Question and Answers Forum

AllQuestion and Answers: Page 78

Question Number 207753 Answers: 1 Comments: 0

Question Number 207752 Answers: 1 Comments: 3

Two ships have the same berth in a port. It is known that the arrival times of the two ships are independent and have the same probability of docking on a Sunday (00.00−24.00) If the berth time of the first ship is 2 hours and the berth time of the second ship is 4 hours, the probability that one ship will have to wait until the berth can be used is □ ((67)/(144)) □ ((67)/(288)) □ (1/4) □((33)/(144))

$$\:\mathrm{Two}\:\mathrm{ships}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{berth}\: \\ $$$$\:\mathrm{in}\:\mathrm{a}\:\mathrm{port}.\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\:\mathrm{arrival}\:\mathrm{times}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{ships}\: \\ $$$$\:\mathrm{are}\:\mathrm{independent}\:\mathrm{and}\:\mathrm{have}\:\mathrm{the}\: \\ $$$$\:\mathrm{same}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{docking}\: \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{Sunday}\:\left(\mathrm{00}.\mathrm{00}−\mathrm{24}.\mathrm{00}\right) \\ $$$$\:\mathrm{If}\:\mathrm{the}\:\mathrm{berth}\:\mathrm{time}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{ship} \\ $$$$\:\mathrm{is}\:\mathrm{2}\:\mathrm{hours}\:\mathrm{and}\:\mathrm{the}\:\mathrm{berth}\:\mathrm{time} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{second}\:\mathrm{ship}\:\mathrm{is}\:\mathrm{4}\:\mathrm{hours},\: \\ $$$$\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{one}\:\mathrm{ship} \\ $$$$\:\mathrm{will}\:\mathrm{have}\:\mathrm{to}\:\mathrm{wait}\:\mathrm{until}\:\mathrm{the} \\ $$$$\:\mathrm{berth}\:\mathrm{can}\:\mathrm{be}\:\mathrm{used}\:\mathrm{is}\: \\ $$$$\:\Box\:\frac{\mathrm{67}}{\mathrm{144}}\:\:\:\:\:\Box\:\frac{\mathrm{67}}{\mathrm{288}}\:\:\:\:\Box\:\frac{\mathrm{1}}{\mathrm{4}}\:\:\:\:\Box\frac{\mathrm{33}}{\mathrm{144}} \\ $$

Question Number 207751 Answers: 0 Comments: 0

It is known that a balanced 6−sided dice originally had 2,3,4,5,6 and 7. The dice wre thrown once and the result was observed. If an odd numbers appears, than the number is replaced with the number 8. However, if an even number appears , the number is replaced with the number 1. Then the dice whose dice have been replaced are thrown again, the probability of an odd dice odd dice appearing is □ (1/3) □ (2/3) □ (1/2) □ 1

$$\:\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{a}\:\mathrm{balanced}\:\mathrm{6}−\mathrm{sided}\: \\ $$$$\:\mathrm{dice}\:\mathrm{originally}\:\mathrm{had}\:\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\:\mathrm{and}\:\mathrm{7}. \\ $$$$\:\mathrm{The}\:\mathrm{dice}\:\mathrm{wre}\:\mathrm{thrown}\:\mathrm{once}\:\mathrm{and}\: \\ $$$$\:\mathrm{the}\:\mathrm{result}\:\mathrm{was}\:\mathrm{observed}.\:\mathrm{If}\:\mathrm{an}\: \\ $$$$\mathrm{odd}\:\mathrm{numbers}\:\mathrm{appears},\:\mathrm{than}\:\mathrm{the}\: \\ $$$$\:\mathrm{number}\:\mathrm{is}\:\mathrm{replaced}\:\mathrm{with}\:\mathrm{the}\: \\ $$$$\:\mathrm{number}\:\mathrm{8}.\:\mathrm{However},\:\mathrm{if}\:\mathrm{an}\:\mathrm{even}\: \\ $$$$\:\mathrm{number}\:\mathrm{appears}\:,\:\mathrm{the}\:\mathrm{number} \\ $$$$\:\mathrm{is}\:\mathrm{replaced}\:\mathrm{with}\:\mathrm{the}\:\mathrm{number}\:\mathrm{1}. \\ $$$$\:\mathrm{Then}\:\mathrm{the}\:\mathrm{dice}\:\mathrm{whose}\:\mathrm{dice}\:\mathrm{have}\: \\ $$$$\:\mathrm{been}\:\mathrm{replaced}\:\mathrm{are}\:\mathrm{thrown}\:\mathrm{again},\: \\ $$$$\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{dice}\: \\ $$$$\:\mathrm{odd}\:\mathrm{dice}\:\mathrm{appearing}\:\mathrm{is}\: \\ $$$$\:\:\Box\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\Box\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\Box\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\Box\:\mathrm{1}\: \\ $$

