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Question Number 21825    Answers: 0   Comments: 2

Find the simplest form of Σ_(k = 1) ^n 2^k [sin^2 (((2kπ)/3)) + (1/4)]

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{simplest}\:\mathrm{form}\:\mathrm{of} \\ $$$$\underset{{k}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{2}^{{k}} \left[\mathrm{sin}^{\mathrm{2}} \:\left(\frac{\mathrm{2}{k}\pi}{\mathrm{3}}\right)\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right] \\ $$

Question Number 21964    Answers: 1   Comments: 0

Question Number 21965    Answers: 1   Comments: 0

integrate ∫sin^3 xdx

$${integrate} \\ $$$$\int{sin}^{\mathrm{3}} {xdx} \\ $$

Question Number 22051    Answers: 0   Comments: 3

I have recently seen a different notation for integration, written as: ∫dxf(x) e.g. ∫dx(x+1)^2 Is this the same as: ∫f(x)dx ⇒ =∫(x+1)^2 dx ???

$$\mathrm{I}\:\mathrm{have}\:\mathrm{recently}\:\mathrm{seen}\:\mathrm{a}\:\mathrm{different}\:\mathrm{notation} \\ $$$$\mathrm{for}\:\mathrm{integration},\:\mathrm{written}\:\mathrm{as}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int{dxf}\left({x}\right) \\ $$$${e}.{g}. \\ $$$$\:\:\:\:\:\:\int{dx}\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\: \\ $$$$\mathrm{Is}\:\mathrm{this}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int{f}\left({x}\right){dx} \\ $$$$\Rightarrow\:\:\:\:=\int\left({x}+\mathrm{1}\right)^{\mathrm{2}} {dx} \\ $$$$??? \\ $$

Question Number 21819    Answers: 0   Comments: 3

Consider the situation shown in figure in which a block P of mass 2 kg is placed over a block Q of mass 4 kg. The combination of the blocks are placed on inclined plane of inclination 37° with horizontal. The coefficient of friction between Block Q and inclined plane is μ_2 and in between the two blocks is μ_1 . The system is released from rest, then when will be the frictional force acting between the block is zero? (p) μ_1 = 0.4; μ_2 = 0 (q) μ_1 = 0.8; μ_2 = 0.8 (r) μ_1 = 0.4; μ_2 = 0.5 (s) μ_1 = 0.5; μ_2 = 0.4 (t) μ_1 = 0; μ_2 = 0.4

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{situation}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{a}\:\mathrm{block}\:{P}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{placed} \\ $$$$\mathrm{over}\:\mathrm{a}\:\mathrm{block}\:{Q}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{4}\:\mathrm{kg}.\:\mathrm{The} \\ $$$$\mathrm{combination}\:\mathrm{of}\:\mathrm{the}\:\mathrm{blocks}\:\mathrm{are}\:\mathrm{placed}\:\mathrm{on} \\ $$$$\mathrm{inclined}\:\mathrm{plane}\:\mathrm{of}\:\mathrm{inclination}\:\mathrm{37}°\:\mathrm{with} \\ $$$$\mathrm{horizontal}.\:\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{friction} \\ $$$$\mathrm{between}\:\mathrm{Block}\:{Q}\:\mathrm{and}\:\mathrm{inclined}\:\mathrm{plane}\:\mathrm{is} \\ $$$$\mu_{\mathrm{2}} \:\mathrm{and}\:\mathrm{in}\:\mathrm{between}\:\mathrm{the}\:\mathrm{two}\:\mathrm{blocks}\:\mathrm{is}\:\mu_{\mathrm{1}} . \\ $$$$\mathrm{The}\:\mathrm{system}\:\mathrm{is}\:\mathrm{released}\:\mathrm{from}\:\mathrm{rest},\:\mathrm{then} \\ $$$$\mathrm{when}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{frictional}\:\mathrm{force} \\ $$$$\mathrm{acting}\:\mathrm{between}\:\mathrm{the}\:\mathrm{block}\:\mathrm{is}\:\mathrm{zero}? \\ $$$$\left(\mathrm{p}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0}.\mathrm{4};\:\mu_{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\left(\mathrm{q}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0}.\mathrm{8};\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\mathrm{8} \\ $$$$\left(\mathrm{r}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0}.\mathrm{4};\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\mathrm{5} \\ $$$$\left(\mathrm{s}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0}.\mathrm{5};\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\mathrm{4} \\ $$$$\left(\mathrm{t}\right)\:\mu_{\mathrm{1}} \:=\:\mathrm{0};\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\mathrm{4} \\ $$

