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Question Number 113110 Answers: 2 Comments: 2

∫_0 ^1 (√(x(x−1)dx))

$$\overset{\mathrm{1}} {\int}_{\mathrm{0}} \sqrt{{x}\left({x}−\mathrm{1}\right){dx}} \\ $$

Question Number 113109 Answers: 1 Comments: 2

Question Number 113107 Answers: 1 Comments: 0

Question Number 113101 Answers: 0 Comments: 2

Prove that GCD ((a,b),b)=(a,b)

$${Prove}\:{that}\:{GCD}\:\left(\left({a},{b}\right),{b}\right)=\left({a},{b}\right) \\ $$

Question Number 113097 Answers: 1 Comments: 0

If 4 women earn as much as 9 boys; 4 men as much as 15 boys; 27 girls as much as 20 women, how many girls will earn the same amount as 24 men?

$$\mathrm{If}\:\mathrm{4}\:\mathrm{women}\:\mathrm{earn}\:\mathrm{as}\:\mathrm{much}\:\mathrm{as}\:\mathrm{9}\:\mathrm{boys};\:\mathrm{4}\:\mathrm{men}\:\mathrm{as}\:\mathrm{much}\:\mathrm{as} \\ $$$$\mathrm{15}\:\mathrm{boys};\:\mathrm{27}\:\mathrm{girls}\:\mathrm{as}\:\mathrm{much}\:\mathrm{as}\:\mathrm{20}\:\mathrm{women},\:\mathrm{how}\:\mathrm{many}\:\mathrm{girls} \\ $$$$\mathrm{will}\:\mathrm{earn}\:\mathrm{the}\:\mathrm{same}\:\mathrm{amount}\:\mathrm{as}\:\mathrm{24}\:\mathrm{men}? \\ $$

Question Number 113092 Answers: 2 Comments: 0

What is the area of a tringle where the sides of triangle are 91 cm, 98 cm, and 105 cm

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{tringle}\:\mathrm{where}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{are}\:\mathrm{91}\:\mathrm{cm},\:\mathrm{98}\:\mathrm{cm},\:\mathrm{and}\:\mathrm{105}\:\mathrm{cm} \\ $$

Question Number 113080 Answers: 1 Comments: 0

The perimeter of a triangle is 84 cm and it′s area is 336 square cm. If the length of one side of triangle is 30 cm, then what is the lengths of the remaining two sides of triangle ?

$$\mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{is} \\ $$$$\mathrm{84}\:\mathrm{cm}\:\mathrm{and}\:\mathrm{it}'\mathrm{s}\:\mathrm{area}\:\mathrm{is}\:\mathrm{336}\:\mathrm{square}\:\mathrm{cm}.\:\mathrm{If}\:\mathrm{the}\: \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{one}\:\mathrm{side}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{30}\:\mathrm{cm},\:\mathrm{then} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the}\:\mathrm{remaining}\:\mathrm{two} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{triangle}\:? \\ $$

Question Number 113091 Answers: 1 Comments: 0

What is the area bounded by the curves arg(z) = (π/3) ; arg(z)= ((2π)/3) and arg(z−2−2i(√3))=π on the complex plane?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curves} \\ $$$$\mathrm{arg}\left(\mathrm{z}\right)\:=\:\frac{\pi}{\mathrm{3}}\:;\:\mathrm{arg}\left(\mathrm{z}\right)=\:\frac{\mathrm{2}\pi}{\mathrm{3}}\:\mathrm{and}\:\mathrm{arg}\left(\mathrm{z}−\mathrm{2}−\mathrm{2i}\sqrt{\mathrm{3}}\right)=\pi \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}? \\ $$

Question Number 113073 Answers: 1 Comments: 0

Find all positive integers n for which 5^n +1 is divisible by 7

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}\:\mathrm{for}\: \\ $$$$\mathrm{which}\:\mathrm{5}^{\mathrm{n}} +\mathrm{1}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7} \\ $$

