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Question Number 24667    Answers: 0   Comments: 0

Question Number 24663    Answers: 1   Comments: 0

solve ∣x−3∣=∣3x+2∣−1

$$\boldsymbol{{solve}}\:\mid{x}−\mathrm{3}\mid=\mid\mathrm{3}{x}+\mathrm{2}\mid−\mathrm{1} \\ $$

Question Number 24661    Answers: 1   Comments: 2

Question Number 24662    Answers: 1   Comments: 0

solve ∣x^2 −4x−5∣=7

$$\boldsymbol{{solve}}\:\mid{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{5}\mid=\mathrm{7} \\ $$$$ \\ $$

Question Number 24651    Answers: 2   Comments: 0

∫sin x+cos y dx

$$\int\mathrm{sin}\:{x}+\mathrm{cos}\:{y}\:{dx} \\ $$

Question Number 24647    Answers: 0   Comments: 0

Question Number 24643    Answers: 1   Comments: 0

Question Number 24635    Answers: 0   Comments: 0

Calculate the electric potential at a point P at a distance of 3m of either charges of +20 μC and − 15μC. which are 25cm apart. Also calculate potential energy of a +3.5μC placed at point P.

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{electric}\:\mathrm{potential}\:\mathrm{at}\:\mathrm{a}\:\mathrm{point}\:\mathrm{P}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{3m}\:\mathrm{of}\:\mathrm{either} \\ $$$$\mathrm{charges}\:\mathrm{of}\:\:+\mathrm{20}\:\mu\mathrm{C}\:\mathrm{and}\:\:−\:\mathrm{15}\mu\mathrm{C}.\:\:\mathrm{which}\:\mathrm{are}\:\mathrm{25cm}\:\mathrm{apart}. \\ $$$$\mathrm{Also}\:\mathrm{calculate}\:\mathrm{potential}\:\mathrm{energy}\:\mathrm{of}\:\mathrm{a}\:\:+\mathrm{3}.\mathrm{5}\mu\mathrm{C}\:\mathrm{placed}\:\mathrm{at}\:\mathrm{point}\:\mathrm{P}. \\ $$

Question Number 24644    Answers: 2   Comments: 0

The density of a non-uniform rod of length 1 m is given by ρ(x) = a(1 + bx^2 ) where a and b are constants and 0 ≤ x ≤ 1. The centre of mass of the rod will be at (1) ((3(2 + b))/(4(3 + b))) (2) ((4(2 + b))/(3(3 + b))) (3) ((3(3 + b))/(4(2 + b))) (4) ((4(3 + b))/(3(2 + b)))

$$\mathrm{The}\:\mathrm{density}\:\mathrm{of}\:\mathrm{a}\:\mathrm{non}-\mathrm{uniform}\:\mathrm{rod}\:\mathrm{of} \\ $$$$\mathrm{length}\:\mathrm{1}\:\mathrm{m}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\rho\left({x}\right)\:=\:{a}\left(\mathrm{1}\:+\:{bx}^{\mathrm{2}} \right) \\ $$$$\mathrm{where}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{constants}\:\mathrm{and} \\ $$$$\mathrm{0}\:\leqslant\:{x}\:\leqslant\:\mathrm{1}.\:\mathrm{The}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rod} \\ $$$$\mathrm{will}\:\mathrm{be}\:\mathrm{at} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{3}\left(\mathrm{2}\:+\:{b}\right)}{\mathrm{4}\left(\mathrm{3}\:+\:{b}\right)} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{4}\left(\mathrm{2}\:+\:{b}\right)}{\mathrm{3}\left(\mathrm{3}\:+\:{b}\right)} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{3}\left(\mathrm{3}\:+\:{b}\right)}{\mathrm{4}\left(\mathrm{2}\:+\:{b}\right)} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{4}\left(\mathrm{3}\:+\:{b}\right)}{\mathrm{3}\left(\mathrm{2}\:+\:{b}\right)} \\ $$

Question Number 24631    Answers: 1   Comments: 0

Question Number 24622    Answers: 0   Comments: 4

Question Number 24612    Answers: 1   Comments: 0

If f(x) = [x] then is fof(x) = f(x)?

$$\mathrm{If}\:{f}\left({x}\right)\:=\:\left[{x}\right]\:{then}\:{is}\:{fof}\left({x}\right)\:=\:{f}\left({x}\right)? \\ $$$$ \\ $$

