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Question Number 144582 Answers: 0 Comments: 0

Q. 1. throw two dice list, the elements of the sample space of the experement. let A be the event that the sum of the fa4es is odd and B be tbe event that at least one occurs. find the subset of the sample space in which i. event A and B occurs togather. ii. event A occurs or event B occurs. iii. event A occours but not B.

Question Number 144579 Answers: 2 Comments: 1

Question Number 144679 Answers: 0 Comments: 2

Question Number 144564 Answers: 1 Comments: 1

Question Number 144563 Answers: 2 Comments: 0

log_4 (x) = log(y) = log_(25) (x+y) Find (x/y) = ?

$$\boldsymbol{{log}}_{\mathrm{4}} \left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{log}}\left(\boldsymbol{{y}}\right)\:=\:\boldsymbol{{log}}_{\mathrm{25}} \left(\boldsymbol{{x}}+\boldsymbol{{y}}\right) \\ $$$${Find}\:\:\frac{{x}}{{y}}\:=\:? \\ $$

Question Number 144558 Answers: 3 Comments: 0

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

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

Question Number 144556 Answers: 2 Comments: 0

lim_(x→0) (((1−cos x(√(cos 2x)))/x^2 )) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}\sqrt{\mathrm{cos}\:\mathrm{2x}}}{\mathrm{x}^{\mathrm{2}} }\right)\:=? \\ $$

Question Number 144554 Answers: 1 Comments: 0

Question Number 144553 Answers: 1 Comments: 0

Solve for natural numbers: (1^4 /z) + (2^4 /(z+1)) + (3^4 /(z+2)) + ... + ((10^4 )/(z+9)) = 3025

$${Solve}\:{for}\:{natural}\:{numbers}: \\ $$$$\frac{\mathrm{1}^{\mathrm{4}} }{\boldsymbol{{z}}}\:+\:\frac{\mathrm{2}^{\mathrm{4}} }{\boldsymbol{{z}}+\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{4}} }{\boldsymbol{{z}}+\mathrm{2}}\:+\:...\:+\:\frac{\mathrm{10}^{\mathrm{4}} }{\boldsymbol{{z}}+\mathrm{9}}\:=\:\mathrm{3025} \\ $$

Question Number 144550 Answers: 0 Comments: 0

Question Number 144549 Answers: 1 Comments: 0

Question Number 144544 Answers: 2 Comments: 1

lim_(x→(π/2)) (cosx)^(cotx)

$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \left({cosx}\right)^{{cotx}} \\ $$

Question Number 144539 Answers: 1 Comments: 0

Let 0°<θ<45°, find the value of sin^2 (45°+θ)+sin^2 (45°−θ)

$$\mathrm{Let}\:\mathrm{0}°<\theta<\mathrm{45}°,\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{sin}^{\mathrm{2}} \left(\mathrm{45}°+\theta\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{45}°−\theta\right) \\ $$

Question Number 144537 Answers: 2 Comments: 0

Let a,b>0 and a+b = 2. Prove that (1) ((a^3 +b^3 )/2)−2(1−ab) ≥ 1 (2) ((a^2 +b^2 )/2)−2(1−ab) ≤ 1

$$\mathrm{Let}\:{a},{b}>\mathrm{0}\:\mathrm{and}\:{a}+{b}\:=\:\mathrm{2}.\:\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} }{\mathrm{2}}−\mathrm{2}\left(\mathrm{1}−{ab}\right)\:\geqslant\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{2}}−\mathrm{2}\left(\mathrm{1}−{ab}\right)\:\leqslant\:\mathrm{1} \\ $$

Question Number 144534 Answers: 2 Comments: 0

Find the shortest distance from the origin to the hyperbola x^2 +8xy+7y^2 =225 ,z=0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{from}\: \\ $$$$\mathrm{the}\:\mathrm{origin}\:\mathrm{to}\:\mathrm{the}\:\mathrm{hyperbola}\: \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{8xy}+\mathrm{7y}^{\mathrm{2}} =\mathrm{225}\:,\mathrm{z}=\mathrm{0}\: \\ $$

Question Number 144532 Answers: 1 Comments: 0

A rectangular box,open at the top is to have a volume of 32 cube feet What must be the dimensions so that the total surface is a minimum?