Question Number 21867    Answers: 0   Comments: 0

The number of points in the cartesian plane with integral coordinates satisfying the inequalities ∣x∣ ≤ 4, ∣y∣ ≤ 4 and ∣x − y∣ ≤ 4 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{points}\:\mathrm{in}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{plane}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coordinates} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{inequalities}\:\mid{x}\mid\:\leqslant\:\mathrm{4},\:\mid{y}\mid\:\leqslant \\ $$$$\mathrm{4}\:\mathrm{and}\:\mid{x}\:−\:{y}\mid\:\leqslant\:\mathrm{4}\:\mathrm{is} \\ $$

Question Number 21815    Answers: 0   Comments: 2

sin18^0 =

$${sin}\mathrm{18}^{\mathrm{0}} = \\ $$

Question Number 21813    Answers: 0   Comments: 0

x_1 +x_2 +x_3 =5 ways=5−1C_(3−1) =4C_2 =6

$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} =\mathrm{5} \\ $$$${ways}=\mathrm{5}−\mathrm{1}{C}_{\mathrm{3}−\mathrm{1}} =\mathrm{4}{C}_{\mathrm{2}} =\mathrm{6} \\ $$

Question Number 21811    Answers: 0   Comments: 1

a_1 =1, a_(n+1) =(a_n /(√(a_n +n+1))) Σ_(n=1) ^∞ a_n =?

$${a}_{\mathrm{1}} =\mathrm{1},\:{a}_{{n}+\mathrm{1}} =\frac{{a}_{{n}} }{\sqrt{{a}_{{n}} +{n}+\mathrm{1}}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} =? \\ $$

Question Number 21810    Answers: 1   Comments: 0

integrate ∫(x^2 /(√(1−x^2 )))dx

$${integrate} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 21802    Answers: 1   Comments: 1

Five balls are to be placed in three boxes. Each can hold all the five balls. In how many different ways can we place the balls so that no box remains empty, if balls are different but boxes are identical?

$$\mathrm{Five}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{three} \\ $$$$\mathrm{boxes}.\:\mathrm{Each}\:\mathrm{can}\:\mathrm{hold}\:\mathrm{all}\:\mathrm{the}\:\mathrm{five}\:\mathrm{balls}. \\ $$$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{different}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{we} \\ $$$$\mathrm{place}\:\mathrm{the}\:\mathrm{balls}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{box}\:\mathrm{remains} \\ $$$$\mathrm{empty},\:\mathrm{if}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{boxes} \\ $$$$\mathrm{are}\:\mathrm{identical}? \\ $$

Question Number 21801    Answers: 0   Comments: 0

There are n straight lines in a plane, no two of which are parallel and no three pass through the same point. Their point of intersection are joined. Then the number of fresh lines thus obtained is

$$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane},\:\mathrm{no} \\ $$$$\mathrm{two}\:\mathrm{of}\:\mathrm{which}\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{and}\:\mathrm{no}\:\mathrm{three} \\ $$$$\mathrm{pass}\:\mathrm{through}\:\mathrm{the}\:\mathrm{same}\:\mathrm{point}.\:\mathrm{Their} \\ $$$$\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{are}\:\mathrm{joined}.\:\mathrm{Then} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fresh}\:\mathrm{lines}\:\mathrm{thus}\:\mathrm{obtained} \\ $$$$\mathrm{is} \\ $$

