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Question Number 22379 Answers: 1 Comments: 0

For each positive integer n, define a_n = 20 + n^2 , and d_n = gcd(a_n , a_(n+1) ). Find the set of all values that are taken by d_n and show by examples that each of these values are attained.

$$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:{n},\:\mathrm{define}\:{a}_{{n}} \:= \\ $$$$\mathrm{20}\:+\:{n}^{\mathrm{2}} ,\:\mathrm{and}\:{d}_{{n}} \:=\:{gcd}\left({a}_{{n}} ,\:{a}_{{n}+\mathrm{1}} \right).\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are}\:\mathrm{taken}\:\mathrm{by} \\ $$$${d}_{{n}} \:\mathrm{and}\:\mathrm{show}\:\mathrm{by}\:\mathrm{examples}\:\mathrm{that}\:\mathrm{each}\:\mathrm{of} \\ $$$$\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \\ $$

Question Number 22372 Answers: 1 Comments: 4

Question Number 22370 Answers: 1 Comments: 3

A uniform flexible chain of length (3/2) m rests on a fixed smooth sphere of radius R = (2/π) m such that one end A of chain is on the top of the sphere while the other end B is hanging freely. Chain is held stationary by a horizontal thread PA. Calculate the acceleration of chain when the horizontal string PA is burnt. (g = 10 m/s^2 )

$$\mathrm{A}\:\mathrm{uniform}\:\mathrm{flexible}\:\mathrm{chain}\:\mathrm{of}\:\mathrm{length}\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{m} \\ $$$$\mathrm{rests}\:\mathrm{on}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{smooth}\:\mathrm{sphere}\:\mathrm{of} \\ $$$$\mathrm{radius}\:{R}\:=\:\frac{\mathrm{2}}{\pi}\:\mathrm{m}\:\mathrm{such}\:\mathrm{that}\:\mathrm{one}\:\mathrm{end}\:{A}\:\mathrm{of} \\ $$$$\mathrm{chain}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere}\:\mathrm{while} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{end}\:{B}\:\mathrm{is}\:\mathrm{hanging}\:\mathrm{freely}.\:\mathrm{Chain} \\ $$$$\mathrm{is}\:\mathrm{held}\:\mathrm{stationary}\:\mathrm{by}\:\mathrm{a}\:\mathrm{horizontal} \\ $$$$\mathrm{thread}\:{PA}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{acceleration} \\ $$$$\mathrm{of}\:\mathrm{chain}\:\mathrm{when}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{string}\:{PA} \\ $$$$\mathrm{is}\:\mathrm{burnt}.\:\left({g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$

Question Number 22368 Answers: 0 Comments: 0

The first and second ionization potentials of helium atoms are 24.58 eV and 54.4 eV per mole respectively. Calculate the energy in kJ required to produce 1 mole of He^(2+) ions.

$$\mathrm{The}\:\mathrm{first}\:\mathrm{and}\:\mathrm{second}\:\mathrm{ionization} \\ $$$$\mathrm{potentials}\:\mathrm{of}\:\mathrm{helium}\:\mathrm{atoms}\:\mathrm{are}\:\mathrm{24}.\mathrm{58}\:\mathrm{eV} \\ $$$$\mathrm{and}\:\mathrm{54}.\mathrm{4}\:\mathrm{eV}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{respectively}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{in}\:\mathrm{kJ}\:\mathrm{required}\:\mathrm{to} \\ $$$$\mathrm{produce}\:\mathrm{1}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{He}^{\mathrm{2}+} \:\mathrm{ions}. \\ $$

Question Number 22365 Answers: 1 Comments: 0

Question Number 22361 Answers: 0 Comments: 0

Question Number 22348 Answers: 1 Comments: 0

The sum of all the solutions of the equation 1 + 2 cosec x = −((sec^2 (x/2))/2) in the interval [0, 4π] is nπ, where n is equal to

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{1}\:+\:\mathrm{2}\:\mathrm{cosec}\:{x}\:=\:−\frac{\mathrm{sec}^{\mathrm{2}} \:\frac{{x}}{\mathrm{2}}}{\mathrm{2}}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{interval}\:\left[\mathrm{0},\:\mathrm{4}\pi\right]\:\mathrm{is}\:{n}\pi,\:\mathrm{where}\:{n}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 22347 Answers: 0 Comments: 0

