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AllQuestion and Answers: Page 1845

Question Number 25109 Answers: 1 Comments: 0

Question Number 25103 Answers: 1 Comments: 0

Question Number 25091 Answers: 1 Comments: 0

A particle of mass m moving with speed u collides perfectly inelastically with a sphere of radius R and same mass, at rest, at an impact parameter d. Find (a) Angle between their final velocities (b) Magnitude of their final velocities

$${A}\:{particle}\:{of}\:{mass}\:{m}\:{moving}\:{with} \\ $$$${speed}\:{u}\:{collides}\:{perfectly}\:{inelastically} \\ $$$${with}\:{a}\:{sphere}\:{of}\:{radius}\:{R}\:{and}\:{same} \\ $$$${mass},\:{at}\:{rest},\:{at}\:{an}\:{impact}\:{parameter} \\ $$$${d}.\:{Find} \\ $$$$\left({a}\right)\:{Angle}\:{between}\:{their}\:{final}\:{velocities} \\ $$$$\left({b}\right)\:{Magnitude}\:{of}\:{their}\:{final} \\ $$$${velocities} \\ $$

Question Number 25094 Answers: 1 Comments: 0

Question Number 25088 Answers: 1 Comments: 1

Q...((x+7)/(x+4))>1, x∈R

$$ \\ $$$$ \\ $$$$ \\ $$$${Q}...\frac{{x}+\mathrm{7}}{{x}+\mathrm{4}}>\mathrm{1},\:\:\:\:\:{x}\in{R} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 25086 Answers: 0 Comments: 4

Question Number 25085 Answers: 1 Comments: 0

If a_n −a_(n−1) =1 for every positive integer greater than 1, then a_1 +a_2 +a_3 +...a_(100) equals (1) 5000 . a_1 (2) 5050 . a_1 (3) 5051 . a_1 (3) 5052 . a_2

$${If}\:{a}_{{n}} −{a}_{{n}−\mathrm{1}} =\mathrm{1}\:{for}\:{every}\:{positive} \\ $$$${integer}\:{greater}\:{than}\:\mathrm{1},\:{then}\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +{a}_{\mathrm{3}} \\ $$$$+...{a}_{\mathrm{100}} \:{equals} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{5000}\:.\:{a}_{\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{5050}\:.\:{a}_{\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{5051}\:.\:{a}_{\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{5052}\:.\:{a}_{\mathrm{2}} \\ $$

Question Number 25068 Answers: 1 Comments: 0

Evaluate lim_(x→(π/2)) (((tan2x)/(x−π/2)))

$${Evaluate}\: \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\left(\frac{{tan}\mathrm{2}{x}}{{x}−\pi/\mathrm{2}}\right) \\ $$

Question Number 25074 Answers: 0 Comments: 1

Question Number 25075 Answers: 1 Comments: 0

Let z_1 and z_2 be two roots of the equation z^2 +az+b=0, z being complex. Further assume that the origin, z_1 and z_2 form an equilateral triangle. Then,

$$\mathrm{Let}\:{z}_{\mathrm{1}} \mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${z}^{\mathrm{2}} +{az}+{b}=\mathrm{0},\:{z}\:\mathrm{being}\:\mathrm{complex}.\:\mathrm{Further} \\ $$$$\mathrm{assume}\:\mathrm{that}\:\mathrm{the}\:\mathrm{origin},\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{form} \\ $$$$\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}.\:\mathrm{Then}, \\ $$

Question Number 25066 Answers: 1 Comments: 0

let a,b,c,x,y and z be complex number such that a=((b+c)/(x−2)) ,b=((c+a)/(y−2)) c=((a+b)/(z−2)). xy +yz +zx=1000 and x+y+z=2016 find the value of xyz.

$${let}\:{a},{b},{c},{x},{y}\:{and}\:{z}\:{be}\:{complex}\:{number} \\ $$$${such}\:{that}\:{a}=\frac{{b}+{c}}{{x}−\mathrm{2}}\:,{b}=\frac{{c}+{a}}{{y}−\mathrm{2}}\:\:\:\:{c}=\frac{{a}+{b}}{{z}−\mathrm{2}}. \\ $$$${xy}\:+{yz}\:+{zx}=\mathrm{1000}\:{and}\:{x}+{y}+{z}=\mathrm{2016} \\ $$$${find}\:{the}\:{value}\:{of}\:{xyz}. \\ $$

Question Number 25076 Answers: 1 Comments: 0

A man can row 6 km/h in still water. When the river is running at 1.2 km/h, it takes]him 1 hour to row to a place and back. How far is the place?

