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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

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

Question Number 117578 Answers: 0 Comments: 1

Question Number 117577 Answers: 1 Comments: 0

If f(x), g(x) and h(x) are three functions, where f(x)=2x^5 −8x^2 +1, f(x−3)=g(3x−2) and g(3x+1)=h(x+3), show h(x)=f(x−5).

$$\mathrm{If}\:{f}\left({x}\right),\:{g}\left({x}\right)\:\mathrm{and}\:{h}\left({x}\right)\:\mathrm{are}\:\mathrm{three}\:\mathrm{functions}, \\ $$$$\mathrm{where}\:{f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{5}} −\mathrm{8}{x}^{\mathrm{2}} +\mathrm{1},\:{f}\left({x}−\mathrm{3}\right)={g}\left(\mathrm{3}{x}−\mathrm{2}\right) \\ $$$$\mathrm{and}\:{g}\left(\mathrm{3}{x}+\mathrm{1}\right)={h}\left({x}+\mathrm{3}\right),\:\mathrm{show}\:{h}\left({x}\right)={f}\left({x}−\mathrm{5}\right). \\ $$

Question Number 117574 Answers: 1 Comments: 0

... advanced integral... Evaluate :: I := ∫_0 ^( ∞) (( 4xln(x))/(x^4 +2x^2 +4 ))dx =?? ... m.n.1970..

$$\:\:\:\:\:\:\:\:...\:{advanced}\:\:{integral}... \\ $$$$\:\:\:\:\:\: \\ $$$$\mathscr{E}{valuate}\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{4}{xln}\left({x}\right)}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{4}\:}{dx}\:=??\: \\ $$$$\:\:\:\:\:...\:{m}.{n}.\mathrm{1970}.. \\ $$$$\: \\ $$

Question Number 117568 Answers: 1 Comments: 1

Suppose the non-constant functions f and g satisfy the following two conditions: I: g(x−y)=g(x)g(y)+f(x)f(y) ∀ x,y∈R II: f(0)=0 Evaluate i. g(0) ii.[f(x)]^2 +[g(x)]^2

$$\mathrm{Suppose}\:\mathrm{the}\:\mathrm{non}-\mathrm{constant}\:\mathrm{functions}\:{f}\:\mathrm{and}\:{g} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{following}\:\mathrm{two}\:\mathrm{conditions}: \\ $$$$\mathrm{I}:\:{g}\left({x}−{y}\right)={g}\left({x}\right){g}\left({y}\right)+{f}\left({x}\right){f}\left({y}\right)\:\forall\:{x},{y}\in\mathbb{R} \\ $$$$\mathrm{II}:\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\mathrm{Evaluate} \\ $$$$\mathrm{i}.\:{g}\left(\mathrm{0}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{ii}.\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{g}\left({x}\right)\right]^{\mathrm{2}} \\ $$

Question Number 117567 Answers: 1 Comments: 0

give A={a,b,c,d,e}; n(p(A))=?

$${give}\:{A}=\left\{{a},{b},{c},{d},{e}\right\};\:{n}\left({p}\left({A}\right)\right)=? \\ $$$$\:\:\: \\ $$

Question Number 117557 Answers: 1 Comments: 0

second derivative x^2 +3y^2 =5

$$\mathrm{second}\:\mathrm{derivative} \\ $$$${x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} =\mathrm{5} \\ $$

Question Number 117555 Answers: 2 Comments: 0

Alternative forms { (((√x)+(√y)=((23)/(12)))),((9x+16y=29)) :}

$${Alternative}\:{forms} \\ $$$$\begin{cases}{\sqrt{{x}}+\sqrt{{y}}=\frac{\mathrm{23}}{\mathrm{12}}}\\{\mathrm{9}{x}+\mathrm{16}{y}=\mathrm{29}}\end{cases} \\ $$$$ \\ $$

Question Number 117552 Answers: 2 Comments: 1

please help cos (π/7) . cos ((2π)/7) . cos ((4π)/7) = ?

$$\mathrm{please}\:\mathrm{help} \\ $$$$\mathrm{cos}\:\frac{\pi}{\mathrm{7}}\:.\:\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{7}}\:.\:\mathrm{cos}\:\frac{\mathrm{4}\pi}{\mathrm{7}}\:=\:? \\ $$

Question Number 117551 Answers: 3 Comments: 0

(a)lim_(x→1) ((1/(2(1−(√x)))) −(1/(3(1−(x)^(1/(3 )) )))) =? (b) lim_(x→∞) ((ln (x+(√(1+x^2 ))) −ln (x+(√(x^2 −1)) ))/((ln (((x+1)/(x−1))))^2 ))=?

$$\left(\mathrm{a}\right)\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{1}−\sqrt{\mathrm{x}}\right)}\:−\frac{\mathrm{1}}{\mathrm{3}\left(\mathrm{1}−\sqrt[{\mathrm{3}\:}]{\mathrm{x}}\:\right)}\right)\:=? \\ $$$$\left(\mathrm{b}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\right)\:−\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\:\right)}{\left(\mathrm{ln}\:\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}\right)\right)^{\mathrm{2}} }=? \\ $$

Question Number 117545 Answers: 4 Comments: 0

lim_(x→0^+ ) (1+tan^2 ((√x)))^(1/(2x))

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left(\sqrt{\mathrm{x}}\right)\right)^{\frac{\mathrm{1}}{\mathrm{2x}}} \\ $$

Question Number 117531 Answers: 0 Comments: 1

The test result of the 100 job applicants are given in the table determinant (((score 53 61 72 85 94)),(( F 12 22 25 32 9))) If 45% of applicants are accepted, what is the score of a person to be accepted?

