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Question Number 176216 Answers: 0 Comments: 0

Question Number 176215 Answers: 2 Comments: 0

A certain amount of money is distributed among A, B and C in the ratio of 2:5:3 and another amount of money among B, D and E is also distributed in the same ratio. If the amount distributed among A, B and C is ⅖ of the amount distributed among B, D and E, what is the ratio in which the amount is distributed among A, C and E?

$$ \\ $$A certain amount of money is distributed among A, B and C in the ratio of 2:5:3 and another amount of money among B, D and E is also distributed in the same ratio. If the amount distributed among A, B and C is ⅖ of the amount distributed among B, D and E, what is the ratio in which the amount is distributed among A, C and E?

Question Number 176213 Answers: 1 Comments: 0

lim_(x→0) ((arctan x)/(arcsin x−x)) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{arctan}\:\mathrm{x}}{\mathrm{arcsin}\:\mathrm{x}−\mathrm{x}}\:=? \\ $$

Question Number 176212 Answers: 0 Comments: 2

A solid rigth triangular prism of length 12cm and a cross section which is an equilateral triangle of 6cm. Find the total surface area.

$$\:\:\mathrm{A}\:\mathrm{solid}\:\mathrm{rigth}\:\mathrm{triangular}\:\mathrm{prism}\:\mathrm{of}\: \\ $$$$\:\:\mathrm{length}\:\mathrm{12cm}\:\mathrm{and}\:\mathrm{a}\:\mathrm{cross}\:\mathrm{section}\: \\ $$$$\:\:\mathrm{which}\:\mathrm{is}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{of} \\ $$$$\:\:\mathrm{6cm}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{total}\:\mathrm{surface}\:\mathrm{area}. \\ $$

Question Number 176211 Answers: 1 Comments: 0

Question Number 176206 Answers: 1 Comments: 0

Question Number 176205 Answers: 1 Comments: 0

In a school election for the positon of SRC president, one candidate obtained 87.5% of the vote cast others obtained 275 votes (a) How many votes did the winner obtain. (b) If the voter turnout was 55% find the total number of eligible voters in the school.

$$\:\mathrm{In}\:\mathrm{a}\:\mathrm{school}\:\mathrm{election}\:\mathrm{for}\:\mathrm{the}\:\mathrm{positon}\: \\ $$$$\:\:\mathrm{of}\:\mathrm{SRC}\:\mathrm{president},\:\mathrm{one}\:\mathrm{candidate} \\ $$$$\:\:\mathrm{obtained}\:\mathrm{87}.\mathrm{5\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vote}\:\mathrm{cast}\: \\ $$$$\:\:\mathrm{others}\:\mathrm{obtained}\:\mathrm{275}\:\mathrm{votes} \\ $$$$\:\:\left(\mathrm{a}\right)\:\mathrm{How}\:\mathrm{many}\:\mathrm{votes}\:\mathrm{did}\:\mathrm{the}\:\mathrm{winner} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{obtain}. \\ $$$$\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{If}\:\mathrm{the}\:\mathrm{voter}\:\mathrm{turnout}\:\mathrm{was}\:\mathrm{55\%}\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\:\mathrm{eligible} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{voters}\:\mathrm{in}\:\mathrm{the}\:\mathrm{school}. \\ $$$$ \\ $$

Question Number 176199 Answers: 1 Comments: 0

Question Number 176195 Answers: 1 Comments: 0

Question Number 176192 Answers: 1 Comments: 0

Suppose ABCD is a rectangle. X and Y are points on BC and CD respectively, such that the area of ABX, CXY, and AYD are 3 cm^2 , 4 cm^2 , and 5 cm^2 respectively. Find the area of AXY.

$$\mathrm{Suppose}\:\mathrm{ABCD}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rectangle}.\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y}\:\mathrm{are}\:\mathrm{points}\:\mathrm{on}\:\mathrm{BC}\:\mathrm{and}\:\mathrm{CD}\:\mathrm{respectively}, \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{ABX},\:\mathrm{CXY},\:\mathrm{and}\:\mathrm{AYD}\:\mathrm{are}\:\mathrm{3}\:\mathrm{cm}^{\mathrm{2}} ,\:\mathrm{4}\:\mathrm{cm}^{\mathrm{2}} ,\:\mathrm{and}\:\mathrm{5}\:\mathrm{cm}^{\mathrm{2}} \:\mathrm{respectively}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{AXY}. \\ $$

Question Number 176189 Answers: 1 Comments: 0

solve for x (1+(1/x))^(x+1) =(1+(1/6))^6 show working

$${solve}\:{for}\:{x} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{x}+\mathrm{1}} =\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{6}}\right)^{\mathrm{6}} \\ $$$${show}\:{working} \\ $$

Question Number 176188 Answers: 1 Comments: 0

∫_( (√2)) ^2 ((sec^2 (sec^(−1) x))/(x(√(x^2 −1))))

$$ \\ $$$$\underset{\:\sqrt{\mathrm{2}}} {\overset{\mathrm{2}} {\int}}\frac{\mathrm{sec}^{\mathrm{2}} \left(\mathrm{sec}^{−\mathrm{1}} \mathrm{x}\right)}{\mathrm{x}\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}} \\ $$$$ \\ $$

