:\mathrm{2}{r}\theta}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{r}\theta−\mathrm{sin}\:^{\mathrm{2}} \theta} \\ $$

Question Number 18322 Answers: 1 Comments: 1

The pulley arrangements are identical. The mass of the rope is negligible. In (a), the mass m is lifted up by attaching a mass (2m) to the other end of the rope. In (b), m is lifted up by pulling the other end of the rope with a constant downward force F = 2mg. In which case, the acceleration of m is more?

$$\mathrm{The}\:\mathrm{pulley}\:\mathrm{arrangements}\:\mathrm{are}\:\mathrm{identical}. \\ $$$$\mathrm{The}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rope}\:\mathrm{is}\:\mathrm{negligible}.\:\mathrm{In} \\ $$$$\left(\mathrm{a}\right),\:\mathrm{the}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{lifted}\:\mathrm{up}\:\mathrm{by}\:\mathrm{attaching} \\ $$$$\mathrm{a}\:\mathrm{mass}\:\left(\mathrm{2}{m}\right)\:\mathrm{to}\:\mathrm{the}\:\mathrm{other}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rope}. \\ $$$$\mathrm{In}\:\left(\mathrm{b}\right),\:{m}\:\mathrm{is}\:\mathrm{lifted}\:\mathrm{up}\:\mathrm{by}\:\mathrm{pulling}\:\mathrm{the} \\ $$$$\mathrm{other}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rope}\:\mathrm{with}\:\mathrm{a}\:\mathrm{constant} \\ $$$$\mathrm{downward}\:\mathrm{force}\:{F}\:=\:\mathrm{2}{mg}.\:\mathrm{In}\:\mathrm{which} \\ $$$$\mathrm{case},\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{of}\:{m}\:\mathrm{is}\:\mathrm{more}? \\ $$

Question Number 18318 Answers: 0 Comments: 3

∫ ((x + sinx)/(cosx)) dx

$$\int\:\frac{\mathrm{x}\:+\:\mathrm{sinx}}{\mathrm{cosx}}\:\mathrm{dx} \\ $$

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Question Number 18320 Answers: 1 Comments: 0

In a triangle ABC with fixed base BC, the vertex A moves such that cos B + cos C = 4 sin^2 (A/2) . If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C respectively, then (1) b + c = 4a (2) b + c = 2a (3) Locus of point A is an ellipse (4) Locus of point A is a pair of straight lines

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC}\:\mathrm{with}\:\mathrm{fixed}\:\mathrm{base}\:{BC}, \\ $$$$\mathrm{the}\:\mathrm{vertex}\:{A}\:\mathrm{moves}\:\mathrm{such}\:\mathrm{that}\:\mathrm{cos}\:{B}\:+ \\ $$$$\mathrm{cos}\:{C}\:=\:\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:\frac{{A}}{\mathrm{2}}\:.\:\mathrm{If}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{denote} \\ $$$$\mathrm{the}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle} \\ $$$$\mathrm{opposite}\:\mathrm{to}\:\mathrm{the}\:\mathrm{angles}\:{A},\:{B}\:\mathrm{and}\:{C} \\ $$$$\mathrm{respectively},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{b}\:+\:{c}\:=\:\mathrm{4}{a} \\ $$$$\left(\mathrm{2}\right)\:{b}\:+\:{c}\:=\:\mathrm{2}{a} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Locus}\:\mathrm{of}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{an}\:\mathrm{ellipse} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Locus}\:\mathrm{of}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{straight} \\ $$$$\mathrm{lines} \\ $$