$$\mathrm{A}\:\mathrm{man}\:\mathrm{can}\:\mathrm{row}\:\mathrm{6}\:\mathrm{km}/\mathrm{h}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water}. \\ $$$$\mathrm{When}\:\mathrm{the}\:\mathrm{river}\:\mathrm{is}\:\mathrm{running}\:\mathrm{at}\:\mathrm{1}.\mathrm{2}\:\mathrm{km}/\mathrm{h}, \\ $$$$\left.\mathrm{it}\:\mathrm{takes}\right]\mathrm{him}\:\mathrm{1}\:\mathrm{hour}\:\mathrm{to}\:\mathrm{row}\:\mathrm{to}\:\mathrm{a}\:\mathrm{place}\: \\ $$$$\mathrm{and}\:\mathrm{back}.\:\mathrm{How}\:\mathrm{far}\:\mathrm{is}\:\mathrm{the}\:\mathrm{place}? \\ $$

Question Number 25058 Answers: 1 Comments: 0

Two objects slide over a frictionless horizontal surface. The first object, mass m_1 = 5 kg, is propelled with a speed u = 4.5 m/s towards the second object, mass m_2 = 5 kg, which is initially at rest. After the collision, both objects have velocities which are directed at θ = 60° on either side of the original line of motion of the first object. What can you say about the elasticity of collision?

$${Two}\:{objects}\:{slide}\:{over}\:{a}\:{frictionless} \\ $$$${horizontal}\:{surface}.\:{The}\:{first}\:{object}, \\ $$$${mass}\:{m}_{\mathrm{1}} \:=\:\mathrm{5}\:{kg},\:{is}\:{propelled}\:{with}\:{a} \\ $$$${speed}\:{u}\:=\:\mathrm{4}.\mathrm{5}\:{m}/{s}\:{towards}\:{the}\:{second} \\ $$$${object},\:{mass}\:{m}_{\mathrm{2}} \:=\:\mathrm{5}\:{kg},\:{which}\:{is} \\ $$$${initially}\:{at}\:{rest}.\:{After}\:{the}\:{collision}, \\ $$$${both}\:{objects}\:{have}\:{velocities}\:{which}\:{are} \\ $$$${directed}\:{at}\:\theta\:=\:\mathrm{60}°\:{on}\:{either}\:{side}\:{of} \\ $$$${the}\:{original}\:{line}\:{of}\:{motion}\:{of}\:{the} \\ $$$${first}\:{object}.\:{What}\:{can}\:{you}\:{say}\:{about} \\ $$$${the}\:{elasticity}\:{of}\:{collision}? \\ $$

Question Number 25054 Answers: 1 Comments: 0

If x:y=5:2, then find the value of (8x+9y)/(8x+27)

$${If}\:{x}:{y}=\mathrm{5}:\mathrm{2},\:{then}\:{find}\:{the}\:{value}\:{of}\:\left(\mathrm{8}{x}+\mathrm{9}{y}\right)/\left(\mathrm{8}{x}+\mathrm{27}\right) \\ $$

Question Number 25053 Answers: 2 Comments: 0

If 2A=3B=4C find the value of A:B:C

$${If}\:\mathrm{2}{A}=\mathrm{3}{B}=\mathrm{4}{C}\:{find}\:{the}\:{value}\:{of}\:{A}:{B}:{C} \\ $$

Question Number 25049 Answers: 1 Comments: 1

If I = Σ_(k=1) ^(98) ∫_k ^(k+1) ((k + 1)/(x(x + 1)))dx, then (1) I > ((49)/(50)) (2) I < ((49)/(50)) (3) I < log_e 99 (4) I > log_e 99

$$\mathrm{If}\:{I}\:=\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{98}} {\sum}}\underset{{k}} {\overset{{k}+\mathrm{1}} {\int}}\frac{{k}\:+\:\mathrm{1}}{{x}\left({x}\:+\:\mathrm{1}\right)}{dx},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{I}\:>\:\frac{\mathrm{49}}{\mathrm{50}} \\ $$$$\left(\mathrm{2}\right)\:{I}\:<\:\frac{\mathrm{49}}{\mathrm{50}} \\ $$$$\left(\mathrm{3}\right)\:{I}\:<\:\mathrm{log}_{{e}} \mathrm{99} \\ $$$$\left(\mathrm{4}\right)\:{I}\:>\:\mathrm{log}_{{e}} \mathrm{99} \\ $$

Question Number 25046 Answers: 1 Comments: 0

Show that (a) N=((10^(143) −1)/9) is composite, and (b) N has two factors each of which is a series of a G.P.

$${Show}\:{that} \\ $$$$\left({a}\right)\:{N}=\frac{\mathrm{10}^{\mathrm{143}} −\mathrm{1}}{\mathrm{9}}\:{is}\:{composite},\:{and} \\ $$$$\left({b}\right)\:{N}\:{has}\:{two}\:{factors}\:{each}\:{of}\:{which}\:{is} \\ $$$${a}\:{series}\:{of}\:{a}\:{G}.{P}. \\ $$

