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Question Number 22014 Answers: 0 Comments: 0

The number of solution(s) of the equation ∣log_e (∣x∣)∣ + ∣x∣ = 10 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solution}\left(\mathrm{s}\right)\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid\mathrm{log}_{{e}} \left(\mid{x}\mid\right)\mid\:+\:\mid{x}\mid\:=\:\mathrm{10}\:\mathrm{is} \\ $$

Question Number 22001 Answers: 1 Comments: 0

if determinant (((z−(z/4))),())=2 then value of determinant ((z),())

$${if}\:\begin{vmatrix}{{z}−\frac{{z}}{\mathrm{4}}}\\{}\end{vmatrix}=\mathrm{2}\:{then}\:{value}\:{of}\:\begin{vmatrix}{{z}}\\{}\end{vmatrix} \\ $$

Question Number 21994 Answers: 0 Comments: 1

Suppose N is an n-digit positive integer such that (a) all the n-digits are distinct; and (b) the sum of any three consecutive digits is divisible by 5. Prove that n is at most 6. Further, show that starting with any digit one can find a six-digit number with these properties.

$$\mathrm{Suppose}\:{N}\:\mathrm{is}\:\mathrm{an}\:{n}-\mathrm{digit}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{the}\:{n}-\mathrm{digits}\:\mathrm{are}\:\mathrm{distinct};\:\mathrm{and} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{digits}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{5}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{n}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most}\:\mathrm{6}.\:\mathrm{Further}, \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{starting}\:\mathrm{with}\:\mathrm{any}\:\mathrm{digit}\:\mathrm{one} \\ $$$$\mathrm{can}\:\mathrm{find}\:\mathrm{a}\:\mathrm{six}-\mathrm{digit}\:\mathrm{number}\:\mathrm{with}\:\mathrm{these} \\ $$$$\mathrm{properties}. \\ $$

Question Number 21990 Answers: 1 Comments: 1

i still search about a general and complete solution about this determine x in N where 7 divise 2^x +3^x note = it is just an exercise in secondary so dont go away... maybe we must use separation of cases methode....

$${i}\:{still}\:{search}\:{about}\:{a}\:{general}\:{and}\: \\ $$$${complete}\:{solution}\:{about}\:{this} \\ $$$${determine}\:{x}\:{in}\:{N}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$${note}\:=\:{it}\:{is}\:{just}\:{an}\:{exercise}\:{in}\:{secondary} \\ $$$${so}\:{dont}\:{go}\:{away}... \\ $$$${maybe}\:{we}\:{must}\:{use}\:{separation}\:{of}\:{cases} \\ $$$${methode}.... \\ $$

Question Number 21973 Answers: 1 Comments: 1

ΔABC with sides a, b, c ∈ Z and ∠A = 3∠B If its circumference is minimum, then find a, b, c

$$\Delta{ABC}\:\mathrm{with}\:\mathrm{sides}\:{a},\:{b},\:{c}\:\in\:\mathbb{Z}\:\mathrm{and}\:\angle{A}\:=\:\mathrm{3}\angle{B} \\ $$$$\mathrm{If}\:\mathrm{its}\:\mathrm{circumference}\:\mathrm{is}\:\mathrm{minimum},\:\mathrm{then} \\ $$$$\mathrm{find}\:{a},\:{b},\:{c} \\ $$

