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Question Number 11150    Answers: 1   Comments: 0

f(x)=(x^2 +x+1)^(sin2x) f′(x)=...???

$${f}\left({x}\right)=\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{{sin}\mathrm{2}{x}} \\ $$$${f}'\left({x}\right)=...??? \\ $$

Question Number 11149    Answers: 2   Comments: 0

∫x^5 ((x^3 +1))^(1/3) dx=...???

$$\int{x}^{\mathrm{5}} \:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}=...??? \\ $$

Question Number 11114    Answers: 2   Comments: 0

Question Number 11101    Answers: 1   Comments: 1

Question Number 11095    Answers: 2   Comments: 1

You have a line of length 1. You place two random points on the line. What is the average distance between the two points? Please show your working.

$$\mathrm{You}\:\mathrm{have}\:\mathrm{a}\:\mathrm{line}\:\mathrm{of}\:\mathrm{length}\:\mathrm{1}. \\ $$$$\mathrm{You}\:\mathrm{place}\:\mathrm{two}\:\mathrm{random}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}. \\ $$$$\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{average}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{points}?\:\mathrm{Please}\:\mathrm{show}\:\mathrm{your}\:\mathrm{working}. \\ $$

Question Number 11147    Answers: 2   Comments: 0

∫cos^4 x sinx dx=....???

$$\int{cos}^{\mathrm{4}} {x}\:{sinx}\:{dx}=....??? \\ $$

Question Number 11089    Answers: 0   Comments: 1

for each n∈N, f_n (x)=nx(1−x^2 )^n for each x, 0≤x≤1 and a_n =∫_0 ^1 f_n (x)dx if s_n =sin(πa_n ), for each n∈N, then li_(n→∼) m s_n =....???

$${for}\:{each}\:{n}\in\mathbb{N},\:{f}_{{n}} \left({x}\right)={nx}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} \\ $$$${for}\:{each}\:{x},\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:{a}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right){dx} \\ $$$${if}\:{s}_{{n}} ={sin}\left(\pi{a}_{{n}} \right),\:{for}\:{each}\:{n}\in\mathbb{N},\:{then} \\ $$$${l}\underset{{n}\rightarrow\sim} {{i}m}\:{s}_{{n}} =....??? \\ $$

Question Number 11088    Answers: 0   Comments: 0

S={(n/(2m)) + ((6m)/n) : n,m∈N} inf(S)=...??? and sup(S)=...???

$${S}=\left\{\frac{{n}}{\mathrm{2}{m}}\:+\:\frac{\mathrm{6}{m}}{{n}}\::\:{n},{m}\in\mathbb{N}\right\} \\ $$$${inf}\left({S}\right)=...???\:{and}\:{sup}\left({S}\right)=...??? \\ $$

Question Number 11087    Answers: 0   Comments: 0

a<(π/2) if M<1 with ∣cos x−cos y∣≤M∣x−y∣ to each x,y∈[0,a] M=....???

$${a}<\frac{\pi}{\mathrm{2}} \\ $$$${if}\:{M}<\mathrm{1}\:{with}\:\mid{cos}\:{x}−{cos}\:{y}\mid\leqslant{M}\mid{x}−{y}\mid \\ $$$${to}\:{each}\:{x},{y}\in\left[\mathrm{0},{a}\right] \\ $$$${M}=....??? \\ $$

Question Number 11078    Answers: 1   Comments: 0

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Question Number 11075    Answers: 1   Comments: 6

Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1 and itself. Find (n/(75))

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{75}\:\mathrm{and}\:\mathrm{has}\:\mathrm{exactly}\:\mathrm{75}\:\mathrm{positive} \\ $$$$\mathrm{integral}\:\mathrm{divisors},\:\mathrm{including}\:\mathrm{1}\:\mathrm{and}\:\mathrm{itself}. \\ $$$$\mathrm{Find}\:\frac{{n}}{\mathrm{75}} \\ $$

