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Question Number 17260 Answers: 1 Comments: 1

for a,b,c>0 prove that (ab+bc+ca)^2 ≥3(a+b+c)abc

$${for}\:{a},{b},{c}>\mathrm{0}\:{prove}\:{that} \\ $$$$\left({ab}+{bc}+{ca}\right)^{\mathrm{2}} \geqslant\mathrm{3}\left({a}+{b}+{c}\right){abc} \\ $$

Question Number 17255 Answers: 1 Comments: 0

∫_0 ^( (Π/2)) ((d(sinx+cosx))/(sinx+cosx))

$$\int_{\mathrm{0}} ^{\:\frac{\Pi}{\mathrm{2}}} \:\frac{\mathrm{d}\left(\mathrm{sinx}+\mathrm{cosx}\right)}{\mathrm{sinx}+\mathrm{cosx}} \\ $$

Question Number 17252 Answers: 0 Comments: 2

The sum of the digits of the number 2^(2000) 5^(2004) is Will it be 13 or 14?

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{2}^{\mathrm{2000}} \mathrm{5}^{\mathrm{2004}} \:\mathrm{is} \\ $$$$\mathrm{Will}\:\mathrm{it}\:\mathrm{be}\:\mathrm{13}\:\mathrm{or}\:\mathrm{14}? \\ $$

Question Number 17328 Answers: 0 Comments: 3

Question Number 17247 Answers: 0 Comments: 0

A carrier based on its anual records notes that its trucks cover 50000 km with a normal distribution with a detour of 12000 km. How many miles can be traveled at least 80% of trucks ?

$${A}\:{carrier}\:{based}\:{on}\:{its}\:{anual}\:{records}\:{notes}\:{that}\:{its}\:{trucks}\:{cover}\:\mathrm{50000} \\ $$$${km}\:{with}\:{a}\:{normal}\:{distribution}\:{with}\:{a}\:{detour}\:{of}\:\mathrm{12000}\:{km}. \\ $$$${How}\:{many}\:{miles}\:{can}\:{be}\:{traveled}\:{at}\:{least}\:\mathrm{80\%}\:{of}\:{trucks}\:? \\ $$

Question Number 17220 Answers: 1 Comments: 0

Show that ∫_a ^( b) f(kx)dx=(1/k)∫_(ka) ^( kb) f(x)dx

$$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{a}} ^{\:\mathrm{b}} {f}\left(\mathrm{kx}\right)\mathrm{dx}=\frac{\mathrm{1}}{\mathrm{k}}\int_{\mathrm{ka}} ^{\:\mathrm{kb}} {f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 17219 Answers: 1 Comments: 1

∫_0 ^( 2a) xy dx=? where x^2 −y^2 =a^2 and y≥0

$$\int_{\mathrm{0}} ^{\:\mathrm{2a}} \mathrm{xy}\:\mathrm{dx}=?\:\:\mathrm{where}\:\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{y}\geqslant\mathrm{0} \\ $$

Question Number 17209 Answers: 1 Comments: 0

Spin only magnetic moment of _(25) Mn^(x+) is (√(15))B.M. Then the value of x is? i did following 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^5 so to get 3 unpaired electron we need to 2 electron so x=2. book says x=4. Why?

$$\mathrm{Spin}\:\mathrm{only}\:\mathrm{magnetic}\:\mathrm{moment} \\ $$$$\mathrm{of}\:_{\mathrm{25}} \mathrm{Mn}^{\mathrm{x}+} \:\mathrm{is}\:\sqrt{\mathrm{15}}\mathrm{B}.\mathrm{M}.\:\mathrm{Then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{is}? \\ $$$$\mathrm{i}\:\mathrm{did}\:\mathrm{following} \\ $$$$\mathrm{1s}^{\mathrm{2}} \mathrm{2s}^{\mathrm{2}} \mathrm{2p}^{\mathrm{6}} \mathrm{3s}^{\mathrm{2}} \mathrm{3p}^{\mathrm{6}} \mathrm{4s}^{\mathrm{2}} \mathrm{3d}^{\mathrm{5}} \\ $$$$\mathrm{so}\:\mathrm{to}\:\mathrm{get}\:\mathrm{3}\:\mathrm{unpaired}\:\mathrm{electron} \\ $$$$\mathrm{we}\:\mathrm{need}\:\mathrm{to}\:\mathrm{2}\:\mathrm{electron}\:\mathrm{so}\:\mathrm{x}=\mathrm{2}. \\ $$$$\mathrm{book}\:\mathrm{says}\:\mathrm{x}=\mathrm{4}.\:\mathrm{Why}? \\ $$

Question Number 17210 Answers: 1 Comments: 0

prove that ∫_0 ^( Π) f(sin x)dx=2×∫_0 ^( (Π/2)) f(sin x)dx

$$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\Pi} {f}\left(\mathrm{sin}\:\mathrm{x}\right)\mathrm{dx}=\mathrm{2}×\int_{\mathrm{0}} ^{\:\frac{\Pi}{\mathrm{2}}} {f}\left(\mathrm{sin}\:\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 17206 Answers: 1 Comments: 0

What will be the vallu of ∫_(−a) ^( a) x^2 y dx ? Where x^2 +y^2 =a^2 and y≥0

$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{vallu}\:\mathrm{of}\:\int_{−\mathrm{a}} ^{\:\mathrm{a}} \mathrm{x}^{\mathrm{2}} \mathrm{y}\:\mathrm{dx}\:\:? \\ $$$$\mathrm{Where}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{y}\geqslant\mathrm{0} \\ $$

Question Number 17205 Answers: 1 Comments: 0

lim_(n→∞) Σ_(r=1) ^(n−1) (1/n)(√((n+r)/(n−r)))

