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Question Number 11991 Answers: 0 Comments: 0

show that: ((2n)/(2n+1))=1−(1/(2n+1)) via power series, and other methods

$$\mathrm{show}\:\mathrm{that}: \\ $$$$\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}} \\ $$$$\mathrm{via}\:\mathrm{power}\:\mathrm{series},\:\mathrm{and}\:\mathrm{other}\:\mathrm{methods} \\ $$

Question Number 11988 Answers: 2 Comments: 3

Question Number 11987 Answers: 1 Comments: 0

The radius of the moon is (1/4), and its mass is (1/(81)) that of the earth. If the acceleration due to gravity on the surface of the earth is 9.8m/s^2 . What is its value on the moon′s surface.

$$\mathrm{The}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{moon}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{4}},\:\mathrm{and}\:\mathrm{its}\:\mathrm{mass}\:\mathrm{is}\:\:\frac{\mathrm{1}}{\mathrm{81}}\:\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\:\mathrm{earth}.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{acceleration}\:\mathrm{due}\:\mathrm{to}\:\mathrm{gravity}\:\mathrm{on}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{the}\:\mathrm{earth}\:\mathrm{is}\:\mathrm{9}.\mathrm{8m}/\mathrm{s}^{\mathrm{2}} .\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{its}\:\mathrm{value}\:\mathrm{on}\:\mathrm{the}\:\mathrm{moon}'\mathrm{s}\:\mathrm{surface}. \\ $$

Question Number 11982 Answers: 1 Comments: 0

∫x^5 ((√(x^3 + 1))) dx

$$\int\mathrm{x}^{\mathrm{5}} \left(\sqrt{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1}}\right)\:\mathrm{dx} \\ $$

Question Number 11977 Answers: 1 Comments: 0

If S_1 , S_2 , S_3 be the sum of n, 2n, 3n terms respectively of an AP, then

$$\mathrm{If}\:\:{S}_{\mathrm{1}} ,\:{S}_{\mathrm{2}} ,\:{S}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:{n},\:\mathrm{2}{n},\:\mathrm{3}{n}\:\mathrm{terms} \\ $$$$\mathrm{respectively}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP},\:\mathrm{then} \\ $$

Question Number 11969 Answers: 2 Comments: 0

A gas occupies 30 dm^3 at s t p, what volume will it occupy at 91°C and 380 mmHg

$$\mathrm{A}\:\mathrm{gas}\:\mathrm{occupies}\:\mathrm{30}\:\mathrm{dm}^{\mathrm{3}} \:\mathrm{at}\:\mathrm{s}\:\mathrm{t}\:\mathrm{p},\:\:\mathrm{what}\:\mathrm{volume}\:\mathrm{will}\:\mathrm{it}\:\mathrm{occupy}\:\mathrm{at}\:\mathrm{91}°\mathrm{C}\: \\ $$$$\mathrm{and}\:\mathrm{380}\:\mathrm{mmHg} \\ $$

Question Number 11968 Answers: 0 Comments: 0

A motorcyclist, passing a road junction , moves due east for 8 seconds at a uniform speed of 5 m/s. He then moves due north for another 6 seconds with the same speed. At the end of 6 seconds his displacement from the road junction is 50 m in the diretion of A) 53°E (B) 37°E (C) 53°W (D) 37°W

