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Question Number 117728 Answers: 3 Comments: 0

Solution (d^2 y/dx^2 ) + 3(dy/dx) − 4y = x^2

$${Solution}\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{3}\frac{{dy}}{{dx}}\:−\:\mathrm{4}{y}\:=\:{x}^{\mathrm{2}} \\ $$

Question Number 117724 Answers: 2 Comments: 5

∫ ((sin^(−1) (x))/x^2 ) dx =?

$$\int\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 117708 Answers: 2 Comments: 3

Find the number of all 5 digit numbers x_1 x_2 x_3 x_4 x_5 with x_1 ≥x_2 ≥x_3 ≥x_4 ≥x_5 .

$${Find}\:{the}\:{number}\:{of}\:{all}\:\mathrm{5}\:{digit} \\ $$$${numbers}\:{x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} {x}_{\mathrm{4}} {x}_{\mathrm{5}} \:{with} \\ $$$${x}_{\mathrm{1}} \geqslant{x}_{\mathrm{2}} \geqslant{x}_{\mathrm{3}} \geqslant{x}_{\mathrm{4}} \geqslant{x}_{\mathrm{5}} . \\ $$

Question Number 117704 Answers: 0 Comments: 7

Given a function ψ:R→R with ψ(θ) = θ^2 −x^2 . Find the value of ((d^2 ψ(θ))/dx^2 ) when θ=8

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\:\psi:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\mathrm{with}\:\psi\left(\theta\right)\:=\:\theta^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{d}^{\mathrm{2}} \psi\left(\theta\right)}{\mathrm{dx}^{\mathrm{2}} }\:\:\mathrm{when}\:\theta=\mathrm{8} \\ $$

Question Number 117688 Answers: 1 Comments: 3

Question Number 117687 Answers: 2 Comments: 0

lim_(x→0) (((ln (cosh x)−ln (cos x))^2 )/( (√(cosh x))+(√(cos x))−2)) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{ln}\:\left(\mathrm{cosh}\:\mathrm{x}\right)−\mathrm{ln}\:\left(\mathrm{cos}\:\mathrm{x}\right)\right)^{\mathrm{2}} }{\:\sqrt{\mathrm{cosh}\:\mathrm{x}}+\sqrt{\mathrm{cos}\:\mathrm{x}}−\mathrm{2}}\:=?\: \\ $$

Question Number 117666 Answers: 0 Comments: 3

Question Number 117655 Answers: 0 Comments: 3

Question Number 117654 Answers: 0 Comments: 1

Question Number 117649 Answers: 1 Comments: 0

Let P(x) be a polynomial function of degree n such that P(k)=(k/(k+1)) for k=0,1,2,...,n. Then P(n+1) is equal to (A) −1 if n is even (B) 1 if n is odd (C) (n/(n+2)) if n is even (D) (n/(n+2)) if n is odd Which among the four proposals is/are correct ?

$$\mathrm{Let}\:\mathrm{P}\left({x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{function}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{P}\left({k}\right)=\frac{{k}}{{k}+\mathrm{1}}\:\:\mathrm{for}\:{k}=\mathrm{0},\mathrm{1},\mathrm{2},...,\mathrm{n}.\:\mathrm{Then}\:\mathrm{P}\left(\mathrm{n}+\mathrm{1}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\: \\ $$$$\left(\mathrm{A}\right)\:−\mathrm{1}\:\mathrm{if}\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{1}\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{odd} \\ $$$$\left(\mathrm{C}\right)\:\frac{{n}}{{n}+\mathrm{2}}\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{even}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\frac{{n}}{{n}+\mathrm{2}}\:\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{odd} \\ $$$$\mathrm{Which}\:\mathrm{among}\:\mathrm{the}\:\mathrm{four}\:\mathrm{proposals}\:\mathrm{is}/\mathrm{are}\:\mathrm{correct}\:? \\ $$

Question Number 117646 Answers: 2 Comments: 1

Question Number 117645 Answers: 1 Comments: 0

Question Number 117641 Answers: 3 Comments: 0

find the solution (√(6−x)) > x−4

$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\sqrt{\mathrm{6}−\mathrm{x}}\:>\:\mathrm{x}−\mathrm{4} \\ $$

Question Number 117638 Answers: 2 Comments: 0

∫_0 ^1 ((2x^(12) +5x^9 )/((x^5 +x^3 +1)^3 )) dx =?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{2x}^{\mathrm{12}} +\mathrm{5x}^{\mathrm{9}} }{\left(\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{dx}\:=?\: \\ $$

Question Number 117637 Answers: 2 Comments: 0

lim_(x→0) (cos x )^(cot x) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{cos}\:\mathrm{x}\:\right)^{\mathrm{cot}\:\mathrm{x}} \:=? \\ $$

Question Number 117673 Answers: 4 Comments: 0

Question Number 117632 Answers: 3 Comments: 0

Solution from 2xy dy = (x^(2 ) − y^2 )dx

$${Solution}\:{from}\:\:\:\mathrm{2}{xy}\:{dy}\:=\:\left({x}^{\mathrm{2}\:} \:−\:{y}^{\mathrm{2}} \right){dx} \\ $$

Question Number 117675 Answers: 2 Comments: 1

Question Number 117620 Answers: 5 Comments: 0

x^4 −⌊5x^2 ⌋+4=0

$${x}^{\mathrm{4}} −\lfloor\mathrm{5}{x}^{\mathrm{2}} \rfloor+\mathrm{4}=\mathrm{0} \\ $$

