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Question Number 21713 Answers: 0 Comments: 7

A constant force F = 20 N acts on a block of mass 2 kg which is connected to two blocks of masses m_1 = 1 kg and m_2 = 2 kg. Calculate the accelerations produced in all the three blocks. Assume pulleys are frictionless and weightless.

$$\mathrm{A}\:\mathrm{constant}\:\mathrm{force}\:{F}\:=\:\mathrm{20}\:\mathrm{N}\:\mathrm{acts}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}\:\mathrm{kg}\:\mathrm{which}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{to} \\ $$$$\mathrm{two}\:\mathrm{blocks}\:\mathrm{of}\:\mathrm{masses}\:{m}_{\mathrm{1}} \:=\:\mathrm{1}\:\mathrm{kg}\:\mathrm{and} \\ $$$${m}_{\mathrm{2}} \:=\:\mathrm{2}\:\mathrm{kg}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{accelerations} \\ $$$$\mathrm{produced}\:\mathrm{in}\:\mathrm{all}\:\mathrm{the}\:\mathrm{three}\:\mathrm{blocks}.\:\mathrm{Assume} \\ $$$$\mathrm{pulleys}\:\mathrm{are}\:\mathrm{frictionless}\:\mathrm{and}\:\mathrm{weightless}. \\ $$

Question Number 21707 Answers: 0 Comments: 3

∫_(π/6) ^(π/3) (1/2)cot^2 2θdθ

$$\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \frac{\mathrm{1}}{\mathrm{2}}{cot}^{\mathrm{2}} \mathrm{2}\theta{d}\theta \\ $$

Question Number 21702 Answers: 0 Comments: 1

∫_(π/6) ^(π/3) 1/2cot^2 2θdθ

$$\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \mathrm{1}/\mathrm{2}{cot}^{\mathrm{2}} \mathrm{2}\theta{d}\theta \\ $$

Question Number 21701 Answers: 2 Comments: 1

∫2cot^2 2t

$$\int\mathrm{2}{cot}^{\mathrm{2}} \mathrm{2}{t} \\ $$

Question Number 21686 Answers: 0 Comments: 0

I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?

$$\mathrm{I}\:\mathrm{have}\:\mathrm{6}\:\mathrm{friends}\:\mathrm{and}\:\mathrm{during}\:\mathrm{a}\:\mathrm{vacation} \\ $$$$\mathrm{I}\:\mathrm{met}\:\mathrm{them}\:\mathrm{during}\:\mathrm{several}\:\mathrm{dinners}.\:\mathrm{I} \\ $$$$\mathrm{found}\:\mathrm{that}\:\mathrm{I}\:\mathrm{dined}\:\mathrm{with}\:\mathrm{all}\:\mathrm{the}\:\mathrm{6}\:\mathrm{exactly} \\ $$$$\mathrm{on}\:\mathrm{1}\:\mathrm{day};\:\mathrm{with}\:\mathrm{every}\:\mathrm{5}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{2}\:\mathrm{days}; \\ $$$$\mathrm{with}\:\mathrm{every}\:\mathrm{4}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{3}\:\mathrm{days};\:\mathrm{with} \\ $$$$\mathrm{every}\:\mathrm{3}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{4}\:\mathrm{days};\:\mathrm{with}\:\mathrm{every}\:\mathrm{2} \\ $$$$\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{5}\:\mathrm{days}.\:\mathrm{Further}\:\mathrm{every}\:\mathrm{friend} \\ $$$$\mathrm{was}\:\mathrm{present}\:\mathrm{at}\:\mathrm{7}\:\mathrm{dinners}\:\mathrm{and}\:\mathrm{every} \\ $$$$\mathrm{friend}\:\mathrm{was}\:\mathrm{absent}\:\mathrm{at}\:\mathrm{7}\:\mathrm{dinners}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{dinners}\:\mathrm{did}\:\mathrm{I}\:\mathrm{have}\:\mathrm{alone}? \\ $$

Question Number 21685 Answers: 0 Comments: 0

In a group of ten persons, each person is asked to write the sum of the ages of all the other 9 persons. If all the ten sums form the 9-element set {82, 83, 84, 85, 87, 90, 91, 92} find the individual ages of the persons (assuming them to be whole numbers of years).