$$\mathrm{A}\:\mathrm{motorcyclist},\:\mathrm{passing}\:\mathrm{a}\:\mathrm{road}\:\mathrm{junction}\:,\:\mathrm{moves}\:\mathrm{due}\:\mathrm{east}\:\mathrm{for}\:\mathrm{8}\:\mathrm{seconds}\:\mathrm{at} \\ $$$$\mathrm{a}\:\mathrm{uniform}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{5}\:\mathrm{m}/\mathrm{s}.\:\mathrm{He}\:\mathrm{then}\:\mathrm{moves}\:\mathrm{due}\:\mathrm{north}\:\mathrm{for}\:\mathrm{another}\:\mathrm{6}\:\mathrm{seconds} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{same}\:\mathrm{speed}.\:\mathrm{At}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{6}\:\mathrm{seconds}\:\mathrm{his}\:\mathrm{displacement}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{road}\:\mathrm{junction}\:\mathrm{is}\:\mathrm{50}\:\mathrm{m}\:\mathrm{in}\:\mathrm{the}\:\mathrm{diretion}\:\mathrm{of} \\ $$$$\left.\mathrm{A}\right)\:\mathrm{53}°\mathrm{E}\:\:\left(\mathrm{B}\right)\:\:\mathrm{37}°\mathrm{E}\:\:\left(\mathrm{C}\right)\:\:\mathrm{53}°\mathrm{W}\:\:\left(\mathrm{D}\right)\:\:\mathrm{37}°\mathrm{W} \\ $$

Question Number 11966 Answers: 0 Comments: 0

If a force of 200N is used to pull a block of mass 30 kg up a plane inclined at 60° to the horizontal at a steady speed . Calculate the percentage efficiency of the incline plane.

$$\mathrm{If}\:\mathrm{a}\:\mathrm{force}\:\mathrm{of}\:\mathrm{200N}\:\mathrm{is}\:\mathrm{used}\:\mathrm{to}\:\mathrm{pull}\:\mathrm{a}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{30}\:\mathrm{kg}\:\mathrm{up}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{inclined} \\ $$$$\mathrm{at}\:\mathrm{60}°\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{at}\:\mathrm{a}\:\mathrm{steady}\:\mathrm{speed}\:.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{percentage}\:\mathrm{efficiency} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{incline}\:\mathrm{plane}. \\ $$

Question Number 11964 Answers: 0 Comments: 0

A load of 60 kg is pushed up a 400 m incline side of a platform 3 m high. what is the velocity ratio of the plane ?

$$\:\mathrm{A}\:\mathrm{load}\:\mathrm{of}\:\mathrm{60}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{pushed}\:\mathrm{up}\:\mathrm{a}\:\mathrm{400}\:\mathrm{m}\:\mathrm{incline}\:\mathrm{side}\:\mathrm{of}\:\mathrm{a}\:\mathrm{platform}\:\mathrm{3}\:\mathrm{m}\:\mathrm{high}.\: \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{plane}\:? \\ $$

Question Number 11958 Answers: 0 Comments: 1

A∈M_(2016×2016) with the entries a_(ij) {_(0, if i+j≠2016) ^(1, if i+j=2016) find the determinant??

$${A}\in{M}_{\mathrm{2016}×\mathrm{2016}} \: \\ $$$${with}\:{the}\:{entries}\:{a}_{{ij}} \left\{_{\mathrm{0},\:{if}\:{i}+{j}\neq\mathrm{2016}} ^{\mathrm{1},\:{if}\:{i}+{j}=\mathrm{2016}} \right. \\ $$$${find}\:{the}\:{determinant}?? \\ $$

Question Number 11956 Answers: 0 Comments: 2

A , 2B ,3C ,4D are positive numbers forming a geometric series prov that : (A + 3C) (B + 2D) > 2

$${A}\:,\:\mathrm{2}{B}\:,\mathrm{3}{C}\:,\mathrm{4}{D}\: \\ $$$${are}\:{positive}\:{numbers}\:{forming}\:{a}\: \\ $$$${geometric}\:{series}\: \\ $$$${prov}\:{that}\:: \\ $$$$\left({A}\:+\:\mathrm{3}{C}\right)\:\left({B}\:+\:\mathrm{2}{D}\right)\:>\:\mathrm{2} \\ $$

Question Number 11943 Answers: 1 Comments: 0

The solution of the equation x^2 + x + 1 = 1 is

$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$${x}^{\mathrm{2}} +\:{x}\:+\:\mathrm{1}\:=\:\mathrm{1}\:\:\:\mathrm{is} \\ $$