Total number of solutions of ∣cot x∣ = cot x + (1/(sin x)), x ∈ [0, 3π] is equal to

$$\mathrm{Total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mid\mathrm{cot}\:{x}\mid\:= \\ $$$$\mathrm{cot}\:{x}\:+\:\frac{\mathrm{1}}{\mathrm{sin}\:{x}},\:{x}\:\in\:\left[\mathrm{0},\:\mathrm{3}\pi\right]\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 22345 Answers: 1 Comments: 1

6^(x+2) =2(3^x ), find x?

$$\mathrm{6}^{\mathrm{x}+\mathrm{2}} =\mathrm{2}\left(\mathrm{3}^{\mathrm{x}} \right),\:\mathrm{find}\:\mathrm{x}? \\ $$

Question Number 22332 Answers: 1 Comments: 0

Question Number 22325 Answers: 0 Comments: 0

If separate samples of argon, methane, nitrogen and carbon dioxide, all at the same initial temperature and pressure and expanded adiabatically reversibally to double their original volumes, then which one of these gases will have high temperature as final temperature?

$$\mathrm{If}\:\mathrm{separate}\:\mathrm{samples}\:\mathrm{of}\:\mathrm{argon},\:\mathrm{methane}, \\ $$$$\mathrm{nitrogen}\:\mathrm{and}\:\mathrm{carbon}\:\mathrm{dioxide},\:\mathrm{all}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{initial}\:\mathrm{temperature}\:\mathrm{and}\:\mathrm{pressure} \\ $$$$\mathrm{and}\:\mathrm{expanded}\:\mathrm{adiabatically}\:\mathrm{reversibally} \\ $$$$\mathrm{to}\:\mathrm{double}\:\mathrm{their}\:\mathrm{original}\:\mathrm{volumes},\:\mathrm{then} \\ $$$$\mathrm{which}\:\mathrm{one}\:\mathrm{of}\:\mathrm{these}\:\mathrm{gases}\:\mathrm{will}\:\mathrm{have}\:\mathrm{high} \\ $$$$\mathrm{temperature}\:\mathrm{as}\:\mathrm{final}\:\mathrm{temperature}? \\ $$

Question Number 22319 Answers: 1 Comments: 8

A string of negligible mass going over a clamped pulley of mass m supports a block of mass M. The force on the pulley by the clamp is given by

$$\mathrm{A}\:\mathrm{string}\:\mathrm{of}\:\mathrm{negligible}\:\mathrm{mass}\:\mathrm{going}\:\mathrm{over}\:\mathrm{a} \\ $$$$\mathrm{clamped}\:\mathrm{pulley}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{supports}\:\mathrm{a} \\ $$$$\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{M}.\:\mathrm{The}\:\mathrm{force}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{pulley}\:\mathrm{by}\:\mathrm{the}\:\mathrm{clamp}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$

Question Number 22316 Answers: 0 Comments: 0

Prove that the coefficient of x^p in the expansion of (a_0 +a_1 x+a_2 x^2 +a_3 x^3 +...+a_k x^k )^n is Σ((n!)/(n_0 !n_1 !n_2 !n_3 !...n_k !))a_0 ^n_0 a_1 ^n_1 a_2 ^n_2 a_3 ^n_3 ...a_k ^n_k where n_0 , n_1 , n_2 , n_3 , ..., n_k are all non- negative integers subject to the conditions n_0 +n_1 +n_2 +n_3 +...+n_k =n and n_1 +2n_2 +3n_3 +...+kn_k =p.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{p}} \:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left({a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +{a}_{\mathrm{3}} {x}^{\mathrm{3}} +...+{a}_{{k}} {x}^{{k}} \right)^{{n}} \\ $$$$\mathrm{is}\:\Sigma\frac{{n}!}{{n}_{\mathrm{0}} !{n}_{\mathrm{1}} !{n}_{\mathrm{2}} !{n}_{\mathrm{3}} !...{n}_{{k}} !}{a}_{\mathrm{0}} ^{{n}_{\mathrm{0}} } {a}_{\mathrm{1}} ^{{n}_{\mathrm{1}} } {a}_{\mathrm{2}} ^{{n}_{\mathrm{2}} } {a}_{\mathrm{3}} ^{{n}_{\mathrm{3}} } ...{a}_{{k}} ^{{n}_{{k}} } \\ $$$$\mathrm{where}\:{n}_{\mathrm{0}} ,\:{n}_{\mathrm{1}} ,\:{n}_{\mathrm{2}} ,\:{n}_{\mathrm{3}} ,\:...,\:{n}_{{k}} \:\mathrm{are}\:\mathrm{all}\:\mathrm{non}- \\ $$$$\mathrm{negative}\:\mathrm{integers}\:\mathrm{subject}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{conditions}\:{n}_{\mathrm{0}} +{n}_{\mathrm{1}} +{n}_{\mathrm{2}} +{n}_{\mathrm{3}} +...+{n}_{{k}} ={n} \\ $$$$\mathrm{and}\:{n}_{\mathrm{1}} +\mathrm{2}{n}_{\mathrm{2}} +\mathrm{3}{n}_{\mathrm{3}} +...+{kn}_{{k}} ={p}. \\ $$