Question Number 25038 Answers: 1 Comments: 0

For a particle of a rotating rigid body, v = rω. So (1) ω ∝ (1/r) (2) ω ∝ v (3) v ∝ r (4) ω is independent of r

$$\mathrm{For}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{a}\:\mathrm{rotating}\:\mathrm{rigid}\:\mathrm{body}, \\ $$$${v}\:=\:{r}\omega.\:\mathrm{So} \\ $$$$\left(\mathrm{1}\right)\:\omega\:\propto\:\left(\mathrm{1}/{r}\right) \\ $$$$\left(\mathrm{2}\right)\:\omega\:\propto\:{v} \\ $$$$\left(\mathrm{3}\right)\:{v}\:\propto\:{r} \\ $$$$\left(\mathrm{4}\right)\:\omega\:\mathrm{is}\:\mathrm{independent}\:\mathrm{of}\:{r} \\ $$

Question Number 25034 Answers: 2 Comments: 0

f(x)=((sin x+sec x)/(1+xtan x)) find f′(x)

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{sec}\:\mathrm{x}}{\mathrm{1}+\mathrm{xtan}\:\mathrm{x}} \\ $$$$ \\ $$$$\mathrm{find}\:\mathrm{f}'\left(\mathrm{x}\right) \\ $$

Question Number 25026 Answers: 2 Comments: 0

lim_(x→1) (((x^(1/3) −1)/(x^(1/4) −1))) Evaluate this

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\frac{{x}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}}{{x}^{\mathrm{1}/\mathrm{4}} −\mathrm{1}}\right) \\ $$$${Evaluate}\:{this} \\ $$$$ \\ $$

Question Number 25025 Answers: 2 Comments: 0

If a^4 + b^4 + c^4 + d^4 = 16, prove that: a^5 + b^5 + c^5 + d^5 ≤ 32 for a, b, c, d ∈ R

$$\mathrm{If}\:\:\:\:\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:+\:\mathrm{d}^{\mathrm{4}} \:=\:\mathrm{16},\:\:\mathrm{prove}\:\mathrm{that}:\:\:\mathrm{a}^{\mathrm{5}} \:+\:\mathrm{b}^{\mathrm{5}} \:+\:\mathrm{c}^{\mathrm{5}} \:+\:\mathrm{d}^{\mathrm{5}} \:\leqslant\:\mathrm{32} \\ $$$$\mathrm{for}\:\:\mathrm{a},\:\mathrm{b},\:\mathrm{c},\:\mathrm{d}\:\in\:\mathbb{R} \\ $$

Question Number 25023 Answers: 1 Comments: 0

Consider the function f(x) which satisfying the functional equation 2f(x) + f(1 − x) = x^2 + 1, ∀ x ∈ R and g(x) = 3f(x) + 1. The range of φ(x) = g(x) + (1/(g(x) + 1)) is

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{which} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{functional}\:\mathrm{equation} \\ $$$$\mathrm{2}{f}\left({x}\right)\:+\:{f}\left(\mathrm{1}\:−\:{x}\right)\:=\:{x}^{\mathrm{2}} \:+\:\mathrm{1},\:\forall\:{x}\:\in\:{R} \\ $$$$\mathrm{and}\:{g}\left({x}\right)\:=\:\mathrm{3}{f}\left({x}\right)\:+\:\mathrm{1}.\:\mathrm{The}\:\mathrm{range}\:\mathrm{of} \\ $$$$\phi\left({x}\right)\:=\:{g}\left({x}\right)\:+\:\frac{\mathrm{1}}{{g}\left({x}\right)\:+\:\mathrm{1}}\:\mathrm{is} \\ $$

Question Number 25013 Answers: 0 Comments: 3

With reference to figure of a cube of edge a and mass m, state whether the following are true or false. (O is the centre of the cube.) (1) The moment of inertia of cube about z-axis is, I_z = I_x + I_y (2) The moment of inertia of cube about z′ is, I_(z′) = I_z + ((ma^2 )/2) (3) The moment of inertia of cube about z′′ is, I_(z′) = I_z + ((ma^2 )/2) (4) I_x = I_y

Question Number 25002 Answers: 1 Comments: 1

Question Number 25001 Answers: 0 Comments: 5

If x, y > 0, then the minimum value of 2x^2 + (2/x) − 2x + 2y^2 + (2/y) − 2y + 2 is equal to

$$\mathrm{If}\:{x},\:{y}\:>\:\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:+\:\frac{\mathrm{2}}{{x}}\:−\:\mathrm{2}{x}\:+\:\mathrm{2}{y}^{\mathrm{2}} \:+\:\frac{\mathrm{2}}{{y}}\:−\:\mathrm{2}{y}\:+\:\mathrm{2}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 25000 Answers: 1 Comments: 0

find x,y from the equation: (1/2)x−yi+(1/(1+i))=((√(1+ω^8 ))+(√(1+ω^(10) )))^4

$${find}\:{x},{y}\:{from}\:{the}\:{equation}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{x}−{yi}+\frac{\mathrm{1}}{\mathrm{1}+{i}}=\left(\sqrt{\mathrm{1}+\omega^{\mathrm{8}} }+\sqrt{\mathrm{1}+\omega^{\mathrm{10}} }\right)^{\mathrm{4}} \\ $$$$ \\ $$

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