Question Number 21800    Answers: 0   Comments: 2

The number of integers which lie between 1 and 10^6 and which have sum of digits equal to 12 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{lie} \\ $$$$\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{10}^{\mathrm{6}} \:\mathrm{and}\:\mathrm{which}\:\mathrm{have}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{digits}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{12}\:\mathrm{is} \\ $$

Question Number 21799    Answers: 0   Comments: 2

There are n white and n red balls marked 1, 2, 3, ....n. The number of ways we can arrange these balls in a row so that neighbouring balls are of different colours is

$$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{white}\:\mathrm{and}\:{n}\:\mathrm{red}\:\mathrm{balls} \\ $$$$\mathrm{marked}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:....{n}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{we}\:\mathrm{can}\:\mathrm{arrange}\:\mathrm{these}\:\mathrm{balls}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{row}\:\mathrm{so}\:\mathrm{that}\:\mathrm{neighbouring}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{of} \\ $$$$\mathrm{different}\:\mathrm{colours}\:\mathrm{is} \\ $$

Question Number 21795    Answers: 1   Comments: 3

help x∈N determine x where 7 divise 2^x +3^x

$${help} \\ $$$${x}\in{N} \\ $$$${determine}\:{x}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$$ \\ $$

Question Number 21793    Answers: 0   Comments: 6

A plank of mass 10 kg rests on a smooth horizontal surface. Two blocks A and B of masses m_A = 2 kg and m_B = 1 kg lies at a distance of 3 m on the plank. The friction coefficient between the blocks and plank are μ_A = 0.3 and μ_B = 0.1. Now a force F = 15 N is applied to the plank in horizontal direction. Find the times (in sec) after which block A collides with B.

$$\mathrm{A}\:\mathrm{plank}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{10}\:\mathrm{kg}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}.\:\mathrm{Two}\:\mathrm{blocks}\:\mathrm{A}\:\mathrm{and} \\ $$$$\mathrm{B}\:\mathrm{of}\:\mathrm{masses}\:\mathrm{m}_{\mathrm{A}} \:=\:\mathrm{2}\:\mathrm{kg}\:\mathrm{and}\:\mathrm{m}_{\mathrm{B}} \:=\:\mathrm{1}\:\mathrm{kg} \\ $$$$\mathrm{lies}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{3}\:\mathrm{m}\:\mathrm{on}\:\mathrm{the}\:\mathrm{plank}. \\ $$$$\mathrm{The}\:\mathrm{friction}\:\mathrm{coefficient}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{and}\:\mathrm{plank}\:\mathrm{are}\:\mu_{\mathrm{A}} \:=\:\mathrm{0}.\mathrm{3}\:\mathrm{and}\:\mu_{\mathrm{B}} \:= \\ $$$$\mathrm{0}.\mathrm{1}.\:\mathrm{Now}\:\mathrm{a}\:\mathrm{force}\:\mathrm{F}\:=\:\mathrm{15}\:\mathrm{N}\:\mathrm{is}\:\mathrm{applied}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{plank}\:\mathrm{in}\:\mathrm{horizontal}\:\mathrm{direction}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{times}\:\left(\mathrm{in}\:\mathrm{sec}\right)\:\mathrm{after}\:\mathrm{which}\:\mathrm{block}\:\mathrm{A} \\ $$$$\mathrm{collides}\:\mathrm{with}\:\mathrm{B}. \\ $$