Question Number 18719 Answers: 0 Comments: 2

If cos^2 x_1 + cos^2 x_2 + cos^2 x_3 + cos^2 x_4 + cos^2 x_5 = 5, then sin x_1 + 2sin x_2 + 3sin x_3 + 4sin x_4 + 5sin x_5 is less than or equal to

$$\mathrm{If}\:\mathrm{cos}^{\mathrm{2}} \:{x}_{\mathrm{1}} \:+\:\mathrm{cos}^{\mathrm{2}} \:{x}_{\mathrm{2}} \:+\:\mathrm{cos}^{\mathrm{2}} \:{x}_{\mathrm{3}} \:+\:\mathrm{cos}^{\mathrm{2}} \:{x}_{\mathrm{4}} \\ $$$$+\:\mathrm{cos}^{\mathrm{2}} \:{x}_{\mathrm{5}} \:=\:\mathrm{5},\:\mathrm{then}\:\mathrm{sin}\:{x}_{\mathrm{1}} \:+\:\mathrm{2sin}\:{x}_{\mathrm{2}} \:+ \\ $$$$\mathrm{3sin}\:{x}_{\mathrm{3}} \:+\:\mathrm{4sin}\:{x}_{\mathrm{4}} \:+\:\mathrm{5sin}\:{x}_{\mathrm{5}} \:\mathrm{is}\:\mathrm{less}\:\mathrm{than} \\ $$$$\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 18299 Answers: 0 Comments: 2

x^x^x = 2, find x

$$\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \:=\:\mathrm{2},\:\:\:\:\:\:\mathrm{find}\:\:\mathrm{x} \\ $$

Question Number 18470 Answers: 1 Comments: 0

Find the partial derivatives for each of the following (a) Z = 3x^2 (5x + 7y)^2 (b) Z = (w − x − y)^2 (3w + 2x − 4y)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{partial}\:\mathrm{derivatives}\:\mathrm{for}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Z}\:=\:\mathrm{3x}^{\mathrm{2}} \left(\mathrm{5x}\:+\:\mathrm{7y}\right)^{\mathrm{2}} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Z}\:=\:\left(\mathrm{w}\:−\:\mathrm{x}\:−\:\mathrm{y}\right)^{\mathrm{2}} \:\left(\mathrm{3w}\:+\:\mathrm{2x}\:−\:\mathrm{4y}\right) \\ $$

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∫_0 ^5 (1/(∫_1 ^8 e^x^(−5) ))dx

$$\int_{\mathrm{0}} ^{\mathrm{5}} \frac{\mathrm{1}}{\int_{\mathrm{1}} ^{\mathrm{8}} \mathrm{e}^{\mathrm{x}^{−\mathrm{5}} } }\mathrm{dx} \\ $$

Question Number 18284 Answers: 1 Comments: 0

If k is any possible number. what is the size of angle between the vectors a(K, k) and b(− 3, 4)

$$\mathrm{If}\:\mathrm{k}\:\mathrm{is}\:\mathrm{any}\:\mathrm{possible}\:\mathrm{number}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{size}\:\mathrm{of}\:\mathrm{angle}\:\mathrm{between}\:\mathrm{the}\:\mathrm{vectors} \\ $$$$\mathrm{a}\left(\mathrm{K},\:\mathrm{k}\right)\:\mathrm{and}\:\mathrm{b}\left(−\:\mathrm{3},\:\mathrm{4}\right) \\ $$

Question Number 18283 Answers: 0 Comments: 0

Given: x^2 + y^2 Show that, (d^2 y/dx^2 ) = ((xy)/(y^2 + x^2 ))

$$\mathrm{Given}:\:\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \\ $$$$\mathrm{Show}\:\mathrm{that},\:\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:=\:\frac{\mathrm{xy}}{\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 18282 Answers: 0 Comments: 0

Two trains 150 m long and 250 m long are travelling at the speed of 30 kmph and 33 kmph respectively on parallel tracks in opposite directions. What is the time taken by these trains to cross each other completely from the moment they meet?

$$\mathrm{Two}\:\mathrm{trains}\:\mathrm{150}\:\mathrm{m}\:\mathrm{long}\:\mathrm{and}\:\mathrm{250}\:\mathrm{m}\:\mathrm{long} \\ $$$$\mathrm{are}\:\mathrm{travelling}\:\mathrm{at}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{30}\:\mathrm{kmph} \\ $$$$\mathrm{and}\:\mathrm{33}\:\mathrm{kmph}\:\mathrm{respectively}\:\mathrm{on}\:\mathrm{parallel} \\ $$$$\mathrm{tracks}\:\mathrm{in}\:\mathrm{opposite}\:\mathrm{directions}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{time}\:\mathrm{taken}\:\mathrm{by}\:\mathrm{these}\:\mathrm{trains}\:\mathrm{to}\:\mathrm{cross} \\ $$$$\mathrm{each}\:\mathrm{other}\:\mathrm{completely}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{moment}\:\mathrm{they}\:\mathrm{meet}? \\ $$