Question Number 207747 Answers: 3 Comments: 0

find the value of y given that y^(y ) =3^(y+9)

$${find}\:{the}\:{value}\:{of}\:{y}\:{given}\:{that}\:{y}^{{y}\:} =\mathrm{3}^{{y}+\mathrm{9}} \\ $$

Question Number 207738 Answers: 2 Comments: 0

Question Number 208158 Answers: 1 Comments: 0

X, Y and Z are points on the sides AB, BC and AC of the triangle ABC, such that AX:XB =4:3, BY:YC=2:3, CZ:ZA=2:1. Find the ratio of the area of the triangle XYZ to that of triangle ABC.

$${X},\:{Y}\:{and}\:{Z}\:{are}\:{points}\:{on}\:{the}\:{sides}\:{AB}, \\ $$$${BC}\:{and}\:{AC}\:{of}\:{the}\:{triangle}\:{ABC},\:{such} \\ $$$${that}\:{AX}:{XB}\:=\mathrm{4}:\mathrm{3},\:{BY}:{YC}=\mathrm{2}:\mathrm{3},\: \\ $$$${CZ}:{ZA}=\mathrm{2}:\mathrm{1}.\:{Find}\:{the}\:{ratio}\:{of}\:{the}\:{area} \\ $$$${of}\:{the}\:{triangle}\:{XYZ}\:{to}\:{that}\:{of}\:{triangle} \\ $$$${ABC}. \\ $$

Question Number 207730 Answers: 0 Comments: 3

Question Number 207731 Answers: 2 Comments: 0

⇃

$$\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 207718 Answers: 1 Comments: 0

∣x−3∣ + ∣x +2∣ = 9 find: x = ?

$$\mid\mathrm{x}−\mathrm{3}\mid\:+\:\mid\mathrm{x}\:+\mathrm{2}\mid\:=\:\mathrm{9} \\ $$$$\mathrm{find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 207717 Answers: 1 Comments: 0

lg^2 (10x) − lg 10x + 1 = 6 − lg x find: x = ?

$$\mathrm{lg}^{\mathrm{2}} \:\left(\mathrm{10x}\right)\:−\:\mathrm{lg}\:\mathrm{10x}\:+\:\mathrm{1}\:=\:\mathrm{6}\:−\:\mathrm{lg}\:\mathrm{x} \\ $$$$\mathrm{find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 207713 Answers: 2 Comments: 0

Question Number 207712 Answers: 1 Comments: 0

Question Number 207710 Answers: 0 Comments: 0

Question Number 207707 Answers: 0 Comments: 0

∫_0 ^(+∞) ((sin^2 (x))/(sin^2 (x)+(xcos (x)+sin (x))^2 ))d(x)

$$\underset{\mathrm{0}} {\overset{+\infty} {\int}}\frac{\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)}{\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)+\left({x}\mathrm{cos}\:\left({x}\right)+\mathrm{sin}\:\left({x}\right)\right)^{\mathrm{2}} }{d}\left({x}\right) \\ $$

Question Number 207699 Answers: 2 Comments: 0

(4/(2x−1)) + ((27)/(3x−1)) + ((125)/(5x−1)) = ((144)/(4x−1)) Find: x = ?

$$\frac{\mathrm{4}}{\mathrm{2x}−\mathrm{1}}\:\:+\:\:\frac{\mathrm{27}}{\mathrm{3x}−\mathrm{1}}\:\:+\:\:\frac{\mathrm{125}}{\mathrm{5x}−\mathrm{1}}\:\:=\:\:\frac{\mathrm{144}}{\mathrm{4x}−\mathrm{1}} \\ $$$$\mathrm{Find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 207692 Answers: 0 Comments: 1

Question Number 207691 Answers: 3 Comments: 0

Question Number 207688 Answers: 0 Comments: 0

Q207657

$${Q}\mathrm{207657} \\ $$

Question Number 207687 Answers: 1 Comments: 0

Question Number 207673 Answers: 1 Comments: 0

$$\:\:\:\:\underline{ \:} \\ $$$$ \\ $$

Question Number 207683 Answers: 2 Comments: 0

Question Number 207665 Answers: 3 Comments: 0

(1/1) (((20)),(( 0)) ) +(1/2) (((20)),(( 1)) ) +(1/3) (((20)),(( 2)) ) +...+(1/(21)) (((20)),((20)) ) =?

$$\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{0}}\end{pmatrix}\:+\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{1}}\end{pmatrix}\:+\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{2}}\end{pmatrix}\:+...+\frac{\mathrm{1}}{\mathrm{21}}\:\begin{pmatrix}{\mathrm{20}}\\{\mathrm{20}}\end{pmatrix}\:=? \\ $$