Question Number 24687    Answers: 1   Comments: 0

A particle starts from rest at t = 0 and moves with uniform acceleration. Then (1) In any time interval starting from t = 0 the space-average of the velocity is (4/3) times of time average velocity (2) If v = v_1 at t = t_1 and v = v_2 at t = t_2 then time average velocity between t_1 and t_2 is ((v_1 + v_2 )/2) (3) Distance travelled in successive equal time intervals are in proportion of 1 : 3 : 5 ... and so on (4) If v_1 , v_2 , v_3 denote the average velocities in three successive intervals of time t_1 , t_2 , t_3 then ((v_1 − v_2 )/(v_2 − v_3 )) = ((t_1 + t_2 )/(t_2 + t_3 ))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{starts}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{and} \\ $$$$\mathrm{moves}\:\mathrm{with}\:\mathrm{uniform}\:\mathrm{acceleration}.\:\mathrm{Then} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{In}\:\mathrm{any}\:\mathrm{time}\:\mathrm{interval}\:\mathrm{starting}\:\mathrm{from} \\ $$$${t}\:=\:\mathrm{0}\:\mathrm{the}\:\mathrm{space}-\mathrm{average}\:\mathrm{of}\:\mathrm{the}\:\mathrm{velocity} \\ $$$$\mathrm{is}\:\frac{\mathrm{4}}{\mathrm{3}}\:\mathrm{times}\:\mathrm{of}\:\mathrm{time}\:\mathrm{average}\:\mathrm{velocity} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{If}\:{v}\:=\:{v}_{\mathrm{1}} \:\mathrm{at}\:{t}\:=\:{t}_{\mathrm{1}} \:\mathrm{and}\:{v}\:=\:{v}_{\mathrm{2}} \:\mathrm{at}\:{t}\:=\:{t}_{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{time}\:\mathrm{average}\:\mathrm{velocity}\:\mathrm{between}\:{t}_{\mathrm{1}} \\ $$$$\mathrm{and}\:{t}_{\mathrm{2}} \:\mathrm{is}\:\frac{{v}_{\mathrm{1}} \:+\:{v}_{\mathrm{2}} }{\mathrm{2}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Distance}\:\mathrm{travelled}\:\mathrm{in}\:\mathrm{successive} \\ $$$$\mathrm{equal}\:\mathrm{time}\:\mathrm{intervals}\:\mathrm{are}\:\mathrm{in}\:\mathrm{proportion} \\ $$$$\mathrm{of}\:\mathrm{1}\::\:\mathrm{3}\::\:\mathrm{5}\:...\:\mathrm{and}\:\mathrm{so}\:\mathrm{on} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{If}\:{v}_{\mathrm{1}} ,\:{v}_{\mathrm{2}} ,\:{v}_{\mathrm{3}} \:\mathrm{denote}\:\mathrm{the}\:\mathrm{average} \\ $$$$\mathrm{velocities}\:\mathrm{in}\:\mathrm{three}\:\mathrm{successive}\:\mathrm{intervals} \\ $$$$\mathrm{of}\:\mathrm{time}\:{t}_{\mathrm{1}} ,\:{t}_{\mathrm{2}} ,\:{t}_{\mathrm{3}} \:\mathrm{then}\:\frac{{v}_{\mathrm{1}} \:−\:{v}_{\mathrm{2}} }{{v}_{\mathrm{2}} \:−\:{v}_{\mathrm{3}} }\:=\:\frac{{t}_{\mathrm{1}} \:+\:{t}_{\mathrm{2}} }{{t}_{\mathrm{2}} \:+\:{t}_{\mathrm{3}} } \\ $$

Question Number 24685    Answers: 0   Comments: 3

The sum of the kinetic and potential energies of a system of objects is conserved (1) Only when no external force acts on the objects (2) Only when the objects move along closed paths (3) Only when the work done by the resultant external force is zero (4) None of these

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{kinetic}\:\mathrm{and}\:\mathrm{potential} \\ $$$$\mathrm{energies}\:\mathrm{of}\:\mathrm{a}\:\mathrm{system}\:\mathrm{of}\:\mathrm{objects}\:\mathrm{is}\:\mathrm{conserved} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Only}\:\mathrm{when}\:\mathrm{no}\:\mathrm{external}\:\mathrm{force}\:\mathrm{acts}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{objects} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{objects}\:\mathrm{move}\:\mathrm{along} \\ $$$$\mathrm{closed}\:\mathrm{paths} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{work}\:\mathrm{done}\:\mathrm{by}\:\mathrm{the} \\ $$$$\mathrm{resultant}\:\mathrm{external}\:\mathrm{force}\:\mathrm{is}\:\mathrm{zero} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these} \\ $$

