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Question Number 145831 Answers: 2 Comments: 0

A box P, contains 4 white, 2 green and 3 blue cards. Another box Q, contains 2 white, 3 green and 2 blue cards. A card is picked at random from P and placed in Q. A card is then picked from Q. Find the probability that the (a) card picked from Q is white. (b) cards picked from P and Q are of the same colour.

$$\:\mathrm{A}\:\mathrm{box}\:\boldsymbol{{P}},\:\mathrm{contains}\:\mathrm{4}\:\mathrm{white},\:\mathrm{2}\:\mathrm{green}\:\mathrm{and}\: \\ $$$$\:\mathrm{3}\:\mathrm{blue}\:\mathrm{cards}.\:\mathrm{Another}\:\mathrm{box}\:\boldsymbol{{Q}},\:\mathrm{contains} \\ $$$$\:\mathrm{2}\:\mathrm{white},\:\mathrm{3}\:\mathrm{green}\:\mathrm{and}\:\mathrm{2}\:\mathrm{blue}\:\mathrm{cards}.\:\mathrm{A}\:\mathrm{card} \\ $$$$\:\mathrm{is}\:\mathrm{picked}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\boldsymbol{{P}}\:\mathrm{and}\:\mathrm{placed} \\ $$$$\:\mathrm{in}\:\boldsymbol{{Q}}.\:\mathrm{A}\:\mathrm{card}\:\mathrm{is}\:\mathrm{then}\:\mathrm{picked}\:\mathrm{from}\:\boldsymbol{{Q}}. \\ $$$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{the} \\ $$$$\:\:\left({a}\right)\:\:\mathrm{card}\:\mathrm{picked}\:\mathrm{from}\:\boldsymbol{{Q}}\:\mathrm{is}\:\mathrm{white}. \\ $$$$\:\:\left({b}\right)\:\:\mathrm{cards}\:\mathrm{picked}\:\mathrm{from}\:\boldsymbol{{P}}\:\mathrm{and}\:\boldsymbol{{Q}}\:\mathrm{are}\:\mathrm{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{the}\:\mathrm{same}\:\mathrm{colour}. \\ $$

Question Number 145828 Answers: 0 Comments: 0

∫(1/(x^α +a))dx

$$\int\frac{\mathrm{1}}{{x}^{\alpha} +{a}}{dx} \\ $$

Question Number 145814 Answers: 0 Comments: 0

Let f(x)={sin (tan^(−1) x)+sin (cot^(−1) x)}^2 −1 where ∣x∣>1 and (dy/dx)=(1/2)(d/dx)(sin^(−1) f(x)). if y((√3))=(Π/6) then y(−(√3))=?

$${Let}\:{f}\left({x}\right)=\left\{\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} {x}\right)+\mathrm{sin}\:\left(\mathrm{cot}^{−\mathrm{1}} {x}\right)\right\}^{\mathrm{2}} −\mathrm{1} \\ $$$${where}\:\mid{x}\mid>\mathrm{1}\:{and}\:\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{\mathrm{2}}\frac{{d}}{{dx}}\left(\mathrm{sin}^{−\mathrm{1}} {f}\left({x}\right)\right). \\ $$$${if}\:{y}\left(\sqrt{\mathrm{3}}\right)=\frac{\Pi}{\mathrm{6}}\:{then}\:{y}\left(−\sqrt{\mathrm{3}}\right)=? \\ $$

Question Number 145822 Answers: 1 Comments: 1

find zeros function f(z)=1−cosz

$${find}\:{zeros}\:{function}\:{f}\left({z}\right)=\mathrm{1}−{cosz} \\ $$

Question Number 145820 Answers: 1 Comments: 3

find resideo e^((z+1)/z)

$${find}\:{resideo}\:{e}^{\left({z}+\mathrm{1}\right)/{z}} \\ $$

Question Number 145819 Answers: 0 Comments: 0

Question Number 145817 Answers: 1 Comments: 1

Question Number 145810 Answers: 1 Comments: 2

find residue f(z)=z^(−3) csc(z^2 )

$${find}\:{residue}\:{f}\left({z}\right)={z}^{−\mathrm{3}} {csc}\left({z}^{\mathrm{2}} \right) \\ $$

Question Number 145806 Answers: 2 Comments: 0

lim_(x→∞) x(tan^(−1) (((x+1)/(x+4)))−(π/4)) =?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{x}\left(\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}+\mathrm{4}}\right)−\frac{\pi}{\mathrm{4}}\right)\:=?\: \\ $$