Question Number 11937 Answers: 1 Comments: 5

∫(dx/(√(5 + 4x − x^2 ))) is this answer correct ? −ln[1/4(x − 5) − ln6(− 1 − x)] + C

$$\int\frac{\mathrm{dx}}{\sqrt{\mathrm{5}\:+\:\mathrm{4x}\:−\:\mathrm{x}^{\mathrm{2}} }}\: \\ $$$$ \\ $$$$ \\ $$$$\mathrm{is}\:\mathrm{this}\:\mathrm{answer}\:\mathrm{correct}\:?\:\:\:\:\:\:\:\:\:\:\:−\mathrm{ln}\left[\mathrm{1}/\mathrm{4}\left(\mathrm{x}\:−\:\mathrm{5}\right)\:−\:\mathrm{ln6}\left(−\:\mathrm{1}\:−\:\mathrm{x}\right)\right]\:+\:\mathrm{C} \\ $$

Question Number 11935 Answers: 3 Comments: 0

((7!)/(6!))+((8!)/(7!))+((9!)/(8!))+...((n!)/((n−1)!))=84 ⇒n=?

$$\frac{\mathrm{7}!}{\mathrm{6}!}+\frac{\mathrm{8}!}{\mathrm{7}!}+\frac{\mathrm{9}!}{\mathrm{8}!}+...\frac{\mathrm{n}!}{\left(\mathrm{n}−\mathrm{1}\right)!}=\mathrm{84}\:\Rightarrow\mathrm{n}=? \\ $$

Question Number 11930 Answers: 1 Comments: 0

Let a and b be two numbers, x be the single arithmetic mean of a and b. Show that the sum of n arithmetic means between a and b is nx.

$$\mathrm{Let}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{be}\:\mathrm{two}\:\mathrm{numbers},\:\mathrm{x}\:\mathrm{be}\:\mathrm{the}\:\mathrm{single}\:\mathrm{arithmetic}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{n}\:\mathrm{arithmetic}\:\mathrm{means}\:\mathrm{between}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{is}\:\mathrm{nx}. \\ $$

Question Number 11979 Answers: 1 Comments: 0

If the sum of first p terms, first q terms and first r terms of an AP be x, y and z respectively. Then (x/p)(q−r) + (y/q)(r−p) + (z/r)(p−q) is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:{p}\:\mathrm{terms},\:\mathrm{first}\:\:{q}\:\mathrm{terms}\:\mathrm{and} \\ $$$$\mathrm{first}\:{r}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{be}\:\:{x},\:{y}\:\:\mathrm{and}\:{z}\: \\ $$$$\mathrm{respectively}.\:\mathrm{Then} \\ $$$$\frac{{x}}{{p}}\left({q}−{r}\right)\:+\:\frac{{y}}{{q}}\left({r}−{p}\right)\:+\:\frac{{z}}{{r}}\left({p}−{q}\right)\:\:\mathrm{is} \\ $$

Question Number 11921 Answers: 1 Comments: 0

given that y=Acos5x + Bsin5x, show that (d^2 y/dx^2 )+25y=0

$${given}\:{that}\:{y}={Acos}\mathrm{5}{x}\:+\:{Bsin}\mathrm{5}{x}, \\ $$$${show}\:{that}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\mathrm{25}{y}=\mathrm{0} \\ $$

Question Number 11920 Answers: 4 Comments: 0

differentiate each function from first principle 1)f (x) = 1+(1/x) 2)f(x) = (1/(2x+3)) 3) f(x)=sin2x 4)f(x)=co2x

$${differentiate}\:{each}\:{function}\:{from}\:{first}\:{principle} \\ $$$$\left.\mathrm{1}\right){f}\:\left({x}\right)\:=\:\mathrm{1}+\frac{\mathrm{1}}{{x}} \\ $$$$\left.\mathrm{2}\right){f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}{x}+\mathrm{3}}\: \\ $$$$\left.\mathrm{3}\right)\:{f}\left({x}\right)={sin}\mathrm{2}{x} \\ $$$$\left.\mathrm{4}\right){f}\left({x}\right)={co}\mathrm{2}{x} \\ $$$$ \\ $$