Question Number 21772    Answers: 2   Comments: 0

integrate ∫2^(4x) dx

$${integrate} \\ $$$$\int\mathrm{2}^{\mathrm{4}{x}} {dx} \\ $$

Question Number 21771    Answers: 0   Comments: 0

The result of 11 chess matches (as win, lose or draw) are to be forecast. Out of all possible forecasts, the number of ways in which 8 correct and 3 incorrect results can be forecast is

$$\mathrm{The}\:\mathrm{result}\:\mathrm{of}\:\mathrm{11}\:\mathrm{chess}\:\mathrm{matches}\:\left(\mathrm{as}\:\mathrm{win},\right. \\ $$$$\left.\mathrm{lose}\:\mathrm{or}\:\mathrm{draw}\right)\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{forecast}.\:\mathrm{Out}\:\mathrm{of} \\ $$$$\mathrm{all}\:\mathrm{possible}\:\mathrm{forecasts},\:\mathrm{the}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{8}\:\mathrm{correct}\:\mathrm{and}\:\mathrm{3}\:\mathrm{incorrect} \\ $$$$\mathrm{results}\:\mathrm{can}\:\mathrm{be}\:\mathrm{forecast}\:\mathrm{is} \\ $$

Question Number 21768    Answers: 0   Comments: 2

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

$$\mathrm{Six}\:\mathrm{cards}\:\mathrm{and}\:\mathrm{six}\:\mathrm{envelopes}\:\mathrm{are}\:\mathrm{numbered} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5},\:\mathrm{6}\:\mathrm{and}\:\mathrm{cards}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed} \\ $$$$\mathrm{in}\:\mathrm{envelopes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{each}\:\mathrm{envelope} \\ $$$$\mathrm{contains}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{card}\:\mathrm{and}\:\mathrm{no}\:\mathrm{card} \\ $$$$\mathrm{is}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{envelope}\:\mathrm{bearing}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{number}\:\mathrm{and}\:\mathrm{moreover}\:\mathrm{the}\:\mathrm{card} \\ $$$$\mathrm{numbered}\:\mathrm{1}\:\mathrm{is}\:\mathrm{always}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{envelope} \\ $$$$\mathrm{numbered}\:\mathrm{2}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways} \\ $$$$\mathrm{it}\:\mathrm{can}\:\mathrm{be}\:\mathrm{done}\:\mathrm{is} \\ $$

Question Number 21766    Answers: 1   Comments: 0

There are 3 apartments A, B and C for rent in a building. Each apartment will accept either 3 or 4 occupants. The number of ways of renting the apartments to 10 students

$$\mathrm{There}\:\mathrm{are}\:\mathrm{3}\:\mathrm{apartments}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{for} \\ $$$$\mathrm{rent}\:\mathrm{in}\:\mathrm{a}\:\mathrm{building}.\:\mathrm{Each}\:\mathrm{apartment}\:\mathrm{will} \\ $$$$\mathrm{accept}\:\mathrm{either}\:\mathrm{3}\:\mathrm{or}\:\mathrm{4}\:\mathrm{occupants}.\:\mathrm{The} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{renting}\:\mathrm{the} \\ $$$$\mathrm{apartments}\:\mathrm{to}\:\mathrm{10}\:\mathrm{students} \\ $$

Question Number 21760    Answers: 1   Comments: 0

There are m points on the line AB and n points on the line AC, excluding the point A. Triangles are formed joining these points (i) When point A is not included, (ii) When point A is included. The ratio of the number of such triangles is

$$\mathrm{There}\:\mathrm{are}\:{m}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AB}\:\mathrm{and} \\ $$$${n}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AC},\:\mathrm{excluding}\:\mathrm{the} \\ $$$$\mathrm{point}\:{A}.\:\mathrm{Triangles}\:\mathrm{are}\:\mathrm{formed}\:\mathrm{joining} \\ $$$$\mathrm{these}\:\mathrm{points} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{When}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{not}\:\mathrm{included}, \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{When}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{included}. \\ $$$$\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{such}\:\mathrm{triangles} \\ $$$$\mathrm{is} \\ $$