$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{box},\mathrm{open}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{top}\:\mathrm{is}\:\mathrm{to}\:\mathrm{have}\:\mathrm{a}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{32}\:\mathrm{cube}\:\mathrm{feet} \\ $$$$\mathrm{What}\:\mathrm{must}\:\mathrm{be}\:\mathrm{the}\:\mathrm{dimensions} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{total}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{a}\:\mathrm{minimum}? \\ $$

Question Number 144530 Answers: 2 Comments: 0

∫_0 ^( π/2) ((cos^2 x)/((2cos x+sin x)^2 )) dx =?

$$\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}{\left(\mathrm{2cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 144528 Answers: 1 Comments: 0

Find the volume of the region bounded by the elliptic paraboloid z = 4−x^2 −(1/4)y^2 and the plane z=0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region}\: \\ $$$$\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{elliptic}\:\mathrm{paraboloid} \\ $$$$\mathrm{z}\:=\:\mathrm{4}−\mathrm{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}\mathrm{y}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{z}=\mathrm{0} \\ $$$$ \\ $$

Question Number 144527 Answers: 2 Comments: 0

..... Calculus (I )..... P:= ((∫_(0 ) ^( (π/2)) ( xcos(x)+1 )e^( sin(x)) dx )/(∫_0 ^( (π/2)) ( xsin(x) −1 )e^( cos(x )) dx))=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:.....\:\:\:\mathrm{Calculus}\:\:\left(\mathrm{I}\:\right)..... \\ $$$$\mathrm{P}:=\:\frac{\int_{\mathrm{0}\:} ^{\:\:\frac{\pi}{\mathrm{2}}} \left(\:{xcos}\left({x}\right)+\mathrm{1}\:\right){e}^{\:{sin}\left({x}\right)} {dx}\:}{\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\:{xsin}\left({x}\right)\:−\mathrm{1}\:\right){e}^{\:{cos}\left({x}\:\right)} {dx}}=? \\ $$

Question Number 144517 Answers: 1 Comments: 0

Question Number 144515 Answers: 1 Comments: 1

S_(15) −S_(25) =150 S_(30) =?

$${S}_{\mathrm{15}} −{S}_{\mathrm{25}} =\mathrm{150} \\ $$$${S}_{\mathrm{30}} =? \\ $$

Question Number 144513 Answers: 1 Comments: 0

Question Number 144509 Answers: 0 Comments: 0

S_n =((Cos1)/1^2 )+((Cos2)/2^2 )+((Cos3)/3^2 )+...+((Cosn)/n^2 ) Indicates that S_(n ) is a cauchy suite?

$$ \\ $$$${S}_{{n}} =\frac{{Cos}\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{{Cos}\mathrm{2}}{\mathrm{2}^{\mathrm{2}} }+\frac{{Cos}\mathrm{3}}{\mathrm{3}^{\mathrm{2}} }+...+\frac{{Cosn}}{{n}^{\mathrm{2}} } \\ $$$${Indicates}\:{that}\:{S}_{{n}\:} \:{is}\:{a}\:{cauchy}\:{suite}? \\ $$

Question Number 144502 Answers: 0 Comments: 2

it is known that after injecting the 1^(st) dosage of a drug in exactly hours p(t)= { ((Ae^(−(t/(24))) , 0≤t≤8)),(((A+Ae^(−(1/3)) )e^(−(((t−8)/(24)))) , 8 ≤ t≤24)) :} A is the initial dosage at t= 0hours a) show that towards the application of the n^(th) dosage , the dosage remained in blood is (((1−e^(−(1/n)) )/(1−e^(−(1/3)) )))×A

$${it}\:{is}\:{known}\:{that}\:{after}\:{injecting}\:{the}\:\mathrm{1}^{{st}} \\ $$$${dosage}\:{of}\:{a}\:{drug}\:{in}\:{exactly}\:{hours} \\ $$$${p}\left({t}\right)=\begin{cases}{{Ae}^{−\frac{{t}}{\mathrm{24}}} ,\:\mathrm{0}\leqslant{t}\leqslant\mathrm{8}}\\{\left({A}+{Ae}^{−\frac{\mathrm{1}}{\mathrm{3}}} \right){e}^{−\left(\frac{{t}−\mathrm{8}}{\mathrm{24}}\right)} ,\:\mathrm{8}\:\leqslant\:{t}\leqslant\mathrm{24}}\end{cases} \\ $$$${A}\:{is}\:{the}\:{initial}\:{dosage}\:{at}\:{t}=\:\mathrm{0}{hours} \\ $$$$\left.{a}\right)\:{show}\:{that}\:{towards}\:{the}\:{application}\:{of} \\ $$$${the}\:{n}^{{th}} \:{dosage}\:,\:{the}\:{dosage}\:{remained}\:{in}\:{blood}\:{is} \\ $$$$\left(\frac{\mathrm{1}−{e}^{−\frac{\mathrm{1}}{{n}}} }{\mathrm{1}−{e}^{−\frac{\mathrm{1}}{\mathrm{3}}} }\right)×{A} \\ $$

Question Number 144500 Answers: 2 Comments: 0

calculate Σ_(n=0) ^∞ (1/(n^2 +4))

$$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} \:+\mathrm{4}} \\ $$

Question Number 144499 Answers: 1 Comments: 0

find ∫_0 ^∞ ((arctan(x^n ))/x^n )dx (n≥2) natural

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{x}^{\mathrm{n}} \right)}{\mathrm{x}^{\mathrm{n}} }\mathrm{dx}\:\:\:\left(\mathrm{n}\geqslant\mathrm{2}\right)\:\mathrm{natural} \\ $$

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