Question Number 21970 Answers: 1 Comments: 0

integrate ∫sec^3 xdx

$${integrate} \\ $$$$\int{sec}^{\mathrm{3}} {xdx} \\ $$

Question Number 21967 Answers: 1 Comments: 1

A block is tied with a thread of length l and moved in a horizontal circle on a rough table. Coefficient of friction between block and table is μ = 0.2. Find tan θ, where θ is the angle between acceleration and frictional force at the instant when speed of particle is v = (√(1.6lg))

$$\mathrm{A}\:\mathrm{block}\:\mathrm{is}\:\mathrm{tied}\:\mathrm{with}\:\mathrm{a}\:\mathrm{thread}\:\mathrm{of}\:\mathrm{length}\:{l} \\ $$$$\mathrm{and}\:\mathrm{moved}\:\mathrm{in}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{circle}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{rough}\:\mathrm{table}.\:\mathrm{Coefficient}\:\mathrm{of}\:\mathrm{friction} \\ $$$$\mathrm{between}\:\mathrm{block}\:\mathrm{and}\:\mathrm{table}\:\mathrm{is}\:\mu\:=\:\mathrm{0}.\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{tan}\:\theta,\:\mathrm{where}\:\theta\:\mathrm{is}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{between}\:\mathrm{acceleration}\:\mathrm{and}\:\mathrm{frictional} \\ $$$$\mathrm{force}\:\mathrm{at}\:\mathrm{the}\:\mathrm{instant}\:\mathrm{when}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{particle}\:\mathrm{is}\:{v}\:=\:\sqrt{\mathrm{1}.\mathrm{6}{lg}} \\ $$

Question Number 21962 Answers: 0 Comments: 2

Let A be a set of 16 positive integers with the property that the product of any two distinct numbers of A will not exceed 1994. Show that there are two numbers a and b in A which are not relatively prime.

$$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{16}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{property}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{numbers}\:\mathrm{of}\:{A}\:\mathrm{will} \\ $$$$\mathrm{not}\:\mathrm{exceed}\:\mathrm{1994}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are} \\ $$$$\mathrm{two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{in}\:{A}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{relatively}\:\mathrm{prime}. \\ $$

Question Number 21977 Answers: 1 Comments: 8

Let n be the number of ways in which 5 boys and 5 girls stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of (m/n) is

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which} \\ $$$$\mathrm{5}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{5}\:\mathrm{girls}\:\mathrm{stand}\:\mathrm{in}\:\mathrm{a}\:\mathrm{queue}\:\mathrm{in} \\ $$$$\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{all}\:\mathrm{the}\:\mathrm{girls}\:\mathrm{stand} \\ $$$$\mathrm{consecutively}\:\mathrm{in}\:\mathrm{the}\:\mathrm{queue}.\:\mathrm{Let}\:\mathrm{m}\:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{5}\:\mathrm{boys} \\ $$$$\mathrm{and}\:\mathrm{5}\:\mathrm{girls}\:\mathrm{can}\:\mathrm{stand}\:\mathrm{in}\:\mathrm{a}\:\mathrm{queue}\:\mathrm{in}\:\mathrm{such} \\ $$$$\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{exactly}\:\mathrm{four}\:\mathrm{girls}\:\mathrm{stand} \\ $$$$\mathrm{consecutively}\:\mathrm{in}\:\mathrm{the}\:\mathrm{queue}.\:\mathrm{Then}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\frac{{m}}{{n}}\:\mathrm{is} \\ $$

Question Number 21940 Answers: 3 Comments: 1

A polynomial function f(x) satisfies f(x)f((1/x)) = 2f(x) + 2f((1/x)); x ≠ 0 and f(2) = 18, then f(3) is equal to

$$\mathrm{A}\:\mathrm{polynomial}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{satisfies} \\ $$$${f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{2}{f}\left({x}\right)\:+\:\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right);\:{x}\:\neq\:\mathrm{0}\:\mathrm{and} \\ $$$${f}\left(\mathrm{2}\right)\:=\:\mathrm{18},\:\mathrm{then}\:{f}\left(\mathrm{3}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 21936 Answers: 1 Comments: 0

A heavy iron bar of weight W is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle θ with the horizontal. What is the weight experienced by the man?