$$\mathrm{In}\:\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{persons},\:\mathrm{each}\:\mathrm{person} \\ $$$$\mathrm{is}\:\mathrm{asked}\:\mathrm{to}\:\mathrm{write}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ages}\:\mathrm{of} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{other}\:\mathrm{9}\:\mathrm{persons}.\:\mathrm{If}\:\mathrm{all}\:\mathrm{the}\:\mathrm{ten} \\ $$$$\mathrm{sums}\:\mathrm{form}\:\mathrm{the}\:\mathrm{9}-\mathrm{element}\:\mathrm{set}\:\left\{\mathrm{82},\:\mathrm{83},\:\mathrm{84},\right. \\ $$$$\left.\mathrm{85},\:\mathrm{87},\:\mathrm{90},\:\mathrm{91},\:\mathrm{92}\right\}\:\mathrm{find}\:\mathrm{the}\:\mathrm{individual} \\ $$$$\mathrm{ages}\:\mathrm{of}\:\mathrm{the}\:\mathrm{persons}\:\left(\mathrm{assuming}\:\mathrm{them}\:\mathrm{to}\right. \\ $$$$\left.\mathrm{be}\:\mathrm{whole}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{years}\right). \\ $$

Question Number 21683 Answers: 1 Comments: 2

Suppose A_1 A_2 ...A_(20) is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but whose sides are not the sides of the polygon?

$$\mathrm{Suppose}\:{A}_{\mathrm{1}} {A}_{\mathrm{2}} ...{A}_{\mathrm{20}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{20}-\mathrm{sided}\:\mathrm{regular} \\ $$$$\mathrm{polygon}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{non}-\mathrm{isosceles} \\ $$$$\left(\mathrm{scalene}\right)\:\mathrm{triangles}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{whose} \\ $$$$\mathrm{vertices}\:\mathrm{are}\:\mathrm{among}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{polygon}\:\mathrm{but}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}? \\ $$

Question Number 21682 Answers: 1 Comments: 0

Prove that the ten′s digit of any power of 3 is even. [e.g. the ten′s digit of 3^6 = 729 is 2].

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{ten}'\mathrm{s}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{any}\:\mathrm{power} \\ $$$$\mathrm{of}\:\mathrm{3}\:\mathrm{is}\:\mathrm{even}.\:\left[\mathrm{e}.\mathrm{g}.\:\mathrm{the}\:\mathrm{ten}'\mathrm{s}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{3}^{\mathrm{6}} \:=\right. \\ $$$$\left.\mathrm{729}\:\mathrm{is}\:\mathrm{2}\right]. \\ $$

Question Number 21670 Answers: 1 Comments: 1

The block of mass 2 kg and 3 kg are placed one over the other. The contact surfaces are rough with coefficient of friction μ_1 = 0.2, μ_2 = 0.06. A force F = (1/2)t N (where t is in second) is applied on upper block in the direction. (Given that g = 10 m/s^2 ) 1. The relative slipping between the blocks occurs at t = 2. Friction force acting between the two blocks at t = 8 s 3. The acceleration time graph for 3 kg block is

$$\mathrm{The}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}\:\mathrm{kg}\:\mathrm{and}\:\mathrm{3}\:\mathrm{kg}\:\mathrm{are} \\ $$$$\mathrm{placed}\:\mathrm{one}\:\mathrm{over}\:\mathrm{the}\:\mathrm{other}.\:\mathrm{The}\:\mathrm{contact} \\ $$$$\mathrm{surfaces}\:\mathrm{are}\:\mathrm{rough}\:\mathrm{with}\:\mathrm{coefficient}\:\mathrm{of} \\ $$$$\mathrm{friction}\:\mu_{\mathrm{1}} \:=\:\mathrm{0}.\mathrm{2},\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\mathrm{06}.\:\mathrm{A}\:\mathrm{force}\:{F}\:= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{t}\:\mathrm{N}\:\left(\mathrm{where}\:{t}\:\mathrm{is}\:\mathrm{in}\:\mathrm{second}\right)\:\mathrm{is}\:\mathrm{applied} \\ $$$$\mathrm{on}\:\mathrm{upper}\:\mathrm{block}\:\mathrm{in}\:\mathrm{the}\:\mathrm{direction}.\:\left(\mathrm{Given}\right. \\ $$$$\left.\mathrm{that}\:{g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$$$\mathrm{1}.\:\mathrm{The}\:\mathrm{relative}\:\mathrm{slipping}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{occurs}\:\mathrm{at}\:{t}\:= \\ $$$$\mathrm{2}.\:\mathrm{Friction}\:\mathrm{force}\:\mathrm{acting}\:\mathrm{between}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{blocks}\:\mathrm{at}\:{t}\:=\:\mathrm{8}\:\mathrm{s} \\ $$$$\mathrm{3}.\:\mathrm{The}\:\mathrm{acceleration}\:\mathrm{time}\:\mathrm{graph}\:\mathrm{for}\:\mathrm{3}\:\mathrm{kg} \\ $$$$\mathrm{block}\:\mathrm{is} \\ $$