Question Number 18281 Answers: 2 Comments: 0

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In an 1800 m race, P beats Q by 50 seconds. In the same race, Q beats R by 40 seconds. If P beats R by 450 m, by what distance does P beat Q ?(in m)

$$\mathrm{In}\:\mathrm{an}\:\mathrm{1800}\:\mathrm{m}\:\mathrm{race},\:\mathrm{P}\:\mathrm{beats}\:\mathrm{Q}\:\mathrm{by}\:\mathrm{50}\: \\ $$$$\mathrm{seconds}.\:\mathrm{In}\:\mathrm{the}\:\mathrm{same}\:\mathrm{race},\:\mathrm{Q}\:\mathrm{beats}\:\mathrm{R}\:\mathrm{by} \\ $$$$\mathrm{40}\:\mathrm{seconds}.\:\mathrm{If}\:\mathrm{P}\:\mathrm{beats}\:\mathrm{R}\:\mathrm{by}\:\mathrm{450}\:\mathrm{m},\:\mathrm{by}\: \\ $$$$\mathrm{what}\:\mathrm{distance}\:\mathrm{does}\:\mathrm{P}\:\mathrm{beat}\:\mathrm{Q}\:?\left(\mathrm{in}\:\mathrm{m}\right) \\ $$

Question Number 18274 Answers: 1 Comments: 0

A balloon moves up vertically such that if a stone is projected with a horizontal velocity u relative to balloon, the stone always hits the ground at a fixed point at a distance ((2u^2 )/g) horizontally away from it. Find the height of the balloon as a function of time.

$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{moves}\:\mathrm{up}\:\mathrm{vertically}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{if}\:\mathrm{a}\:\mathrm{stone}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{velocity}\:{u}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{balloon}, \\ $$$$\mathrm{the}\:\mathrm{stone}\:\mathrm{always}\:\mathrm{hits}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{at}\:\mathrm{a} \\ $$$$\mathrm{fixed}\:\mathrm{point}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\frac{\mathrm{2}{u}^{\mathrm{2}} }{{g}}\:\mathrm{horizontally} \\ $$$$\mathrm{away}\:\mathrm{from}\:\mathrm{it}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{height}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{balloon}\:\mathrm{as}\:\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:\mathrm{time}. \\ $$

Question Number 18271 Answers: 0 Comments: 3

There are two parallel planes, each inclined to the horizontal at an angle θ. A particle is projected from a point mid way between the foot of the two planes so that it grazes one of the planes and strikes the other at right angle. Find the angle of projection of the projectile.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{two}\:\mathrm{parallel}\:\mathrm{planes},\:\mathrm{each} \\ $$$$\mathrm{inclined}\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\theta. \\ $$$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{mid} \\ $$$$\mathrm{way}\:\mathrm{between}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{planes} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{it}\:\mathrm{grazes}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{planes}\:\mathrm{and} \\ $$$$\mathrm{strikes}\:\mathrm{the}\:\mathrm{other}\:\mathrm{at}\:\mathrm{right}\:\mathrm{angle}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{projection}\:\mathrm{of}\:\mathrm{the}\:\mathrm{projectile}. \\ $$

Question Number 18269 Answers: 0 Comments: 3

The flow velocity of a river increases linearly with the distance (r) from its bank and has its maximum value v_0 in the middle of the river. The velocity near the bank is zero. A boat which can move with speed u in still water moves in the river in such a way that it is always perpendicular to the flow of current. Find (i) The distance along the bank through which boat is carried away by the flow current, when the boat crosses the river. (ii) The equation of trajectory for the coordinate system shown. Assume that the swimmer starts from origin.

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