Question Number 11915 Answers: 2 Comments: 2

Question Number 11914 Answers: 0 Comments: 0

Turevlenebilir bir f fonksiyonu icin f(x+y)=f(x)+f(y)+2xy ve f′(0)=−3 old.gore f′(2)=? czm∵ f(x+y)=f(x)+f(y)+2xy y yi sabit kabul edersek f′(x+y)=f′(x)+2y x=0,y=2 icn f′(2)=f′(0)+2.2 f′(2)=−3+4=1

$${Turevlenebilir}\:{bir}\:{f}\:{fonksiyonu}\:{icin} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+\mathrm{2}{xy}\:{ve}\:{f}'\left(\mathrm{0}\right)=−\mathrm{3} \\ $$$${old}.{gore}\:{f}'\left(\mathrm{2}\right)=? \\ $$$${czm}\because\:\:{f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+\mathrm{2}{xy} \\ $$$${y}\:{yi}\:{sabit}\:{kabul}\:{edersek} \\ $$$${f}'\left({x}+{y}\right)={f}'\left({x}\right)+\mathrm{2}{y} \\ $$$${x}=\mathrm{0},{y}=\mathrm{2}\:{icn} \\ $$$${f}'\left(\mathrm{2}\right)={f}'\left(\mathrm{0}\right)+\mathrm{2}.\mathrm{2} \\ $$$${f}'\left(\mathrm{2}\right)=−\mathrm{3}+\mathrm{4}=\mathrm{1} \\ $$

Question Number 11913 Answers: 0 Comments: 0

∫x^(x^x ) dx

$$\int\mathrm{x}^{\mathrm{x}^{\mathrm{x}} \:\:} \:\mathrm{dx} \\ $$

Question Number 11902 Answers: 1 Comments: 1

Assuming it rained at a constant rate, and the rain fell at angle θ to the ground (see diagram), determine if walking or running causes you to get more/less wet, or of it makes no difference for: 1. θ=90° (downwards) 2. θ<90° (the rain is moving on the same direction as you) 3. θ>90° (the rain is moving/blowing into you)

$$\mathrm{Assuming}\:\mathrm{it}\:\mathrm{rained}\:\mathrm{at}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{rate}, \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{rain}\:\mathrm{fell}\:\mathrm{at}\:\mathrm{angle}\:\theta\:\mathrm{to}\:\mathrm{the}\:\mathrm{ground} \\ $$$$\left(\mathrm{see}\:\mathrm{diagram}\right),\:\mathrm{determine}\:\mathrm{if}\:\mathrm{walking}\:\mathrm{or} \\ $$$$\mathrm{running}\:\mathrm{causes}\:\mathrm{you}\:\mathrm{to}\:\mathrm{get}\:\mathrm{more}/\mathrm{less}\:\mathrm{wet}, \\ $$$$\mathrm{or}\:\mathrm{of}\:\mathrm{it}\:\mathrm{makes}\:\mathrm{no}\:\mathrm{difference}\:\mathrm{for}: \\ $$$$\: \\ $$$$\mathrm{1}.\:\:\:\theta=\mathrm{90}°\:\:\left(\mathrm{downwards}\right) \\ $$$$\mathrm{2}.\:\theta<\mathrm{90}°\:\:\left(\mathrm{the}\:\mathrm{rain}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{on}\:\mathrm{the}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\mathrm{same}\:\mathrm{direction}\:\mathrm{as}\:\mathrm{you}\right) \\ $$$$\mathrm{3}.\:\theta>\mathrm{90}°\:\:\left(\mathrm{the}\:\mathrm{rain}\:\mathrm{is}\:\mathrm{moving}/\mathrm{blowing}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\mathrm{into}\:\mathrm{you}\right) \\ $$

Question Number 11901 Answers: 0 Comments: 0

((7cos^2 x+sin^2 x−3)/(2cos^2 x−sin^2 x))=? czm∵ ((7cos^2 x+1−cos^2 x−3)/(2cos^2 x−sin^2 x)) ((6cos^2 x−2)/(2cos^2 x−(1−cos^2 x)))=((6cos^2 x−2)/(3cos^2 x−1)) ((2(3cos^2 x−1))/(3cos^2 x−1))=2

