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Question Number 111536 Answers: 2 Comments: 2

Let 2,3,5,6,7,10,11,... be increasing sequence of positive integers that are neither the square nor cube of an integer. Find the 2016th term of this sequence.

$$\mathrm{Let}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{10},\mathrm{11},...\:\mathrm{be}\:\mathrm{increasing} \\ $$$$\mathrm{sequence}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{neither}\:\mathrm{the}\:\mathrm{square}\:\mathrm{nor}\:\mathrm{cube}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{integer}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{2016th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{sequence}. \\ $$

Question Number 111535 Answers: 1 Comments: 0

If f(x)=ax^2 −c satisfy −4≤f(1)≤−1 and −1≤f(2)≤5, then A. 7≤f(3)≤26 B. −1≤f(3)≤20 C. −4≤f(3)≤15 D. ((−28)/3)≤f(3)≤((35)/3) E. (8/3)≤f(3)≤((13)/3)

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ax}^{\mathrm{2}} −\mathrm{c}\:\mathrm{satisfy}\:−\mathrm{4}\leqslant\mathrm{f}\left(\mathrm{1}\right)\leqslant−\mathrm{1} \\ $$$$\mathrm{and}\:−\mathrm{1}\leqslant\mathrm{f}\left(\mathrm{2}\right)\leqslant\mathrm{5},\:\mathrm{then} \\ $$$$ \\ $$$$\mathrm{A}.\:\mathrm{7}\leqslant\mathrm{f}\left(\mathrm{3}\right)\leqslant\mathrm{26}\:\mathrm{B}.\:−\mathrm{1}\leqslant\mathrm{f}\left(\mathrm{3}\right)\leqslant\mathrm{20}\:\mathrm{C}. \\ $$$$−\mathrm{4}\leqslant\mathrm{f}\left(\mathrm{3}\right)\leqslant\mathrm{15}\:\mathrm{D}.\:\frac{−\mathrm{28}}{\mathrm{3}}\leqslant\mathrm{f}\left(\mathrm{3}\right)\leqslant\frac{\mathrm{35}}{\mathrm{3}}\:\mathrm{E}. \\ $$$$\frac{\mathrm{8}}{\mathrm{3}}\leqslant\mathrm{f}\left(\mathrm{3}\right)\leqslant\frac{\mathrm{13}}{\mathrm{3}} \\ $$

Question Number 111534 Answers: 2 Comments: 0

If x=((1+(√(2016)))/2), then 4x^3 −2019x−2017 equals?

$$\mathrm{If}\:\mathrm{x}=\frac{\mathrm{1}+\sqrt{\mathrm{2016}}}{\mathrm{2}},\:\mathrm{then} \\ $$$$\mathrm{4x}^{\mathrm{3}} −\mathrm{2019x}−\mathrm{2017}\:\mathrm{equals}? \\ $$

Question Number 111533 Answers: 1 Comments: 2

The mean,median and mode of the data values 90,54,x,123,62,78,58,81 are all equal. What is the value of x?

$$\mathrm{The}\:\mathrm{mean},\mathrm{median}\:\mathrm{and}\:\mathrm{mode}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{data}\:\mathrm{values}\:\mathrm{90},\mathrm{54},\mathrm{x},\mathrm{123},\mathrm{62},\mathrm{78},\mathrm{58},\mathrm{81} \\ $$$$\mathrm{are}\:\mathrm{all}\:\mathrm{equal}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}? \\ $$

Question Number 111528 Answers: 3 Comments: 0

(√(bemath)) lim_(x→π/2) (1−sin x)^(cos x) ?

$$\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\:\:\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\left(\mathrm{1}−\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{cos}\:\mathrm{x}} \:? \\ $$

Question Number 111520 Answers: 0 Comments: 0

find nature of Σ_(n=1) ^∞ (n^p /(n!)) (p natural )

$$\mathrm{find}\:\mathrm{nature}\:\mathrm{of}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{n}^{\mathrm{p}} }{\mathrm{n}!}\:\:\:\left(\mathrm{p}\:\mathrm{natural}\:\right) \\ $$

Question Number 111513 Answers: 2 Comments: 0

((cos^2 x))^(1/(3 )) + ((sin^2 x))^(1/(3 )) = (2)^(1/(3 )) find cos^2 (2x) ?

$$\:\:\sqrt[{\mathrm{3}\:}]{\mathrm{cos}\:^{\mathrm{2}} {x}}\:+\:\sqrt[{\mathrm{3}\:}]{\mathrm{sin}\:^{\mathrm{2}} {x}}\:=\:\sqrt[{\mathrm{3}\:}]{\mathrm{2}} \\ $$$${find}\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{2}{x}\right)\:? \\ $$

Question Number 111503 Answers: 2 Comments: 0

Find four values of n satisfying 1≤n≤2000 and 2^n =n^2 (mod 1024)

$$\mathrm{Find}\:\mathrm{four}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\:\mathrm{satisfying} \\ $$$$\mathrm{1}\leqslant\mathrm{n}\leqslant\mathrm{2000}\:\mathrm{and}\:\mathrm{2}^{\mathrm{n}} =\mathrm{n}^{\mathrm{2}} \left(\mathrm{mod}\:\mathrm{1024}\right) \\ $$

Question Number 111499 Answers: 0 Comments: 0

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Question Number 111498 Answers: 8 Comments: 0

(√(bemath )) (1)lim_(x→∞) ((2x^2 −x^3 ))^(1/(3 )) + x ? (2) lim_(x→1) ((1/x))^(1/(sin πx)) ? (3) ∫_0 ^x^2 f(t) dt = x cos (πx) . Find f (4).

