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Question Number 12023 Answers: 3 Comments: 0

Solve simultaneously x + y − (√(xy)) = 3 .......... (i) (√(x + 1)) + (√(y + 1)) = 4 .......... (ii)

$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:−\:\sqrt{\mathrm{xy}}\:=\:\mathrm{3}\:\:\:\:\:..........\:\left(\mathrm{i}\right) \\ $$$$\sqrt{\mathrm{x}\:+\:\mathrm{1}}\:+\:\sqrt{\mathrm{y}\:+\:\mathrm{1}}\:=\:\mathrm{4}\:\:\:\:\:..........\:\left(\mathrm{ii}\right) \\ $$

Question Number 12019 Answers: 1 Comments: 0

Question Number 12018 Answers: 0 Comments: 0

Question Number 12017 Answers: 0 Comments: 0

How Can we expand (a+b)^(1/2) and (a+b)^(−n) ?

$${How}\:{Can}\:{we}\:{expand}\:\left({a}+{b}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{and} \\ $$$$\left({a}+{b}\right)^{−{n}} \:? \\ $$

Question Number 12015 Answers: 1 Comments: 0

prove that for all x ∈R, e^x ≥x^e

$${prove}\:{that}\:{for}\:{all}\:{x}\:\in{R}, \\ $$$${e}^{{x}} \geqslant{x}^{{e}} \\ $$

Question Number 12013 Answers: 1 Comments: 0

The slope of a curve is, 7x + 3 and it passes through the point (2, 4), Find the equation of the point

$$\mathrm{The}\:\mathrm{slope}\:\mathrm{of}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{is},\:\mathrm{7x}\:+\:\mathrm{3}\:\:\mathrm{and}\:\mathrm{it}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{2},\:\mathrm{4}\right), \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point} \\ $$

Question Number 12008 Answers: 1 Comments: 1

Question Number 11998 Answers: 0 Comments: 4

Question Number 11994 Answers: 0 Comments: 1

4x=60

$$\mathrm{4}{x}=\mathrm{60} \\ $$

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

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

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