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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

Question Number 21611 Answers: 1 Comments: 0

Sin5° =?

$$\mathrm{Sin5}°\:=? \\ $$

Question Number 21693 Answers: 0 Comments: 15

How many six-digit numbers contain exactly four different digits?

$${How}\:{many}\:{six}-{digit}\:{numbers}\:{contain} \\ $$$${exactly}\:{four}\:{different}\:{digits}? \\ $$

Question Number 21604 Answers: 1 Comments: 0

A particle will leave a vertical circle of radius r, when its velocity at the lowest point of the circle (v_L ) is (a) (√(2gr)) (b) (√(5gr)) (c) (√(3gr)) (d) (√(6gr))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{will}\:\mathrm{leave}\:\mathrm{a}\:\mathrm{vertical}\:\mathrm{circle}\:\mathrm{of} \\ $$$$\mathrm{radius}\:{r},\:\mathrm{when}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{at}\:\mathrm{the}\:\mathrm{lowest} \\ $$$$\mathrm{point}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\left({v}_{{L}} \right)\:\mathrm{is} \\ $$$$\left({a}\right)\:\sqrt{\mathrm{2}{gr}} \\ $$$$\left({b}\right)\:\sqrt{\mathrm{5}{gr}} \\ $$$$\left({c}\right)\:\sqrt{\mathrm{3}{gr}} \\ $$$$\left({d}\right)\:\sqrt{\mathrm{6}{gr}} \\ $$

Question Number 21636 Answers: 1 Comments: 1

Question Number 21598 Answers: 0 Comments: 0

Prove that ((n^2 !)/((n!)^n )) is an integer, n ∈ N.

$${Prove}\:{that}\:\frac{{n}^{\mathrm{2}} !}{\left({n}!\right)^{{n}} }\:{is}\:{an}\:{integer},\:{n}\:\in\:{N}. \\ $$

Question Number 21588 Answers: 0 Comments: 1

Show that if G is a finite group of even order, then G has an odd number of elements of order 2.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{if}\:{G}\:\mathrm{is}\:\mathrm{a}\:\mathrm{finite}\:\mathrm{group}\:\mathrm{of} \\ $$$$\mathrm{even}\:\mathrm{order},\:\mathrm{then}\:{G}\:\mathrm{has}\:\mathrm{an}\:\mathrm{odd} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{order}\:\mathrm{2}. \\ $$

Question Number 21587 Answers: 1 Comments: 1

If n objects are arranged in a row, then find the number of ways of selecting three of these objects so that no two of them are next to each other.

$$\mathrm{If}\:{n}\:\mathrm{objects}\:\mathrm{are}\:\mathrm{arranged}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{selecting} \\ $$$$\mathrm{three}\:\mathrm{of}\:\mathrm{these}\:\mathrm{objects}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{of} \\ $$$$\mathrm{them}\:\mathrm{are}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$

Question Number 21591 Answers: 1 Comments: 1

5cos^5 tsin t

$$\mathrm{5cos}\:^{\mathrm{5}} {t}\mathrm{sin}\:{t} \\ $$

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