Question Number 117608 Answers: 1 Comments: 0

solve (d^2 y/dt^2 )+w^2 x=0

$${solve} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }+{w}^{\mathrm{2}} {x}=\mathrm{0} \\ $$

Question Number 117606 Answers: 2 Comments: 0

find sina+sin(a+b)+sin(a+2b)++..+sin{a+(n−1)b}

$${find} \\ $$$${sina}+{sin}\left({a}+{b}\right)+{sin}\left({a}+\mathrm{2}{b}\right)++..+{sin}\left\{{a}+\left({n}−\mathrm{1}\right){b}\right\} \\ $$

Question Number 117602 Answers: 2 Comments: 0

(d^2 y/dx^2 )+a^2 y=cosax

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+{a}^{\mathrm{2}} {y}={cosax} \\ $$

Question Number 117603 Answers: 0 Comments: 0

Let f : R→R be a function satisfying the following : (a) f(−x)=−f(x) (b) f(x+1)=f(x)+1 (c) f((1/x))=((f(x))/x^2 ) for all x≠0 Show that (i)f(x)=x for all x,y∈R (ii) f(x+y)=f(x)+f(y) for all x,y∈R (iii) f(xy)=f(x)f(y) for all x,y∈R (iv) f((x/y))=((f(x))/(f(y))) for all x,y∈R with y≠0

$$\mathrm{Let}\:\mathrm{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{following}\:: \\ $$$$\left(\mathrm{a}\right)\:{f}\left(−{x}\right)=−{f}\left({x}\right) \\ $$$$\left(\mathrm{b}\right)\:{f}\left({x}+\mathrm{1}\right)={f}\left({x}\right)+\mathrm{1} \\ $$$$\left(\mathrm{c}\right)\:{f}\left(\frac{\mathrm{1}}{{x}}\right)=\frac{{f}\left({x}\right)}{{x}^{\mathrm{2}} }\:\mathrm{for}\:\mathrm{all}\:{x}\neq\mathrm{0} \\ $$$$\mathrm{Show}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right){f}\left({x}\right)={x}\:\mathrm{for}\:\mathrm{all}\:{x},\mathrm{y}\in\mathbb{R} \\ $$$$\left(\mathrm{ii}\right)\:{f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left(\mathrm{y}\right)\:\mathrm{for}\:\mathrm{all}\:{x},\mathrm{y}\in\mathbb{R} \\ $$$$\left(\mathrm{iii}\right)\:{f}\left({xy}\right)={f}\left({x}\right){f}\left(\mathrm{y}\right)\:\mathrm{for}\:\mathrm{all}\:{x},\mathrm{y}\in\mathbb{R} \\ $$$$\left(\mathrm{iv}\right)\:{f}\left(\frac{{x}}{\mathrm{y}}\right)=\frac{{f}\left({x}\right)}{{f}\left(\mathrm{y}\right)}\:\mathrm{for}\:\mathrm{all}\:{x},\mathrm{y}\in\mathbb{R}\:\mathrm{with}\:\mathrm{y}\neq\mathrm{0} \\ $$

Question Number 117597 Answers: 0 Comments: 1

Let A, B, and C be three sets and X be the set of all elements which belong to exactly two of the sets A,B and C. Prove that X is equal to (A∪B∪C)−[AΔ(BΔC)]

$$\mathrm{Let}\:\mathrm{A},\:\mathrm{B},\:\mathrm{and}\:\mathrm{C}\:\mathrm{be}\:\mathrm{three}\:\mathrm{sets}\:\mathrm{and}\:\mathrm{X}\:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all} \\ $$$$\mathrm{elements}\:\mathrm{which}\:\mathrm{belong}\:\mathrm{to}\:\mathrm{exactly}\:\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sets}\:\mathrm{A},\mathrm{B} \\ $$$$\mathrm{and}\:\mathrm{C}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{X}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{A}\cup\mathrm{B}\cup\mathrm{C}\right)−\left[\mathrm{A}\Delta\left(\mathrm{B}\Delta\mathrm{C}\right)\right] \\ $$

Question Number 117594 Answers: 1 Comments: 0

A rope 5m long is fastened to two hooks 4.0m apart on a horizontal ceiling.to the rope is attached a 10kg mass so that the segments of the rope are 3.0m and 2.0m.compute the tensionin each segment

$${A}\:{rope}\:\mathrm{5}{m}\:{long}\:{is}\:{fastened}\:{to}\:{two}\:{hooks}\: \\ $$$$\mathrm{4}.\mathrm{0}{m}\:{apart}\:{on}\:{a}\:{horizontal} \\ $$$${ceiling}.{to}\:{the}\:{rope}\:{is}\:{attached}\:{a}\:\mathrm{10}{kg}\: \\ $$$${mass}\:{so}\:{that}\:{the}\:{segments}\:{of}\:{the}\:{rope} \\ $$$${are}\:\mathrm{3}.\mathrm{0}{m}\:{and}\:\mathrm{2}.\mathrm{0}{m}.{compute}\:{the} \\ $$$${tensionin}\:{each}\:{segment} \\ $$

Question Number 117585 Answers: 1 Comments: 4

solution (dy/dx) = sin x + e^(2x) + x^2

$${solution}\:\:\:\:\frac{{dy}}{{dx}}\:=\:{sin}\:{x}\:+\:{e}^{\mathrm{2}{x}} \:+\:{x}^{\mathrm{2}} \\ $$

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