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A small particle of mass m is projected at an angle θ with the x-axis with an initial velocity v_0 in the x-y plane as shown in the Figure. At a time t < ((v_0 sin θ)/g), the angular momentum of the particle is

$$\mathrm{A}\:\mathrm{small}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{projected} \\ $$$$\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:{x}-\mathrm{axis}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{initial}\:\mathrm{velocity}\:{v}_{\mathrm{0}} \:\mathrm{in}\:\mathrm{the}\:{x}-{y}\:\mathrm{plane}\:\mathrm{as} \\ $$$$\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Figure}.\:\mathrm{At}\:\mathrm{a}\:\mathrm{time} \\ $$$${t}\:<\:\frac{{v}_{\mathrm{0}} \:\mathrm{sin}\:\theta}{{g}},\:\mathrm{the}\:\mathrm{angular}\:\mathrm{momentum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{particle}\:\mathrm{is} \\ $$

Question Number 20372 Answers: 2 Comments: 0

The two roots of an equation x^3 − 9x^2 + 14x + 24 = 0 are in the ratio 3 : 2. Find the roots.

$$\mathrm{The}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equation}\:{x}^{\mathrm{3}} \:−\:\mathrm{9}{x}^{\mathrm{2}} \\ $$$$+\:\mathrm{14}{x}\:+\:\mathrm{24}\:=\:\mathrm{0}\:\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{3}\::\:\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{roots}. \\ $$

Question Number 20364 Answers: 1 Comments: 0

Question Number 20367 Answers: 1 Comments: 0

Let f(x) = x^3 + 3x^2 + 9x + 6sinx, then find the number of real roots of the equation (1/(x − f(1))) + (2/(x − f(2))) + (3/(x − f(3))) = 0.

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{9}{x}\:+\:\mathrm{6sin}{x},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\frac{\mathrm{1}}{{x}\:−\:{f}\left(\mathrm{1}\right)}\:+\:\frac{\mathrm{2}}{{x}\:−\:{f}\left(\mathrm{2}\right)}\:+\:\frac{\mathrm{3}}{{x}\:−\:{f}\left(\mathrm{3}\right)}\:=\:\mathrm{0}. \\ $$

Question Number 20578 Answers: 0 Comments: 2

Tinkutara and Ajfour please how do you do the following using lekh diagram: (i)introduction of dotted lines (ii)writing of letters (iii)shading (iv)putting colours in a diagram (v)draw live figures like birds thanks for the help

$$\:{Tinkutara}\:{and}\:\:{Ajfour}\:{please}\:{how} \\ $$$${do}\:{you}\:{do}\:{the}\:{following}\:{using} \\ $$$${lekh}\:{diagram}: \\ $$$$\left({i}\right){introduction}\:{of}\:{dotted}\:{lines} \\ $$$$\left({ii}\right){writing}\:{of}\:{letters} \\ $$$$\left({iii}\right){shading} \\ $$$$\left({iv}\right){putting}\:{colours}\:{in}\:{a}\:{diagram} \\ $$$$\left({v}\right){draw}\:{live}\:{figures}\:{like}\:{birds} \\ $$$$ \\ $$$${thanks}\:{for}\:{the}\:{help} \\ $$

Question Number 20576 Answers: 1 Comments: 0

sec(A−3Π/2)

$${sec}\left({A}−\mathrm{3}\Pi/\mathrm{2}\right) \\ $$

Question Number 20349 Answers: 0 Comments: 3

A ball rolled on ice with a velocity of 14 ms^(−1) comes to rest after travelling 40 m. Find the coefficient of friction. (Given, g = 9.8 m/s^2 )

$$\mathrm{A}\:\mathrm{ball}\:\mathrm{rolled}\:\mathrm{on}\:\mathrm{ice}\:\mathrm{with}\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of} \\ $$$$\mathrm{14}\:\mathrm{ms}^{−\mathrm{1}} \:\mathrm{comes}\:\mathrm{to}\:\mathrm{rest}\:\mathrm{after}\:\mathrm{travelling} \\ $$$$\mathrm{40}\:\mathrm{m}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{friction}. \\ $$$$\left(\mathrm{Given},\:{g}\:=\:\mathrm{9}.\mathrm{8}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$