Question Number 21662 Answers: 1 Comments: 0

Question Number 21661 Answers: 0 Comments: 2

Question Number 21679 Answers: 1 Comments: 0

∫((secθ dθ)/(1−secθ))

$$\int\frac{\mathrm{sec}\theta\:\mathrm{d}\theta}{\mathrm{1}−\mathrm{sec}\theta} \\ $$

Question Number 21656 Answers: 0 Comments: 4

Let A(x) is a cubic polynomial and B(x) = (x −1)(x − 2)(x − 3) Find how many C(x) so that B(C(x)) = B(x) . A(x)

$$\mathrm{Let}\:{A}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{cubic}\:\mathrm{polynomial}\:\mathrm{and}\:{B}\left({x}\right)\:=\:\left({x}\:−\mathrm{1}\right)\left({x}\:−\:\mathrm{2}\right)\left({x}\:−\:\mathrm{3}\right) \\ $$$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:{C}\left({x}\right)\:\mathrm{so}\:\mathrm{that} \\ $$$${B}\left({C}\left({x}\right)\right)\:=\:{B}\left({x}\right)\:.\:{A}\left({x}\right) \\ $$

Question Number 21655 Answers: 1 Comments: 0

(((2017)),(( 0)) ) + (((2017)),(( 2)) ) + (((2017)),(( 4)) ) + (((2017)),(( 6)) ) + ... + (((2017)),((2016)) ) is equal to ...

$$\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{4}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{6}}\end{pmatrix}\:+\:...\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\mathrm{2016}}\end{pmatrix} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21654 Answers: 0 Comments: 0

Find the minimum value of Q that satisfy: ∣xy(x^2 − y^2 ) + yz(y^2 − z^2 ) + zx(z^2 − x^2 )∣ ≤ Q(x^2 + y^2 + z^2 )^2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{Q}\:\mathrm{that}\:\mathrm{satisfy}: \\ $$$$\mid{xy}\left({x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \right)\:+\:{yz}\left({y}^{\mathrm{2}} \:−\:{z}^{\mathrm{2}} \right)\:+\:{zx}\left({z}^{\mathrm{2}} \:−\:{x}^{\mathrm{2}} \right)\mid\:\leqslant\:{Q}\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{2}} \\ $$

Question Number 21653 Answers: 0 Comments: 0

Find all pair of solutions (x,y) that satisfy the equation: ((7x^2 − 13xy + 7y^2 ))^(1/3) = ∣x − y∣ + 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{solutions}\:\left({x},{y}\right)\:\mathrm{that}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{13}{xy}\:+\:\mathrm{7}{y}^{\mathrm{2}} }\:=\:\mid{x}\:−\:{y}\mid\:+\:\mathrm{1} \\ $$

Question Number 21651 Answers: 0 Comments: 4

How many four-digit numbers are there whose decimal notation contains not more than two distinct digits?

$${How}\:{many}\:{four}-{digit}\:{numbers}\:{are} \\ $$$${there}\:{whose}\:{decimal}\:{notation}\:{contains} \\ $$$${not}\:{more}\:{than}\:{two}\:{distinct}\:{digits}? \\ $$

Question Number 21650 Answers: 0 Comments: 0

The three distinct successive terms of an A.P are the first,second and fourth terms of a G.P. If the sum to infinity of a G.P is 3+(√5) , find the first term.

$${The}\:{three}\:{distinct}\:{successive}\:{terms}\:{of}\:{an}\:{A}.{P}\:{are} \\ $$$${the}\:{first},{second}\:{and}\:{fourth}\:{terms}\:{of}\:{a}\:{G}.{P}.\:{If}\:{the}\: \\ $$$${sum}\:{to}\:{infinity}\:{of}\:{a}\:{G}.{P}\:{is}\:\mathrm{3}+\sqrt{\mathrm{5}}\:,\:{find}\: \\ $$$${the}\:{first}\:{term}. \\ $$$$ \\ $$

