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Question Number 111541 Answers: 3 Comments: 0

How many natural numbers less than 1000 have the sum of their digits equal to 5?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{less}\:\mathrm{than} \\ $$$$\mathrm{1000}\:\mathrm{have}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}\:\mathrm{equal} \\ $$$$\mathrm{to}\:\mathrm{5}? \\ $$

Question Number 111539 Answers: 1 Comments: 0

In a trapezoid ABCD, sides AB and CD are parallel and side BC=CD=(√5). If DC^ B =120° and BA^ D=60°. Find the area of ABCD.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{trapezoid}\:\mathrm{ABCD},\:\mathrm{sides}\:\mathrm{AB}\:\mathrm{and} \\ $$$$\mathrm{CD}\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{and}\:\mathrm{side}\:\mathrm{BC}=\mathrm{CD}=\sqrt{\mathrm{5}}. \\ $$$$\mathrm{If}\:\mathrm{D}\hat {\mathrm{C}B}\:=\mathrm{120}°\:\mathrm{and}\:\mathrm{B}\hat {\mathrm{A}D}=\mathrm{60}°.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{ABCD}. \\ $$

Question Number 111537 Answers: 2 Comments: 0

What is the minimum value obtained when an arbitrary number of three different non−zero digits is divided by the sum of its digits?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{obtained} \\ $$$$\mathrm{when}\:\mathrm{an}\:\mathrm{arbitrary}\:\mathrm{number}\:\mathrm{of}\:\mathrm{three} \\ $$$$\mathrm{different}\:\mathrm{non}−\mathrm{zero}\:\mathrm{digits}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}? \\ $$$$ \\ $$

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 ))))

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

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