Question Number 176187 Answers: 1 Comments: 0

∫(dy/(tan^(−1) y(1+y^2 )))

$$\int\frac{\mathrm{dy}}{\mathrm{tan}^{−\mathrm{1}} \mathrm{y}\left(\mathrm{1}+\mathrm{y}^{\mathrm{2}} \right)} \\ $$

Question Number 176185 Answers: 0 Comments: 1

∫(e^(sin^(−1) x) /( (√(1−x_ ^2 )))) dx

$$\int\frac{\mathrm{e}^{\mathrm{sin}^{−\mathrm{1}} \mathrm{x}} }{\:\sqrt{\mathrm{1}−\mathrm{x}_{} ^{\mathrm{2}} }}\:\mathrm{dx} \\ $$

Question Number 176180 Answers: 4 Comments: 0

∫(dx/((x+1)(√(x^2 +2x))))

$$\int\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}}} \\ $$$$ \\ $$

Question Number 176785 Answers: 2 Comments: 1

(6/(1+c^x ))+(1/(1+c^(−x) ))=y Express ((11)/(1+c^x ))+(1/(1+c^(−x) )) in terms of y

$$\frac{\mathrm{6}}{\mathrm{1}+{c}^{{x}} }+\frac{\mathrm{1}}{\mathrm{1}+{c}^{−{x}} }={y} \\ $$$$\mathrm{Express}\:\:\frac{\mathrm{11}}{\mathrm{1}+{c}^{{x}} }+\frac{\mathrm{1}}{\mathrm{1}+{c}^{−{x}} }\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{y} \\ $$

Question Number 176174 Answers: 1 Comments: 0

a,b,c ∈R_(+ ) ^∗ and abc=1. Prove ((c (√(a^3 +b^3 )))/(a^2 +b^2 )) + ((b (√(a^3 +c^3 )))/(a^2 +c^2 )) + ((a(√(b^3 +c^3 )))/(b^2 +c^2 )) ≥ (3/( (√2)))

$$ \\ $$$$\:\mathrm{a},{b},{c}\:\in\mathbb{R}_{+\:} ^{\ast} \:\:\mathrm{and}\:{abc}=\mathrm{1}.\:\mathrm{Prove} \\ $$$$\:\:\:\frac{{c}\:\sqrt{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} }}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:+\:\frac{{b}\:\sqrt{{a}^{\mathrm{3}} +{c}^{\mathrm{3}} }}{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }\:+\:\frac{{a}\sqrt{{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }}{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\:\geqslant\:\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

Question Number 176168 Answers: 1 Comments: 0

Two fair dice are rolled once. Let X be the random variable representing the sum of the numbers that show up on the two dice. Find X.

$$\mathrm{Two}\:\mathrm{fair}\:\mathrm{dice}\:\mathrm{are}\:\mathrm{rolled}\:\mathrm{once}.\:\mathrm{Let}\:{X} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{random}\:\mathrm{variable}\:\mathrm{representing} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{show} \\ $$$$\mathrm{up}\:\mathrm{on}\:\mathrm{the}\:\mathrm{two}\:\mathrm{dice}.\:\mathrm{Find}\:{X}. \\ $$

Question Number 176164 Answers: 1 Comments: 0

Proof that : (√((1 − cos x)/(1 + cos x))) + (√((1 + cos x)/(1 − cos x))) = 2 ∙ cosec x

$$\:{Proof}\:{that}\:: \\ $$$$\: \\ $$$$\:\sqrt{\frac{\mathrm{1}\:−\:\mathrm{cos}\:{x}}{\mathrm{1}\:+\:\mathrm{cos}\:{x}}}\:+\:\sqrt{\frac{\mathrm{1}\:+\:\mathrm{cos}\:{x}}{\mathrm{1}\:−\:\mathrm{cos}\:{x}}}\:=\:\mathrm{2}\:\centerdot\:\mathrm{cosec}\:{x} \\ $$$$\: \\ $$

Question Number 176163 Answers: 0 Comments: 0

Question Number 176160 Answers: 1 Comments: 0

∫(x^3 /(x^4 +x^2 +1))dx

$$\int\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$

Question Number 176159 Answers: 1 Comments: 0

Question Number 176155 Answers: 2 Comments: 3

Find how many distinct integers are there in this sequence: ⌊((1^2 +1)/(100))⌋, ⌊((2^2 +2)/(100))⌋, ⌊((3^2 +3)/(100))⌋, ..., ⌊((100^2 +100)/(100))⌋ where ⌊x⌋ is the greatest integer that is less than or equal to x

$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:\mathrm{distinct}\:\mathrm{integers}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{this}\:\mathrm{sequence}: \\ $$$$\lfloor\frac{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{3}}{\mathrm{100}}\rfloor,\:...,\:\lfloor\frac{\mathrm{100}^{\mathrm{2}} +\mathrm{100}}{\mathrm{100}}\rfloor \\ $$$$\mathrm{where}\:\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:{x} \\ $$

Question Number 176154 Answers: 1 Comments: 0

lim_(x→0) ((tan x−x)/(x−sin x)) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{x}}{\mathrm{x}−\mathrm{sin}\:\mathrm{x}}\:=? \\ $$

Question Number 176147 Answers: 1 Comments: 0

Question Number 176146 Answers: 1 Comments: 0

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