Question Number 113070 Answers: 1 Comments: 0

If n∈Z^+ , prove that (1/(2(√1)))+(1/(3(√2)))+(1/(4(√3)))+...(1/((n+1)(√n)))<2

$$\mathrm{If}\:{n}\in\mathbb{Z}^{+} ,\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{1}}}+\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{3}}}+...\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\sqrt{{n}}}<\mathrm{2} \\ $$$$ \\ $$

Question Number 113063 Answers: 1 Comments: 0

Question Number 113060 Answers: 3 Comments: 0

(1) (√(dy/dx)) = ((d((√y)))/dx) (2) x^2 ≡ 73 (mod 216) (3) If 2^x_1 +2^x_2 +2^x_3 +...+2^x_n =80,000 where x_1 ,x_2 ,x_3 ,...,x_n are distinct whole number . find the value of n

$$\:\left(\mathrm{1}\right)\:\sqrt{\frac{\mathrm{dy}}{\mathrm{dx}}}\:=\:\frac{\mathrm{d}\left(\sqrt{\mathrm{y}}\right)}{\mathrm{dx}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{x}^{\mathrm{2}} \:\equiv\:\mathrm{73}\:\left(\mathrm{mod}\:\mathrm{216}\right) \\ $$$$\left(\mathrm{3}\right)\:\mathrm{If}\:\mathrm{2}^{\mathrm{x}_{\mathrm{1}} } +\mathrm{2}^{\mathrm{x}_{\mathrm{2}} } +\mathrm{2}^{\mathrm{x}_{\mathrm{3}} } +...+\mathrm{2}^{\mathrm{x}_{\mathrm{n}} } =\mathrm{80},\mathrm{000} \\ $$$$\:\mathrm{where}\:\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} ,\mathrm{x}_{\mathrm{3}} ,...,\mathrm{x}_{\mathrm{n}} \:\mathrm{are}\:\mathrm{distinct} \\ $$$$\mathrm{whole}\:\mathrm{number}\:.\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n} \\ $$

Question Number 113059 Answers: 2 Comments: 0

Question Number 113015 Answers: 2 Comments: 0

Question Number 113006 Answers: 1 Comments: 0

a+(1/(b+(1/(c+(1/(d+...))))))=(2)^(1/3) If a,b,c,d,... are positive integers, then what is the value of ′b′?

$$\mathrm{a}+\frac{\mathrm{1}}{\mathrm{b}+\frac{\mathrm{1}}{\mathrm{c}+\frac{\mathrm{1}}{\mathrm{d}+...}}}=\sqrt[{\mathrm{3}}]{\mathrm{2}} \\ $$$$\mathrm{If}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},...\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:'\mathrm{b}'? \\ $$

Question Number 113005 Answers: 1 Comments: 0

Two different two−digit natural numbers are written beside each other such that the larger number is written on the left. When the absolute difference of the two numbers is subtracted from the four−digit number so formed, the number obtained is 5481. What is the sum of the two−digit numbers?

$$\mathrm{Two}\:\mathrm{different}\:\mathrm{two}−\mathrm{digit}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{written}\:\mathrm{beside}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{larger}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{written}\:\mathrm{on}\:\mathrm{the}\:\mathrm{left}.\:\mathrm{When}\:\mathrm{the} \\ $$$$\mathrm{absolute}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:\mathrm{is}\:\mathrm{subtracted}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{number}\:\mathrm{so}\:\mathrm{formed},\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{obtained}\:\mathrm{is}\:\mathrm{5481}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}−\mathrm{digit}\:\mathrm{numbers}? \\ $$

Question Number 113004 Answers: 2 Comments: 0

prove that _0 ∫^( ∞) cos(x^2 )dx = _0 ∫^( ∞) sin(x^2 )dx =((√π)/(2(√2)))

$${prove}\:{that} \\ $$$$\:_{\mathrm{0}} \int^{\:\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}\:=\:\:_{\mathrm{0}} \int^{\:\infty} {sin}\left({x}^{\mathrm{2}} \right){dx}\:=\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{\mathrm{2}}} \\ $$