Question Number 207664 Answers: 1 Comments: 0

If p+q+r = 0 and A=(((p^2 +q^2 +r^2 )^2 )/((pq)^2 +(qr)^2 +(rp)^2 )) B= ((q^2 −pr)/(p^2 +q^2 +r^2 )) . Find A^2 −4B

$$\:\:\:\:\mathrm{If}\:\mathrm{p}+\mathrm{q}+\mathrm{r}\:=\:\mathrm{0}\:\mathrm{and}\: \\ $$$$\:\:\:\:\mathrm{A}=\frac{\left(\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left(\mathrm{pq}\right)^{\mathrm{2}} +\left(\mathrm{qr}\right)^{\mathrm{2}} +\left(\mathrm{rp}\right)^{\mathrm{2}} }\: \\ $$$$\:\:\:\mathrm{B}=\:\frac{\mathrm{q}^{\mathrm{2}} −\mathrm{pr}}{\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} }\:. \\ $$$$\:\:\mathrm{Find}\:\mathrm{A}^{\mathrm{2}} −\mathrm{4B}\: \\ $$

Question Number 207663 Answers: 0 Comments: 0

Given x_1 +x_3 +...+x_(2023) = 25−(x_2 +x_4 +...+x_(2022) ) x_1 ^(2 ) + x_3 ^2 +...+x_(2023) ^2 = 125−(x_2 ^2 +x_4 ^2 +...+x_(2022) ^2 ) −2≤x_i ≤1 , i=1,2,3,...,2023 x_i integer number Find minimum value of x_1 ^3 +x_2 ^3 +x_3 ^3 +x_4 ^3 +...+x_(2023) ^3 □−100 □−71 □−50 □−16

$$\:\:\mathrm{Given}\: \\ $$$$\:\:\:\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{3}} +...+\mathrm{x}_{\mathrm{2023}} \:=\:\mathrm{25}−\left(\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{4}} +...+\mathrm{x}_{\mathrm{2022}} \right) \\ $$$$\:\:\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}\:} +\:\mathrm{x}_{\mathrm{3}} ^{\mathrm{2}} +...+\mathrm{x}_{\mathrm{2023}} ^{\mathrm{2}} \:=\:\mathrm{125}−\left(\mathrm{x}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{4}} ^{\mathrm{2}} +...+\mathrm{x}_{\mathrm{2022}} ^{\mathrm{2}} \right) \\ $$$$\:\:−\mathrm{2}\leqslant\mathrm{x}_{\mathrm{i}} \leqslant\mathrm{1}\:,\:\mathrm{i}=\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{2023}\: \\ $$$$\:\:\:\mathrm{x}_{\mathrm{i}} \:\mathrm{integer}\:\mathrm{number}\: \\ $$$$\:\mathrm{Find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\:\mathrm{x}_{\mathrm{1}} ^{\mathrm{3}} +\mathrm{x}_{\mathrm{2}} ^{\mathrm{3}} +\mathrm{x}_{\mathrm{3}} ^{\mathrm{3}} +\mathrm{x}_{\mathrm{4}} ^{\mathrm{3}} +...+\mathrm{x}_{\mathrm{2023}} ^{\mathrm{3}} \: \\ $$$$\:\:\Box−\mathrm{100}\: \\ $$$$\:\:\Box−\mathrm{71} \\ $$$$\:\:\Box−\mathrm{50} \\ $$$$\:\:\Box−\mathrm{16}\: \\ $$

Question Number 207661 Answers: 2 Comments: 0

P=1+(1/3)+(1/5)+(1/7)+...+(1/(2023)) Q= (1/(1×2023))+(1/(3×2021))+(1/(5×2019))+...+(1/(2023×1)) (P/Q)=?

$$\:\mathrm{P}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{7}}+...+\frac{\mathrm{1}}{\mathrm{2023}} \\ $$$$\:\mathrm{Q}=\:\frac{\mathrm{1}}{\mathrm{1}×\mathrm{2023}}+\frac{\mathrm{1}}{\mathrm{3}×\mathrm{2021}}+\frac{\mathrm{1}}{\mathrm{5}×\mathrm{2019}}+...+\frac{\mathrm{1}}{\mathrm{2023}×\mathrm{1}} \\ $$$$\:\:\frac{\mathrm{P}}{\mathrm{Q}}=? \\ $$

Question Number 207657 Answers: 0 Comments: 1

Pg 73 Pg 74 Pg 75 Pg 76 Pg 77 Pg 78 Pg 79 Pg 80 Pg 81 Pg 82

Contact: info@tinkutara.com