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{\mathrm{r}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{\mathrm{n}}\sqrt{\frac{\mathrm{n}+\mathrm{r}}{\mathrm{n}−\mathrm{r}}} \\ $$

Question Number 17204 Answers: 2 Comments: 0

∫_0 ^( (Π/2)) sinθ cosθ(a^2 sin^2 θ+b^2 cos^2 θ)^(1/2) dθ

$$\int_{\mathrm{0}} ^{\:\frac{\Pi}{\mathrm{2}}} \mathrm{sin}\theta\:\mathrm{cos}\theta\left(\mathrm{a}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta+\mathrm{b}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{d}\theta \\ $$

Question Number 17203 Answers: 0 Comments: 3

∫_0 ^( 1) cot^(−1) (1−x+x^2 )dx

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

Question Number 17281 Answers: 0 Comments: 0

Question Number 17187 Answers: 3 Comments: 0

Question Number 17180 Answers: 0 Comments: 1

Question Number 17179 Answers: 1 Comments: 0

Question Number 17177 Answers: 1 Comments: 0

Question Number 17167 Answers: 1 Comments: 0

∫x^2 sin^(−1) 3x dx

$$\int\mathrm{x}^{\mathrm{2}} \mathrm{sin}^{−\mathrm{1}} \mathrm{3x}\:\mathrm{dx} \\ $$

Question Number 17158 Answers: 0 Comments: 4

Please solve Q. 16069. Ask from me the solution if needed and please explain it.

$$\mathrm{Please}\:\mathrm{solve}\:\mathrm{Q}.\:\mathrm{16069}.\:\mathrm{Ask}\:\mathrm{from}\:\mathrm{me}\:\mathrm{the} \\ $$$$\mathrm{solution}\:\mathrm{if}\:\mathrm{needed}\:\mathrm{and}\:\mathrm{please}\:\mathrm{explain}\:\mathrm{it}. \\ $$

Question Number 17153 Answers: 1 Comments: 0

If m, n ∈ N(n > m), then number of solutions of the equation n∣sin x∣ = m∣sin x∣ in [0, 2π] is

$$\mathrm{If}\:{m},\:{n}\:\in\:{N}\left({n}\:>\:{m}\right),\:\mathrm{then}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${n}\mid\mathrm{sin}\:{x}\mid\:=\:{m}\mid\mathrm{sin}\:{x}\mid\:\mathrm{in}\:\left[\mathrm{0},\:\mathrm{2}\pi\right]\:\mathrm{is} \\ $$

Question Number 17152 Answers: 1 Comments: 0

If sinA = sinB and cosA = cosB, then (1) A = B + nπ, n ∈ I (2) A = B − nπ, n ∈ I (3) A = 2nπ + B, n ∈ I (4) A = nπ − B, n ∈ I

$$\mathrm{If}\:\mathrm{sin}{A}\:=\:\mathrm{sin}{B}\:\mathrm{and}\:\mathrm{cos}{A}\:=\:\mathrm{cos}{B},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{A}\:=\:{B}\:+\:{n}\pi,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{2}\right)\:{A}\:=\:{B}\:−\:{n}\pi,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{3}\right)\:{A}\:=\:\mathrm{2}{n}\pi\:+\:{B},\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{4}\right)\:{A}\:=\:{n}\pi\:−\:{B},\:{n}\:\in\:{I} \\ $$

Question Number 17151 Answers: 1 Comments: 0

The solution of the equation cos^2 θ − 2cosθ = 4sinθ − sin2θ where θ ∈ [0, π] is

$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{cos}^{\mathrm{2}} \theta\:−\:\mathrm{2cos}\theta\:=\:\mathrm{4sin}\theta\:−\:\mathrm{sin2}\theta\:\mathrm{where} \\ $$$$\theta\:\in\:\left[\mathrm{0},\:\pi\right]\:\mathrm{is} \\ $$

Question Number 17150 Answers: 0 Comments: 3

If the equation cos x + 3 cos (2Kx) = 4 has exactly one solution, then (1) K is a rational number of the form (P/(P + 1)), P ≠ −1 (2) K is irrational number whose rational approximation does not exceed 2 (3) K is irrational number (4) K is a rational number of the form (P/(P − 1)), P ≠ 1

$$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{cos}\:{x}\:+\:\mathrm{3}\:\mathrm{cos}\:\left(\mathrm{2}{Kx}\right)\:=\:\mathrm{4} \\ $$$$\mathrm{has}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{solution},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{K}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form} \\ $$$$\frac{{P}}{{P}\:+\:\mathrm{1}},\:{P}\:\neq\:−\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{K}\:\mathrm{is}\:\mathrm{irrational}\:\mathrm{number}\:\mathrm{whose} \\ $$$$\mathrm{rational}\:\mathrm{approximation}\:\mathrm{does}\:\mathrm{not} \\ $$$$\mathrm{exceed}\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:{K}\:\mathrm{is}\:\mathrm{irrational}\:\mathrm{number} \\ $$$$\left(\mathrm{4}\right)\:{K}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form} \\ $$$$\frac{{P}}{{P}\:−\:\mathrm{1}},\:{P}\:\neq\:\mathrm{1} \\ $$

Question Number 17137 Answers: 0 Comments: 0

Solve the differential equation 2x[ye^x − 1]dx + e^y dy = 0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}\: \\ $$$$\mathrm{2x}\left[\mathrm{ye}^{\mathrm{x}} \:−\:\mathrm{1}\right]\mathrm{dx}\:+\:\mathrm{e}^{\mathrm{y}} \:\mathrm{dy}\:=\:\mathrm{0} \\ $$

Question Number 17129 Answers: 1 Comments: 0

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