Question Number 145827 Answers: 2 Comments: 0

Use Abel summation to evaluate :: Σ_(n=1) ^∞ (1/((2n−1)∙2^n ))=(1/( (√2)))ln((√2)+1)

$$\mathrm{Use}\:\mathrm{Abel}\:\mathrm{summation}\:\mathrm{to}\:\mathrm{evaluate}\::: \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2n}−\mathrm{1}\right)\centerdot\mathrm{2}^{\mathrm{n}} }=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{ln}\left(\sqrt{\mathrm{2}}+\mathrm{1}\right) \\ $$

Question Number 145829 Answers: 1 Comments: 0

Question Number 145803 Answers: 1 Comments: 1

Question Number 145801 Answers: 2 Comments: 0

Question Number 145800 Answers: 0 Comments: 0

Question Number 145799 Answers: 0 Comments: 0

Question Number 145791 Answers: 1 Comments: 2

find lim_(x→0) ∫_0 ^x ((e^t +e^(−t) −2)/(1−cosx))dx

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\:\int_{\mathrm{0}} ^{\mathrm{x}} \:\frac{\mathrm{e}^{\mathrm{t}} +\mathrm{e}^{−\mathrm{t}} −\mathrm{2}}{\mathrm{1}−\mathrm{cosx}}\mathrm{dx} \\ $$

Question Number 145783 Answers: 2 Comments: 0

Question Number 145781 Answers: 1 Comments: 0

consider f(x)=Ax^2 +Bx+C with A>0. Show that f(x)≥0 ∀x iff B^2 −4AC≤0

$${consider}\:{f}\left({x}\right)={Ax}^{\mathrm{2}} +{Bx}+{C} \\ $$$${with}\:{A}>\mathrm{0}.\:{Show}\:{that}\:{f}\left({x}\right)\geqslant\mathrm{0}\:\forall{x}\:\:{iff}\: \\ $$$${B}^{\mathrm{2}} −\mathrm{4}{AC}\leqslant\mathrm{0} \\ $$

Question Number 145780 Answers: 1 Comments: 0

find the volume of the solid generated by the region bounded by y=(√x) , 0≤x≤1 and X−axis

$${find}\:{the}\:{volume}\:{of}\:{the}\:{solid}\: \\ $$$${generated}\:{by}\:{the}\:{region}\:{bounded}\:{by} \\ $$$${y}=\sqrt{{x}}\:,\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:{X}−{axis} \\ $$

Question Number 145779 Answers: 1 Comments: 0

lim_(x→0) ∫_0 ^1 (e^t +e^(−t) −2)(dt/(1−cosx))

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\int_{\mathrm{0}} ^{\mathrm{1}} \left({e}^{{t}} +{e}^{−{t}} −\mathrm{2}\right)\frac{{dt}}{\mathrm{1}−{cosx}} \\ $$

Question Number 145777 Answers: 2 Comments: 0

∫(1/( (√(1−9x^2 ))))dx

$$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{9}{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 145776 Answers: 2 Comments: 0

∫_0 ^(π/2) e^x cosxdx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{x}} {cosxdx} \\ $$

Question Number 145775 Answers: 2 Comments: 0

∫((2x+1)/( (√(x^2 +4x+5))))dx

$$\int\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{5}}}{dx} \\ $$

Question Number 145772 Answers: 0 Comments: 0

a;b;c>0 ; a^2 +b^2 +c^2 =2 prove: (a^6 +b^6 +c^6 )^3 ≥ (a^5 +b^5 +c^5 )^4

$${a};{b};{c}>\mathrm{0}\:;\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{2}\:{prove}: \\ $$$$\left({a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} \right)^{\mathrm{3}} \:\geqslant\:\left({a}^{\mathrm{5}} +{b}^{\mathrm{5}} +{c}^{\mathrm{5}} \right)^{\mathrm{4}} \\ $$

Question Number 145774 Answers: 1 Comments: 0

find the area bounded by y=2x, y=(x/2) and?xy=2

$${find}\:{the}\:{area}\:{bounded}\:{by}\:{y}=\mathrm{2}{x},\:{y}=\frac{{x}}{\mathrm{2}}\:{and}?{xy}=\mathrm{2} \\ $$

Question Number 145773 Answers: 1 Comments: 0

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