$$\mathrm{The}\:\mathrm{test}\:\mathrm{result}\:\mathrm{of}\:\mathrm{the}\:\mathrm{100}\:\mathrm{job}\:\mathrm{applicants} \\ $$$$\mathrm{are}\:\mathrm{given}\:\mathrm{in}\:\mathrm{the}\:\mathrm{table} \\ $$$$\begin{vmatrix}{\mathrm{score}\:\:\:\mathrm{53}\:\:\:\mathrm{61}\:\:\:\:\mathrm{72}\:\:\:\:\mathrm{85}\:\:\:\:\mathrm{94}}\\{\:\:\:\:\mathrm{F}\:\:\:\:\:\:\mathrm{12}\:\:\:\mathrm{22}\:\:\:\:\mathrm{25}\:\:\:\:\mathrm{32}\:\:\:\:\:\:\mathrm{9}}\end{vmatrix} \\ $$$$\mathrm{If}\:\mathrm{45\%}\:\mathrm{of}\:\mathrm{applicants}\:\mathrm{are}\:\mathrm{accepted},\:\mathrm{what} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{score}\:\mathrm{of}\:\mathrm{a}\:\mathrm{person}\:\mathrm{to}\:\mathrm{be}\:\mathrm{accepted}? \\ $$

Question Number 117543 Answers: 2 Comments: 0

If vector a^→ +b^→ +c^→ =0 ∣a^→ ∣=7, ∣b^→ ∣=3 and ∣c^→ ∣=5 find the angle vector a^→ and c^→ ?

$$\mathrm{If}\:\mathrm{vector}\:\overset{\rightarrow} {\mathrm{a}}+\overset{\rightarrow} {\mathrm{b}}+\overset{\rightarrow} {\mathrm{c}}=\mathrm{0} \\ $$$$\mid\overset{\rightarrow} {\mathrm{a}}\mid=\mathrm{7},\:\mid\overset{\rightarrow} {\mathrm{b}}\mid=\mathrm{3}\:\mathrm{and}\:\mid\overset{\rightarrow} {\mathrm{c}}\mid=\mathrm{5} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{vector}\:\overset{\rightarrow} {\mathrm{a}}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{c}}\:? \\ $$

Question Number 117527 Answers: 1 Comments: 0

∫_( 0) ^( 1) xsec(2x)dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \mathrm{xsec}\left(\mathrm{2x}\right)\mathrm{dx} \\ $$

Question Number 117518 Answers: 1 Comments: 0

f(x+1)+xf(1−x)=x^2 f^(−1) (x) =?

$$\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)+\mathrm{xf}\left(\mathrm{1}−\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} \\ $$$$\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\:=? \\ $$

Question Number 117511 Answers: 3 Comments: 0

Question Number 117498 Answers: 3 Comments: 0

lim_(x→0) ((sin (x−sin x))/( (√(1+x^3 ))−1)) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\mathrm{x}−\mathrm{sin}\:\mathrm{x}\right)}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }−\mathrm{1}}\:? \\ $$

Question Number 117500 Answers: 1 Comments: 0

In ΔABC, (a/(cos A))=(b/(cos B))=(c/(cos C)), then ΔABC is A. irregular sides acute-angled triangle B. obtuse-angled triangle C. right-angled triangle D. equilateral triangle E. isoceles triangle

$$\mathrm{In}\:\Delta\mathrm{ABC},\:\frac{{a}}{\mathrm{cos}\:{A}}=\frac{{b}}{\mathrm{cos}\:{B}}=\frac{{c}}{\mathrm{cos}\:{C}}, \\ $$$$\mathrm{then}\:\Delta\mathrm{ABC}\:\mathrm{is}\: \\ $$$${A}.\:\mathrm{irregular}\:\mathrm{sides}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$${B}.\:\mathrm{obtuse}-\mathrm{angled}\:\mathrm{triangle} \\ $$$${C}.\:\mathrm{right}-\mathrm{angled}\:\mathrm{triangle} \\ $$$${D}.\:\mathrm{equilateral}\:\mathrm{triangle} \\ $$$${E}.\:\mathrm{isoceles}\:\mathrm{triangle} \\ $$

Question Number 117497 Answers: 1 Comments: 0

find the solution set of the equation sec 3θ = sec θ

$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{equation}\:\mathrm{sec}\:\mathrm{3}\theta\:=\:\mathrm{sec}\:\theta \\ $$

Question Number 117496 Answers: 2 Comments: 0

∫ ((sec^2 θ tan^2 θ)/( (√(9−tan^2 θ)))) dθ =?

$$\int\:\frac{\mathrm{sec}\:^{\mathrm{2}} \theta\:\mathrm{tan}\:^{\mathrm{2}} \theta}{\:\sqrt{\mathrm{9}−\mathrm{tan}\:^{\mathrm{2}} \theta}}\:\mathrm{d}\theta\:=? \\ $$

Question Number 117493 Answers: 1 Comments: 0

∫_c (3xy−e^(sin x) )dx+(7x+(√(y^4 +1)) )dy C : triangle with vertex (0,0),(0,1) and (1,0)

$$\int_{\mathrm{c}} \left(\mathrm{3xy}−\mathrm{e}^{\mathrm{sin}\:\mathrm{x}} \right)\mathrm{dx}+\left(\mathrm{7x}+\sqrt{\mathrm{y}^{\mathrm{4}} +\mathrm{1}}\:\right)\mathrm{dy} \\ $$$$\mathrm{C}\::\:\mathrm{triangle}\:\mathrm{with}\:\mathrm{vertex}\:\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{0},\mathrm{1}\right) \\ $$$$\mathrm{and}\:\left(\mathrm{1},\mathrm{0}\right) \\ $$

Question Number 117486 Answers: 1 Comments: 0

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