Question Number 24628    Answers: 0   Comments: 1

Question Number 24605    Answers: 3   Comments: 1

Question Number 24604    Answers: 2   Comments: 0

x^2 −xsin x−cos x=0

$${x}^{\mathrm{2}} −{x}\mathrm{sin}\:{x}−\mathrm{cos}\:{x}=\mathrm{0} \\ $$

Question Number 24598    Answers: 2   Comments: 0

if y=x^3 +x^2 +3x.... find its turning point

$${if}\:{y}={x}^{\mathrm{3}} +{x}^{\mathrm{2}} +\mathrm{3}{x}.... \\ $$$${find}\:{its}\:{turning}\:{point} \\ $$

Question Number 24641    Answers: 1   Comments: 1

Question Number 24576    Answers: 1   Comments: 1

Question Number 24569    Answers: 1   Comments: 3

Question Number 24565    Answers: 1   Comments: 0

y=ax^3 +bx^2 +cx+d , then prove that the equation y=0 has only one real root if a[(9ad−bc)^2 −4(b^2 −3ac)(c^2 −3bd)] > 0 provided b^2 > 3ac .

$$\:\:\boldsymbol{{y}}=\boldsymbol{{ax}}^{\mathrm{3}} +\boldsymbol{{bx}}^{\mathrm{2}} +\boldsymbol{{cx}}+\boldsymbol{{d}}\:,\:{then} \\ $$$${prove}\:{that}\:{the}\:{equation}\:{y}=\mathrm{0} \\ $$$${has}\:{only}\:{one}\:{real}\:{root}\:{if} \\ $$$$\:\boldsymbol{{a}}\left[\left(\mathrm{9}\boldsymbol{{ad}}−\boldsymbol{{bc}}\right)^{\mathrm{2}} −\mathrm{4}\left(\boldsymbol{{b}}^{\mathrm{2}} −\mathrm{3}\boldsymbol{{ac}}\right)\left(\boldsymbol{{c}}^{\mathrm{2}} −\mathrm{3}\boldsymbol{{bd}}\right)\right] \\ $$$$\:\:\:\:>\:\mathrm{0}\:\:\:\:\:{provided}\:\:\:\boldsymbol{{b}}^{\mathrm{2}} \:>\:\mathrm{3}\boldsymbol{{ac}}\:. \\ $$

Question Number 24555    Answers: 1   Comments: 3

Question Number 24554    Answers: 1   Comments: 4

Show that: tan^(−1) ((p/(p + 2q))) + tan^(−1) ((p/(p + q))) = (π/2)

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{p}}{\mathrm{p}\:+\:\mathrm{2q}}\right)\:+\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{p}}{\mathrm{p}\:+\:\mathrm{q}}\right)\:=\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 24549    Answers: 0   Comments: 5

Question Number 24548    Answers: 1   Comments: 0

prove that Σ_(n=1) ^r {n(n−(r/2))^2 }= r∙Σ_(n=1) ^(r/2) n^2 where r = 2k ; k ∈ N

$${prove}\:{that}\: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{{r}} {\sum}}\left\{{n}\left({n}−\frac{{r}}{\mathrm{2}}\right)^{\mathrm{2}} \right\}=\:{r}\centerdot\underset{{n}=\mathrm{1}} {\overset{{r}/\mathrm{2}} {\sum}}{n}^{\mathrm{2}} \\ $$$$\:{where}\:\:\:{r}\:=\:\mathrm{2}{k}\:;\:{k}\:\in\:\mathbb{N} \\ $$

Question Number 24542    Answers: 0   Comments: 2

Prove that coefficient of x^n in ((a+bx+cx^2 )/e^x ) is (((−1)^n )/(n!))[cn^2 −(b+c)n+a]

$${Prove}\:{that}\:{coefficient}\:{of}\:{x}^{{n}} \:{in} \\ $$$$\frac{{a}+{bx}+{cx}^{\mathrm{2}} }{{e}^{{x}} }\:{is}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\left[{cn}^{\mathrm{2}} −\left({b}+{c}\right){n}+{a}\right] \\ $$

Question Number 24540    Answers: 2   Comments: 1

Prove that (i) Σ_(n=0) ^∞ (n^2 /(n!))=2e. (ii) Σ_(n=0) ^∞ (n^3 /(n!))=5e. (iii) Σ_(n=0) ^∞ (n^4 /(n!))=15e.

$${Prove}\:{that} \\ $$$$\left({i}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{2}} }{{n}!}=\mathrm{2}{e}. \\ $$$$\left({ii}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{3}} }{{n}!}=\mathrm{5}{e}. \\ $$$$\left({iii}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{4}} }{{n}!}=\mathrm{15}{e}. \\ $$

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