Question Number 113003 Answers: 1 Comments: 2

The digits of a three−digit number A are written in the reverse order to form another three−digit number B. If B>A and B−A is perfectly divisible by 7. Find the range of values of A.

$$\mathrm{The}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{a}\:\mathrm{three}−\mathrm{digit}\:\mathrm{number}\:\mathrm{A} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{reverse}\:\mathrm{order}\:\mathrm{to} \\ $$$$\mathrm{form}\:\mathrm{another}\:\mathrm{three}−\mathrm{digit}\:\mathrm{number}\:\mathrm{B}. \\ $$$$\mathrm{If}\:\mathrm{B}>\mathrm{A}\:\mathrm{and}\:\mathrm{B}−\mathrm{A}\:\mathrm{is}\:\mathrm{perfectly} \\ $$$$\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of} \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{A}. \\ $$

Question Number 113002 Answers: 1 Comments: 0

After distributing sweets equally among 25 children, 8 sweets remained. Had the number of children been 28, 22 sweets would have been left after equally distributing. What was the total number of sweets?

$$\mathrm{After}\:\mathrm{distributing}\:\mathrm{sweets}\:\mathrm{equally} \\ $$$$\mathrm{among}\:\mathrm{25}\:\mathrm{children},\:\mathrm{8}\:\mathrm{sweets} \\ $$$$\mathrm{remained}.\:\mathrm{Had}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{children} \\ $$$$\mathrm{been}\:\mathrm{28},\:\mathrm{22}\:\mathrm{sweets}\:\mathrm{would}\:\mathrm{have}\:\mathrm{been} \\ $$$$\mathrm{left}\:\mathrm{after}\:\mathrm{equally}\:\mathrm{distributing}.\:\mathrm{What} \\ $$$$\mathrm{was}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{sweets}? \\ $$

Question Number 113001 Answers: 0 Comments: 1

N! is completely divisible by 13^(52) . What is the sum of the digits of the smallest such number N?

$$\mathrm{N}!\:\mathrm{is}\:\mathrm{completely}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{13}^{\mathrm{52}} . \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{smallest}\:\mathrm{such}\:\mathrm{number}\:\mathrm{N}? \\ $$

Question Number 113000 Answers: 1 Comments: 0

The first 23 natural numbers are written in an increasing order beside each other to form a single number. What is the remainder when this number is divided by 18?

$$\mathrm{The}\:\mathrm{first}\:\mathrm{23}\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{order} \\ $$$$\mathrm{beside}\:\mathrm{each}\:\mathrm{other}\:\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{number}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\mathrm{this}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{18}? \\ $$

Question Number 112999 Answers: 1 Comments: 0

Product of divisors of 7056 ?

$$\mathrm{Product}\:\mathrm{of}\:\mathrm{divisors}\:\mathrm{of}\:\mathrm{7056}\:? \\ $$

Question Number 112998 Answers: 1 Comments: 0

If x+y+z=1 and x,y,z are positive real numbers, then the least value of ((1/x)−1)((1/y)−1)((1/z)−1) is

$$\mathrm{If}\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{1}\:\mathrm{and}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{real}\:\mathrm{numbers},\:\mathrm{then}\:\mathrm{the}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of} \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{x}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{y}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{z}}−\mathrm{1}\right)\:\mathrm{is}\: \\ $$

Question Number 112997 Answers: 1 Comments: 3

∫(( tan x dx)/( (√(sec^3 x + 1)))) = ?

$$\int\frac{\:{tan}\:{x}\:{dx}}{\:\sqrt{{sec}^{\mathrm{3}} \:{x}\:+\:\mathrm{1}}}\:=\:? \\ $$

Question Number 112996 Answers: 1 Comments: 0

Find the last digit of the sum 19^(81) +4^(9k) , k∈N

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{19}^{\mathrm{81}} +\mathrm{4}^{\mathrm{9k}} ,\:\mathrm{k}\in\mathrm{N} \\ $$

Question Number 112992 Answers: 1 Comments: 1

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