Question Number 20346 Answers: 0 Comments: 3

To paint the side of a building, painter normally hoists himself up by pulling on the rope A as in figure. The painter and platform together weigh 200 N. The rope B can withstand 300 N. Find the maximum acceleration of the painter.

$$\mathrm{To}\:\mathrm{paint}\:\mathrm{the}\:\mathrm{side}\:\mathrm{of}\:\mathrm{a}\:\mathrm{building},\:\mathrm{painter} \\ $$$$\mathrm{normally}\:\mathrm{hoists}\:\mathrm{himself}\:\mathrm{up}\:\mathrm{by}\:\mathrm{pulling}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{rope}\:{A}\:\mathrm{as}\:\mathrm{in}\:\mathrm{figure}.\:\mathrm{The}\:\mathrm{painter}\:\mathrm{and} \\ $$$$\mathrm{platform}\:\mathrm{together}\:\mathrm{weigh}\:\mathrm{200}\:\mathrm{N}.\:\mathrm{The} \\ $$$$\mathrm{rope}\:{B}\:\mathrm{can}\:\mathrm{withstand}\:\mathrm{300}\:\mathrm{N}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{maximum}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the}\:\mathrm{painter}. \\ $$

Question Number 20344 Answers: 1 Comments: 3

Determine the speed with which block B rises in figure if the end of the cord at A is pulled down with a speed of 2 m/s.

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{with}\:\mathrm{which}\:\mathrm{block} \\ $$$${B}\:\mathrm{rises}\:\mathrm{in}\:\mathrm{figure}\:\mathrm{if}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cord}\:\mathrm{at} \\ $$$${A}\:\mathrm{is}\:\mathrm{pulled}\:\mathrm{down}\:\mathrm{with}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{2}\:\mathrm{m}/\mathrm{s}. \\ $$

Question Number 20368 Answers: 0 Comments: 2

If α is a real root of 2x^3 − 3x^2 + 6x + 6 = 0, then find [α] where [∙] denotes the greatest integer function.

$$\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{root}\:\mathrm{of}\:\mathrm{2}{x}^{\mathrm{3}} \:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{6}{x}\:+\:\mathrm{6}\:=\:\mathrm{0}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\left[\alpha\right]\:\mathrm{where}\:\left[\centerdot\right]\:\mathrm{denotes}\:\mathrm{the} \\ $$$$\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}. \\ $$

Question Number 20338 Answers: 0 Comments: 0

Why Al_2 O_3 is amphoteric while B_2 O_3 is acidic?

$$\mathrm{Why}\:\mathrm{Al}_{\mathrm{2}} \mathrm{O}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{amphoteric}\:\mathrm{while}\:\mathrm{B}_{\mathrm{2}} \mathrm{O}_{\mathrm{3}} \\ $$$$\mathrm{is}\:\mathrm{acidic}? \\ $$

Question Number 20337 Answers: 0 Comments: 0

Why oxidising character of F_2 > Cl_2 ?

$$\mathrm{Why}\:\mathrm{oxidising}\:\mathrm{character}\:\mathrm{of}\:\mathrm{F}_{\mathrm{2}} \:>\:\mathrm{Cl}_{\mathrm{2}} ? \\ $$

Question Number 20335 Answers: 0 Comments: 0

Covalent radius of an element having 82 electrons in extranuclear part and 82 protons in the nucleus is 146 A^o . Calculate the electronegativity on Allred Rochow scale of that element.

$$\mathrm{Covalent}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{an}\:\mathrm{element}\:\mathrm{having} \\ $$$$\mathrm{82}\:\mathrm{electrons}\:\mathrm{in}\:\mathrm{extranuclear}\:\mathrm{part}\:\mathrm{and}\:\mathrm{82} \\ $$$$\mathrm{protons}\:\mathrm{in}\:\mathrm{the}\:\mathrm{nucleus}\:\mathrm{is}\:\mathrm{146}\:\overset{\mathrm{o}} {\mathrm{A}}.\:\mathrm{Calculate} \\ $$$$\mathrm{the}\:\mathrm{electronegativity}\:\mathrm{on}\:\mathrm{Allred}\:\mathrm{Rochow} \\ $$$$\mathrm{scale}\:\mathrm{of}\:\mathrm{that}\:\mathrm{element}. \\ $$