Question Number 11900 Answers: 1 Comments: 0

Calculate. cos(𝛑/7)×cos((4𝛑)/7)×cos((5𝛑)/7).

$$\boldsymbol{\mathrm{Calculate}}. \\ $$$$\boldsymbol{\mathrm{cos}}\frac{\boldsymbol{\pi}}{\mathrm{7}}×\boldsymbol{\mathrm{cos}}\frac{\mathrm{4}\boldsymbol{\pi}}{\mathrm{7}}×\boldsymbol{\mathrm{cos}}\frac{\mathrm{5}\boldsymbol{\pi}}{\mathrm{7}}. \\ $$

Question Number 11899 Answers: 0 Comments: 0

f′(x)={_(3 ;x>2) ^(2x ; x≤2) f(2)=1 ise f(1)+f(3)=? czm∵ f(x)={_(3x+c_2 ; x>2) ^(x^2 +c_1 ;x≤2) f(2)=x^2 +c_1 dir 1=4+c_1 => c_1 =−3 1=2.3+c_2 => c_2 =−5 f(x)={_(3x−5 x>2) ^(x^2 −3 ;x≤2) f(1)=1^2 −3=−2, f(3)=3.3−5=4 f(1)+f(3)=−2+4=2

$${f}'\left({x}\right)=\left\{_{\mathrm{3}\:\:\:\:\:;{x}>\mathrm{2}} ^{\mathrm{2}{x}\:\:;\:{x}\leqslant\mathrm{2}} \right. \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{1}\:{ise}\:{f}\left(\mathrm{1}\right)+{f}\left(\mathrm{3}\right)=? \\ $$$${czm}\because\:{f}\left({x}\right)=\left\{_{\mathrm{3}{x}+{c}_{\mathrm{2}} \:;\:{x}>\mathrm{2}} ^{{x}^{\mathrm{2}} +{c}_{\mathrm{1}} \:;{x}\leqslant\mathrm{2}} \right. \\ $$$${f}\left(\mathrm{2}\right)={x}^{\mathrm{2}} +{c}_{\mathrm{1}} \:{dir} \\ $$$$\mathrm{1}=\mathrm{4}+{c}_{\mathrm{1}} \:=>\:{c}_{\mathrm{1}} =−\mathrm{3} \\ $$$$\mathrm{1}=\mathrm{2}.\mathrm{3}+{c}_{\mathrm{2}} \:=>\:{c}_{\mathrm{2}} =−\mathrm{5} \\ $$$${f}\left({x}\right)=\left\{_{\mathrm{3}{x}−\mathrm{5}\:\:\:\:{x}>\mathrm{2}} ^{{x}^{\mathrm{2}} −\mathrm{3}\:\:\:;{x}\leqslant\mathrm{2}} \right. \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1}^{\mathrm{2}} −\mathrm{3}=−\mathrm{2},\:{f}\left(\mathrm{3}\right)=\mathrm{3}.\mathrm{3}−\mathrm{5}=\mathrm{4} \\ $$$${f}\left(\mathrm{1}\right)+{f}\left(\mathrm{3}\right)=−\mathrm{2}+\mathrm{4}=\mathrm{2} \\ $$$$ \\ $$

Question Number 11889 Answers: 1 Comments: 0

The number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{8}\: \\ $$$$\mathrm{different}\:\mathrm{flowers}\:\mathrm{can}\:\mathrm{be}\:\mathrm{strung}\:\mathrm{to} \\ $$$$\mathrm{form}\:\mathrm{a}\:\mathrm{garland}\:\mathrm{so}\:\mathrm{that}\:\mathrm{4}\:\mathrm{particular} \\ $$$$\mathrm{flowers}\:\mathrm{are}\:\mathrm{never}\:\mathrm{separated}\:\mathrm{is} \\ $$

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