Question Number 11074    Answers: 1   Comments: 0

(2/((x − 1)^(2 ) )) + (4/((y + 2)^2 )) + (5/z^2 ) = (9/4) (4/((x − 1)^2 )) − (2/((x + 2)^2 )) − (1/z^2 ) = (1/2) (3/((z − 1)^2 )) + (6/((y + 2)^2 )) − (2/z^2 ) = 1 (z − 1)^2 + (y + 2)^2 + z^2 = ???

$$\frac{\mathrm{2}}{\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}\:} }\:+\:\frac{\mathrm{4}}{\left({y}\:+\:\mathrm{2}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{5}}{{z}^{\mathrm{2}} }\:=\:\frac{\mathrm{9}}{\mathrm{4}} \\ $$$$\frac{\mathrm{4}}{\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} }\:−\:\frac{\mathrm{2}}{\left({x}\:+\:\mathrm{2}\right)^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{{z}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\frac{\mathrm{3}}{\left({z}\:−\:\mathrm{1}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{6}}{\left({y}\:+\:\mathrm{2}\right)^{\mathrm{2}} }\:−\:\frac{\mathrm{2}}{{z}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$$ \\ $$$$\left({z}\:−\:\mathrm{1}\right)^{\mathrm{2}} \:+\:\left({y}\:+\:\mathrm{2}\right)^{\mathrm{2}} \:\:+\:{z}^{\mathrm{2}} \:=\:??? \\ $$

Question Number 11072    Answers: 0   Comments: 0

Question Number 11071    Answers: 0   Comments: 0

Question Number 11070    Answers: 1   Comments: 0

m∈Z , x^2 −(m+1)x+m^2 −16=0 f(x_1 )=f(x_2 )=0 ,x_1 <0, 0<x_2 ⇒Σm=?

$${m}\in{Z}\:,\:{x}^{\mathrm{2}} −\left({m}+\mathrm{1}\right){x}+{m}^{\mathrm{2}} −\mathrm{16}=\mathrm{0} \\ $$$${f}\left({x}_{\mathrm{1}} \right)={f}\left({x}_{\mathrm{2}} \right)=\mathrm{0}\:\:,{x}_{\mathrm{1}} <\mathrm{0},\:\mathrm{0}<{x}_{\mathrm{2}} \:\Rightarrow\Sigma{m}=? \\ $$

Question Number 11069    Answers: 0   Comments: 1

Question Number 11067    Answers: 0   Comments: 4

if E={f∣f:R→R continuous function with f(x)∈Q, ∀x∈R}, then E=....???

$${if}\:{E}=\left\{{f}\mid{f}:\mathbb{R}\rightarrow\mathbb{R}\:{continuous}\:{function}\:{with}\:{f}\left({x}\right)\in\mathbb{Q},\:\forall{x}\in\mathbb{R}\right\}, \\ $$$${then}\:{E}=....??? \\ $$

Question Number 11061    Answers: 0   Comments: 2

A 3 dimentional parabola f(x, y, z) has a focus P(x, y, z)=(p, q, r). A rectangular box with side lengths a, b, and c (see diagram), has a point P on the front center point of its face. If the origin lies on the opposite face, in the middle, what is the function of the curve? Note Origin O(x, y, z)=(0, 0, 0) O(x, y, z) = (p−c, q, r)

$$\mathrm{A}\:\mathrm{3}\:\mathrm{dimentional}\:\mathrm{parabola}\:{f}\left({x},\:{y},\:{z}\right) \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{focus}\:{P}\left({x},\:{y},\:{z}\right)=\left({p},\:{q},\:{r}\right). \\ $$$$\: \\ $$$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{box}\:\mathrm{with}\:\mathrm{side}\:\mathrm{lengths} \\ $$$${a},\:{b},\:\mathrm{and}\:{c}\:\left(\mathrm{see}\:\mathrm{diagram}\right),\:\mathrm{has}\:\mathrm{a}\:\mathrm{point}\:{P} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{front}\:\mathrm{center}\:\mathrm{point}\:\mathrm{of}\:\mathrm{its}\:\mathrm{face}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{face}, \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{middle},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{function}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{curve}? \\ $$$$\: \\ $$$$\boldsymbol{\mathrm{Note}} \\ $$$$\mathrm{Origin}\:{O}\left({x},\:{y},\:{z}\right)=\left(\mathrm{0},\:\mathrm{0},\:\mathrm{0}\right) \\ $$$${O}\left({x},\:{y},\:{z}\right)\:=\:\left({p}−{c},\:{q},\:{r}\right) \\ $$