Question Number 22304 Answers: 1 Comments: 0

A particle moves in a straight line along x-axis. At t = 0, it is released from rest at x = a. Acceleration of the particle varies as (d^2 x/dt^2 ) = −(k/x^2 ) , where k is a positive constant. Time required by particle to reach the origin will be

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{along} \\ $$$${x}-\mathrm{axis}.\:\mathrm{At}\:{t}\:=\:\mathrm{0},\:\mathrm{it}\:\mathrm{is}\:\mathrm{released}\:\mathrm{from} \\ $$$$\mathrm{rest}\:\mathrm{at}\:{x}\:=\:{a}.\:\mathrm{Acceleration}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{varies}\:\mathrm{as}\:\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }\:=\:−\frac{{k}}{{x}^{\mathrm{2}} }\:,\:\mathrm{where}\:{k} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{constant}. \\ $$$$\mathrm{Time}\:\mathrm{required}\:\mathrm{by}\:\mathrm{particle}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the} \\ $$$$\mathrm{origin}\:\mathrm{will}\:\mathrm{be} \\ $$

Question Number 22302 Answers: 1 Comments: 0

A small bead of mass m is given an initial velocity of magnitude v_0 on a horizontal circular wire. If the coefficient of kinetic friction is μ_k , the determine the distance travelled before the collar comes to rest. (Given that radius of circular wire is R).

$$\mathrm{A}\:\mathrm{small}\:\mathrm{bead}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{given}\:\mathrm{an} \\ $$$$\mathrm{initial}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{magnitude}\:{v}_{\mathrm{0}} \:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{circular}\:\mathrm{wire}.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{coefficient}\:\mathrm{of}\:\mathrm{kinetic}\:\mathrm{friction}\:\mathrm{is}\:\mu_{\mathrm{k}} ,\:\mathrm{the} \\ $$$$\mathrm{determine}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{travelled}\:\mathrm{before} \\ $$$$\mathrm{the}\:\mathrm{collar}\:\mathrm{comes}\:\mathrm{to}\:\mathrm{rest}.\:\left(\mathrm{Given}\:\mathrm{that}\right. \\ $$$$\left.\mathrm{radius}\:\mathrm{of}\:\mathrm{circular}\:\mathrm{wire}\:\mathrm{is}\:{R}\right). \\ $$

Question Number 22286 Answers: 1 Comments: 5

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire B is (a) always radially outwards (b) always radially inwards (c) radially outwards initially and radially inwards later (d) radially inwards initially and radially outwards later.

$$\mathrm{A}\:\mathrm{wire},\:\mathrm{which}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{hole} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{small}\:\mathrm{bead},\:\mathrm{is}\:\mathrm{bent}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{of} \\ $$$$\mathrm{quarter}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}.\:\mathrm{The}\:\mathrm{wire}\:\mathrm{is}\:\mathrm{fixed} \\ $$$$\mathrm{vertically}\:\mathrm{on}\:\mathrm{ground}.\:\mathrm{The}\:\mathrm{bead}\:\mathrm{is} \\ $$$$\mathrm{released}\:\mathrm{from}\:\mathrm{near}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire} \\ $$$$\mathrm{and}\:\mathrm{it}\:\mathrm{slides}\:\mathrm{along}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{without} \\ $$$$\mathrm{friction}.\:\mathrm{As}\:\mathrm{the}\:\mathrm{bead}\:\mathrm{moves}\:\mathrm{from}\:{A}\:\mathrm{to}\:{B}, \\ $$$$\mathrm{the}\:\mathrm{force}\:\mathrm{it}\:\mathrm{applies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{wire}\:{B}\:\mathrm{is} \\ $$$$\left({a}\right)\:\mathrm{always}\:\mathrm{radially}\:\mathrm{outwards} \\ $$$$\left({b}\right)\:\mathrm{always}\:\mathrm{radially}\:\mathrm{inwards} \\ $$$$\left({c}\right)\:\mathrm{radially}\:\mathrm{outwards}\:\mathrm{initially}\:\mathrm{and} \\ $$$$\mathrm{radially}\:\mathrm{inwards}\:\mathrm{later} \\ $$$$\left({d}\right)\:\mathrm{radially}\:\mathrm{inwards}\:\mathrm{initially}\:\mathrm{and} \\ $$$$\mathrm{radially}\:\mathrm{outwards}\:\mathrm{later}. \\ $$