$$\:\:\:\:\:\:\sqrt{\mathrm{bemath}\:} \\ $$$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{3}\:}]{\mathrm{2x}^{\mathrm{2}} −\mathrm{x}^{\mathrm{3}} }\:+\:\mathrm{x}\:? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)^{\frac{\mathrm{1}}{\mathrm{sin}\:\pi\mathrm{x}}} \:? \\ $$$$\left(\mathrm{3}\right)\:\underset{\mathrm{0}} {\overset{\mathrm{x}^{\mathrm{2}} } {\int}}\:\mathrm{f}\left(\mathrm{t}\right)\:\mathrm{dt}\:=\:\mathrm{x}\:\mathrm{cos}\:\left(\pi\mathrm{x}\right)\:.\:\mathrm{Find}\:\mathrm{f}\:\left(\mathrm{4}\right). \\ $$

Question Number 111497 Answers: 0 Comments: 1

Question Number 111466 Answers: 1 Comments: 2

Σ_(n=1) ^∞ (n^n /(n!))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{{n}} }{{n}!} \\ $$

Question Number 111458 Answers: 0 Comments: 6

Question Number 111450 Answers: 0 Comments: 1

Question Number 111447 Answers: 1 Comments: 1

Question Number 111442 Answers: 3 Comments: 0

lim_(x→∞) (((x+a)/(x−a)))^x ?

$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{x}+\mathrm{a}}{\mathrm{x}−\mathrm{a}}\right)^{\mathrm{x}} ? \\ $$

Question Number 111441 Answers: 1 Comments: 0

solve { ((y′′ −2y′+2y=sinht)),((y′(0)=1 , y(0)=1)) :}

$${solve} \\ $$$$ \\ $$$$\begin{cases}{{y}''\:−\mathrm{2}{y}'+\mathrm{2}{y}={sinht}}\\{{y}'\left(\mathrm{0}\right)=\mathrm{1}\:,\:{y}\left(\mathrm{0}\right)=\mathrm{1}}\end{cases} \\ $$

Question Number 111432 Answers: 2 Comments: 2

Question Number 111429 Answers: 1 Comments: 0

please evaluate : .... I=∫_0 ^( (π/2)) ((1/(ln(tan(x)))) + (1/(1−tan(x))))dx =??? ::: M. N.july 1970 :::

$$\:\:\:\:\:\:{please}\:\:{evaluate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:....\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\frac{\mathrm{1}}{{ln}\left({tan}\left({x}\right)\right)}\:+\:\frac{\mathrm{1}}{\mathrm{1}−{tan}\left({x}\right)}\right){dx}\:=??? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\::::\:\:\:\:\mathscr{M}.\:\mathscr{N}.{july}\:\mathrm{1970}\:::: \\ $$$$\:\: \\ $$

Question Number 111428 Answers: 0 Comments: 1

Σ_(n=1) ^∞ (n^3 /(n!))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{3}} }{{n}!} \\ $$

Question Number 111427 Answers: 0 Comments: 1

4sin^6 α + 4cos^6 α − 3cos^2 2α

$$\mathrm{4sin}\:^{\mathrm{6}} \alpha\:+\:\mathrm{4cos}\:^{\mathrm{6}} \alpha\:−\:\mathrm{3cos}\:^{\mathrm{2}} \mathrm{2}\alpha \\ $$

Question Number 111426 Answers: 2 Comments: 0

Question Number 111414 Answers: 2 Comments: 0

(1) lim_(x→0) ((sin x−ln (e^x cos x))/(x sin x)) (2) lim_(x→1) ((1−x+ln (x))/(1−(√(2x−x^2 ))))

$$\:\left(\mathrm{1}\right)\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{ln}\:\left(\mathrm{e}^{\mathrm{x}} \:\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{x}\:\mathrm{sin}\:\mathrm{x}} \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{x}+\mathrm{ln}\:\left(\mathrm{x}\right)}{\mathrm{1}−\sqrt{\mathrm{2x}−\mathrm{x}^{\mathrm{2}} }} \\ $$

Question Number 111393 Answers: 0 Comments: 6

What is the sum of the coefficients in the expansion of (2015v−2015u+1)^(2015) ?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{2015v}−\mathrm{2015u}+\mathrm{1}\right)^{\mathrm{2015}} ? \\ $$

Question Number 111394 Answers: 1 Comments: 2

A teacher conducts a test for five students. He provides the marking scheme and asked them to exchange their scripts such that none of them marks his own script. How many ways can the students carry out the marking?

$$\mathrm{A}\:\mathrm{teacher}\:\mathrm{conducts}\:\mathrm{a}\:\mathrm{test}\:\mathrm{for}\:\mathrm{five} \\ $$$$\mathrm{students}.\:\mathrm{He}\:\mathrm{provides}\:\mathrm{the}\:\mathrm{marking} \\ $$$$\mathrm{scheme}\:\mathrm{and}\:\mathrm{asked}\:\mathrm{them}\:\mathrm{to}\:\mathrm{exchange} \\ $$$$\mathrm{their}\:\mathrm{scripts}\:\mathrm{such}\:\mathrm{that}\:\mathrm{none}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{marks}\:\mathrm{his}\:\mathrm{own}\:\mathrm{script}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{ways} \\ $$$$\mathrm{can}\:\mathrm{the}\:\mathrm{students}\:\mathrm{carry}\:\mathrm{out}\:\mathrm{the} \\ $$$$\mathrm{marking}? \\ $$

Question Number 111391 Answers: 2 Comments: 0

Compute cos(Π/(12))

$$\mathrm{Compute}\:\mathrm{cos}\frac{\Pi}{\mathrm{12}} \\ $$

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