Question Number 11050    Answers: 1   Comments: 0

The function f(x)=acosx+b where a<0 has a maximum value of 8 and a minimum value of −2.Find a+b.

$${The}\:{function}\:{f}\left({x}\right)={acosx}+{b}\:{where} \\ $$$${a}<\mathrm{0}\:{has}\:{a}\:{maximum}\:{value}\:{of}\:\mathrm{8}\:{and} \\ $$$${a}\:{minimum}\:{value}\:{of}\:−\mathrm{2}.\boldsymbol{{F}}{ind}\:{a}+{b}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 11049    Answers: 1   Comments: 1

If n^2 + 3n + 1 is divisible by 3n + 10 Find out all possible solution for n

$$\mathrm{If}\:\:{n}^{\mathrm{2}} \:+\:\mathrm{3}{n}\:+\:\mathrm{1}\:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}{n}\:+\:\mathrm{10} \\ $$$$\mathrm{Find}\:\mathrm{out}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{solution}\:\mathrm{for}\:{n}\: \\ $$

Question Number 11048    Answers: 4   Comments: 0

Which one is largest? (without using calculator) 31^(11) or 17^(14) ??

$$\mathrm{Which}\:\mathrm{one}\:\mathrm{is}\:\mathrm{largest}?\:\left(\mathrm{without}\:\mathrm{using}\:\mathrm{calculator}\right) \\ $$$$\mathrm{31}^{\mathrm{11}} \:\mathrm{or}\:\mathrm{17}^{\mathrm{14}} \:\:?? \\ $$

Question Number 11042    Answers: 1   Comments: 0

∫cos(x)(√(((sin(x)+1)/(sin(x)−1)) )) dx

$$\int{cos}\left({x}\right)\sqrt{\frac{{sin}\left({x}\right)+\mathrm{1}}{{sin}\left({x}\right)−\mathrm{1}}\:}\:{dx} \\ $$

Question Number 11040    Answers: 0   Comments: 0

Question Number 11036    Answers: 2   Comments: 1

li_(n→∼) m Σ_(k=1) ^n ((8n^2 )/(n^4 +1)) =....?

$${l}\underset{{n}\rightarrow\sim} {{i}m}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{8}{n}^{\mathrm{2}} }{{n}^{\mathrm{4}} +\mathrm{1}}\:=....? \\ $$

Question Number 11035    Answers: 1   Comments: 0

A= [(1,(−1)),((−4),(−2)) ] A^(2016) =.....?

$${A}=\begin{bmatrix}{\mathrm{1}}&{−\mathrm{1}}\\{−\mathrm{4}}&{−\mathrm{2}}\end{bmatrix} \\ $$$${A}^{\mathrm{2016}} =.....? \\ $$

Question Number 11034    Answers: 0   Comments: 1

u= [(1,(−1),2) ] A= [(3,2),(1,3),(0,1) ] v= [(2,(−1)) ] uAv^t =....?

$${u}=\begin{bmatrix}{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{2}}\end{bmatrix} \\ $$$${A}=\begin{bmatrix}{\mathrm{3}}&{\mathrm{2}}\\{\mathrm{1}}&{\mathrm{3}}\\{\mathrm{0}}&{\mathrm{1}}\end{bmatrix} \\ $$$${v}=\begin{bmatrix}{\mathrm{2}}&{−\mathrm{1}}\end{bmatrix} \\ $$$${uAv}^{{t}} =....? \\ $$

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