Question Number 22279 Answers: 0 Comments: 0

Question Number 22299 Answers: 0 Comments: 0

If a certain mass of gas is made to undergo separately adiabatic and isothermal expansions to the same pressure, starting from the same initial conditions of temperature and pressure, then as compared to that of isothermal expansion, in the case of adiabatic expansion, the final (1) Volume and temperature will be higher (2) Volume and temperature will be lower (3) Temperature will be lower but the final volume will be higher (4) Volume will be lower but the final temperature will be higher

$$\mathrm{If}\:\mathrm{a}\:\mathrm{certain}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{gas}\:\mathrm{is}\:\mathrm{made}\:\mathrm{to} \\ $$$$\mathrm{undergo}\:\mathrm{separately}\:\mathrm{adiabatic}\:\mathrm{and} \\ $$$$\mathrm{isothermal}\:\mathrm{expansions}\:\mathrm{to}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{pressure},\:\mathrm{starting}\:\mathrm{from}\:\mathrm{the}\:\mathrm{same}\:\mathrm{initial} \\ $$$$\mathrm{conditions}\:\mathrm{of}\:\mathrm{temperature}\:\mathrm{and}\:\mathrm{pressure}, \\ $$$$\mathrm{then}\:\mathrm{as}\:\mathrm{compared}\:\mathrm{to}\:\mathrm{that}\:\mathrm{of}\:\mathrm{isothermal} \\ $$$$\mathrm{expansion},\:\mathrm{in}\:\mathrm{the}\:\mathrm{case}\:\mathrm{of}\:\mathrm{adiabatic} \\ $$$$\mathrm{expansion},\:\mathrm{the}\:\mathrm{final} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Volume}\:\mathrm{and}\:\mathrm{temperature}\:\mathrm{will}\:\mathrm{be} \\ $$$$\mathrm{higher} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Volume}\:\mathrm{and}\:\mathrm{temperature}\:\mathrm{will}\:\mathrm{be} \\ $$$$\mathrm{lower} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Temperature}\:\mathrm{will}\:\mathrm{be}\:\mathrm{lower}\:\mathrm{but}\:\mathrm{the} \\ $$$$\mathrm{final}\:\mathrm{volume}\:\mathrm{will}\:\mathrm{be}\:\mathrm{higher} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Volume}\:\mathrm{will}\:\mathrm{be}\:\mathrm{lower}\:\mathrm{but}\:\mathrm{the}\:\mathrm{final} \\ $$$$\mathrm{temperature}\:\mathrm{will}\:\mathrm{be}\:\mathrm{higher} \\ $$

Question Number 22273 Answers: 0 Comments: 1

Ant rided 30 meters in 12 seconds. How fast are rided ant?

$${Ant}\:{rided}\:\mathrm{30}\:{meters}\:{in}\:\mathrm{12}\:{seconds}.\:{How}\:{fast}\:{are}\:{rided}\:{ant}? \\ $$

Question Number 23106 Answers: 0 Comments: 0

Is 2(x+1) had a x=3

$${Is}\:\mathrm{2}\left({x}+\mathrm{1}\right)\:{had}\:{a}\:{x}=\mathrm{3} \\ $$

Question Number 22269 Answers: 1 Comments: 0

Product costs 599.99 zloty. Person have 433.94 zloty. How muh zloty is a rest or missing?

$${Product}\:{costs}\:\mathrm{599}.\mathrm{99}\:{zloty}.\:{Person}\:{have}\:\mathrm{433}.\mathrm{94}\:{zloty}. \\ $$$${How}\:{muh}\:{zloty}\:{is}\:{a}\:{rest}\:{or}\:{missing}? \\ $$

Question Number 22268 Answers: 1 Comments: 2

if tanx = (1/3) or other (except standard values) how to find x