Question Number 21680 Answers: 1 Comments: 0

∫(√(secθ)) dθ

$$\int\sqrt{\mathrm{sec}\theta}\:\mathrm{d}\theta \\ $$

Question Number 21634 Answers: 1 Comments: 0

∫3tan2t

$$\int\mathrm{3}{tan}\mathrm{2}{t} \\ $$

Question Number 21708 Answers: 1 Comments: 0

∫_0 ^(0.5) 2tan^2 2tdt

$$\int_{\mathrm{0}} ^{\mathrm{0}.\mathrm{5}} \mathrm{2}{tan}^{\mathrm{2}} \mathrm{2}{tdt} \\ $$

Question Number 21628 Answers: 1 Comments: 0

One end of a massless spring of constant 100 N/m and natural length 0.5 m is fixed and the other end is connected to a particle of mass 0.5 kg lying on a frictionless horizontal table. The spring remains horizontal. If the mass is made to rotate at an angular velocity of 2 rad/s, find the elongation of the spring.

$$\mathrm{One}\:\mathrm{end}\:\mathrm{of}\:\mathrm{a}\:\mathrm{massless}\:\mathrm{spring}\:\mathrm{of}\:\mathrm{constant} \\ $$$$\mathrm{100}\:\mathrm{N}/\mathrm{m}\:\mathrm{and}\:\mathrm{natural}\:\mathrm{length}\:\mathrm{0}.\mathrm{5}\:\mathrm{m}\:\mathrm{is} \\ $$$$\mathrm{fixed}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{end}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{to} \\ $$$$\mathrm{a}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{0}.\mathrm{5}\:\mathrm{kg}\:\mathrm{lying}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{frictionless}\:\mathrm{horizontal}\:\mathrm{table}.\:\mathrm{The}\:\mathrm{spring} \\ $$$$\mathrm{remains}\:\mathrm{horizontal}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{mass}\:\mathrm{is}\:\mathrm{made} \\ $$$$\mathrm{to}\:\mathrm{rotate}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angular}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{2} \\ $$$$\mathrm{rad}/{s},\:\mathrm{find}\:\mathrm{the}\:\mathrm{elongation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{spring}. \\ $$

Question Number 21626 Answers: 0 Comments: 1

Is it also possible to import text or fomulars from other apps using the android−clipboard?

$$\mathrm{Is}\:\mathrm{it}\:\mathrm{also}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{import}\:\mathrm{text}\:\mathrm{or}\:\mathrm{fomulars} \\ $$$$\mathrm{from}\:\mathrm{other}\:\mathrm{apps}\:\mathrm{using}\:\mathrm{the}\:\mathrm{android}−\mathrm{clipboard}? \\ $$

Question Number 21622 Answers: 2 Comments: 0

$$\mathrm{If}\:\mathrm{sec}\:{x}\:+\:\mathrm{tan}\:{x}\:=\:\mathrm{2012} \\ $$$$\mathrm{then}\:\mathrm{2011}\left(\mathrm{cosec}\:{x}\:+\:\mathrm{cot}\:{x}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left({A}\right)\:\mathrm{2011} \\ $$$$\left({B}\right)\:\mathrm{2012} \\ $$$$\left({C}\right)\:\mathrm{2013} \\ $$$$\left({D}\right)\:\frac{\mathrm{2011}}{\mathrm{2013}} \\ $$$$\left({E}\right)\:\frac{\mathrm{2013}}{\mathrm{2012}} \\ $$

Question Number 21643 Answers: 0 Comments: 1

(1/(1 + 1^2 + 1^4 )) + (2/(1 + 2^2 + 2^4 )) + (3/(1 + 3^2 + 3^4 )) + ... + ((2012)/(1 + 2012^2 + 2012^4 ))

$$\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{1}^{\mathrm{2}} \:+\:\mathrm{1}^{\mathrm{4}} }\:+\:\frac{\mathrm{2}}{\mathrm{1}\:+\:\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{2}^{\mathrm{4}} }\:+\:\frac{\mathrm{3}}{\mathrm{1}\:+\:\mathrm{3}^{\mathrm{2}} \:+\:\mathrm{3}^{\mathrm{4}} }\:+\:...\:+\:\frac{\mathrm{2012}}{\mathrm{1}\:+\:\mathrm{2012}^{\mathrm{2}} \:+\:\mathrm{2012}^{\mathrm{4}} } \\ $$

Question Number 21612 Answers: 0 Comments: 0

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