Question Number 20334 Answers: 0 Comments: 0

Choose the correct regarding E.N. (1) B > Al > Ga > In (2) B > Al = Ga = In (3) B > In > Ga = Al (4) B > In > Ga > Al

$$\mathrm{Choose}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{regarding}\:\mathrm{E}.\mathrm{N}. \\ $$$$\left(\mathrm{1}\right)\:\mathrm{B}\:>\:\mathrm{Al}\:>\:\mathrm{Ga}\:>\:\mathrm{In} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{B}\:>\:\mathrm{Al}\:=\:\mathrm{Ga}\:=\:\mathrm{In} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{B}\:>\:\mathrm{In}\:>\:\mathrm{Ga}\:=\:\mathrm{Al} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{B}\:>\:\mathrm{In}\:>\:\mathrm{Ga}\:>\:\mathrm{Al} \\ $$

Question Number 20326 Answers: 0 Comments: 1

The number of positive inegral solutions of abc = 30 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{inegral} \\ $$$$\mathrm{solutions}\:\mathrm{of}\:\:\:{abc}\:=\:\mathrm{30}\:\mathrm{is} \\ $$

Question Number 20316 Answers: 0 Comments: 8

If at a height of 40 m, the direction of motion of a projectile makes an angle π/4 with the horizontal, then its initial velocity and angle of projection are, respectively (a) 30, (1/2)cos^(−1) (−(4/5)) (b) 30, (1/2)cos^(−1) (−(1/2)) (c) 50, (1/2)cos^(−1) (−(8/(25))) (d) 60, (1/2)cos^(−1) (−(1/4))

$$\mathrm{If}\:\mathrm{at}\:\mathrm{a}\:\mathrm{height}\:\mathrm{of}\:\mathrm{40}\:\mathrm{m},\:\mathrm{the}\:\mathrm{direction}\:\mathrm{of} \\ $$$$\mathrm{motion}\:\mathrm{of}\:\mathrm{a}\:\mathrm{projectile}\:\mathrm{makes}\:\mathrm{an}\:\mathrm{angle} \\ $$$$\pi/\mathrm{4}\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal},\:\mathrm{then}\:\mathrm{its}\:\mathrm{initial} \\ $$$$\mathrm{velocity}\:\mathrm{and}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{projection}\:\mathrm{are}, \\ $$$$\mathrm{respectively} \\ $$$$\left({a}\right)\:\mathrm{30},\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{−\mathrm{1}} \left(−\frac{\mathrm{4}}{\mathrm{5}}\right) \\ $$$$\left({b}\right)\:\mathrm{30},\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{−\mathrm{1}} \left(−\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$\left({c}\right)\:\mathrm{50},\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{−\mathrm{1}} \left(−\frac{\mathrm{8}}{\mathrm{25}}\right) \\ $$$$\left({d}\right)\:\mathrm{60},\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{−\mathrm{1}} \left(−\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$

Question Number 20309 Answers: 0 Comments: 0

∫x^x dx

$$\int{x}^{{x}} {dx} \\ $$

Question Number 20308 Answers: 1 Comments: 0

For what value of k, (x + y + z)^2 + k(x^2 + y^2 + z^2 ) can be resolved into linear rational factors?

$$\mathrm{For}\:\mathrm{what}\:\mathrm{value}\:\mathrm{of}\:{k},\:\left({x}\:+\:{y}\:+\:{z}\right)^{\mathrm{2}} \:+ \\ $$$${k}\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{resolved}\:\mathrm{into} \\ $$$$\mathrm{linear}\:\mathrm{rational}\:\mathrm{factors}? \\ $$

Question Number 20307 Answers: 2 Comments: 0

Show that a(b − c)x^2 + b(c − a)xy + c(a − b)y^2 will be a perfect square if a, b, c are in H.P.

$$\mathrm{Show}\:\mathrm{that}\:{a}\left({b}\:−\:{c}\right){x}^{\mathrm{2}} \:+\:{b}\left({c}\:−\:{a}\right){xy}\:+ \\ $$$${c}\left({a}\:−\:{b}\right){y}^{\mathrm{2}} \:\mathrm{will}\:\mathrm{be}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{if}\:{a}, \\ $$$${b},\:{c}\:\mathrm{are}\:\mathrm{in}\:\mathrm{H}.\mathrm{P}. \\ $$

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