Question Number 21759    Answers: 1   Comments: 0

A block of mass 1 kg is pushed against a rough vertical wall with a force of 20 N, coefficient of static friction being (1/4). Another horizontal force of 10 N is applied on the block in a direction parallel to the wall. Will the block move? If yes, with what acceleration? If no, find the frictional force exerted by wall on the block. (g = 10 m/s^2 )

$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{1}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{pushed}\:\mathrm{against} \\ $$$$\mathrm{a}\:\mathrm{rough}\:\mathrm{vertical}\:\mathrm{wall}\:\mathrm{with}\:\mathrm{a}\:\mathrm{force}\:\mathrm{of}\:\mathrm{20} \\ $$$$\mathrm{N},\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{static}\:\mathrm{friction}\:\mathrm{being}\:\frac{\mathrm{1}}{\mathrm{4}}. \\ $$$$\mathrm{Another}\:\mathrm{horizontal}\:\mathrm{force}\:\mathrm{of}\:\mathrm{10}\:\mathrm{N}\:\mathrm{is} \\ $$$$\mathrm{applied}\:\mathrm{on}\:\mathrm{the}\:\mathrm{block}\:\mathrm{in}\:\mathrm{a}\:\mathrm{direction} \\ $$$$\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{wall}.\:\mathrm{Will}\:\mathrm{the}\:\mathrm{block}\:\mathrm{move}? \\ $$$$\mathrm{If}\:\mathrm{yes},\:\mathrm{with}\:\mathrm{what}\:\mathrm{acceleration}?\:\mathrm{If}\:\mathrm{no}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{frictional}\:\mathrm{force}\:\mathrm{exerted}\:\mathrm{by}\:\mathrm{wall} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{block}.\:\left({g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$

Question Number 21756    Answers: 0   Comments: 0

Prove that the product for all nth roots of unity is equal to zero, except n=1. Note: U_n ={e^(2kπi/n) ∣ k∈{1, 2, ..., n}} x^n =1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{for}\:\mathrm{all} \\ $$$${n}\mathrm{th}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{zero}, \\ $$$$\mathrm{except}\:{n}=\mathrm{1}. \\ $$$$\: \\ $$$$\mathrm{Note}: \\ $$$${U}_{{n}} =\left\{{e}^{\mathrm{2}{k}\pi{i}/{n}} \:\mid\:{k}\in\left\{\mathrm{1},\:\mathrm{2},\:...,\:{n}\right\}\right\} \\ $$$${x}^{{n}} =\mathrm{1} \\ $$

Question Number 21754    Answers: 1   Comments: 0

Q. In 2014, country X had 783 miles of paved roads. Starting in 2015, the country has been building 8 miles of new paved roads each year. At this rate, how many miles of paved roads will country X have in 2030?

$$\mathrm{Q}.\:\mathrm{In}\:\mathrm{2014},\:\mathrm{country}\:\mathrm{X}\:\mathrm{had}\:\mathrm{783}\:\mathrm{miles}\:\mathrm{of}\:\mathrm{paved}\:\mathrm{roads}.\:\mathrm{Starting}\:\mathrm{in}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{2015},\:\mathrm{the}\:\mathrm{country}\:\mathrm{has}\:\:\mathrm{been}\:\mathrm{building}\:\:\mathrm{8}\:\:\mathrm{miles}\:\mathrm{of}\:\mathrm{new}\:\mathrm{paved} \\ $$$$\:\:\:\:\:\:\mathrm{roads}\:\mathrm{each}\:\mathrm{year}.\:\mathrm{At}\:\mathrm{this}\:\mathrm{rate},\:\:\mathrm{how}\:\:\mathrm{many}\:\mathrm{miles}\:\mathrm{of}\:\mathrm{paved}\:\mathrm{roads} \\ $$$$\:\:\:\:\:\:\mathrm{will}\:\mathrm{country}\:\mathrm{X}\:\mathrm{have}\:\mathrm{in}\:\mathrm{2030}? \\ $$

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