$$\mathrm{A}\:\mathrm{heavy}\:\mathrm{iron}\:\mathrm{bar}\:\mathrm{of}\:\mathrm{weight}\:{W}\:\mathrm{is}\:\mathrm{having} \\ $$$$\mathrm{its}\:\mathrm{one}\:\mathrm{end}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{shoulder}\:\mathrm{of}\:\mathrm{a}\:\mathrm{man}.\:\mathrm{The}\:\mathrm{rod} \\ $$$$\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{weight}\:\mathrm{experienced}\:\mathrm{by}\:\mathrm{the} \\ $$$$\mathrm{man}? \\ $$

Question Number 21933 Answers: 0 Comments: 11

Let n_1 < n_2 < n_3 < n_4 < n_5 be positive integers such that n_1 + n_2 + n_3 + n_4 + n_5 = 20. Then the number of such distinct arrangements (n_1 , n_2 , n_3 , n_4 , n_5 ) is

$$\mathrm{Let}\:{n}_{\mathrm{1}} \:<\:{n}_{\mathrm{2}} \:<\:{n}_{\mathrm{3}} \:<\:{n}_{\mathrm{4}} \:<\:{n}_{\mathrm{5}} \:\mathrm{be}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:{n}_{\mathrm{1}} \:+\:{n}_{\mathrm{2}} \:+\:{n}_{\mathrm{3}} \:+\:{n}_{\mathrm{4}} \:+ \\ $$$${n}_{\mathrm{5}} \:=\:\mathrm{20}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{such} \\ $$$$\mathrm{distinct}\:\mathrm{arrangements}\:\left({n}_{\mathrm{1}} ,\:{n}_{\mathrm{2}} ,\:{n}_{\mathrm{3}} ,\:{n}_{\mathrm{4}} ,\:{n}_{\mathrm{5}} \right) \\ $$$$\mathrm{is} \\ $$

Question Number 21931 Answers: 0 Comments: 2

The number of ways of distributing six identical mathematics books and six identical physics books among three students such that each student gets atleast one mathematics book and atleast one physics book is ((5.5!)/k), then k is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{distributing}\:\mathrm{six} \\ $$$$\mathrm{identical}\:\mathrm{mathematics}\:\mathrm{books}\:\mathrm{and}\:\mathrm{six} \\ $$$$\mathrm{identical}\:\mathrm{physics}\:\mathrm{books}\:\mathrm{among}\:\mathrm{three} \\ $$$$\mathrm{students}\:\mathrm{such}\:\mathrm{that}\:\mathrm{each}\:\mathrm{student}\:\mathrm{gets} \\ $$$$\mathrm{atleast}\:\mathrm{one}\:\mathrm{mathematics}\:\mathrm{book}\:\mathrm{and} \\ $$$$\mathrm{atleast}\:\mathrm{one}\:\mathrm{physics}\:\mathrm{book}\:\mathrm{is}\:\frac{\mathrm{5}.\mathrm{5}!}{{k}},\:\mathrm{then}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21930 Answers: 0 Comments: 3

An eight digit number is formed from 1, 2, 3, 4 such that product of all digits is always 3072, the total number of ways is (23.^8 C_k ), where the value of k is

$$\mathrm{An}\:\mathrm{eight}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{is}\:\mathrm{formed}\:\mathrm{from} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\mathrm{such}\:\mathrm{that}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{digits} \\ $$$$\mathrm{is}\:\mathrm{always}\:\mathrm{3072},\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{is}\:\left(\mathrm{23}.\:^{\mathrm{8}} {C}_{{k}} \right),\:\mathrm{where}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21929 Answers: 0 Comments: 2

There are 8 Hindi novels and 6 English novels. 4 Hindi novels and 3 English novels are selected and arranged in a row such that they are alternate then no. of ways is

$$\mathrm{There}\:\mathrm{are}\:\mathrm{8}\:\mathrm{Hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{6}\:\mathrm{English} \\ $$$$\mathrm{novels}.\:\mathrm{4}\:\mathrm{Hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{3}\:\mathrm{English} \\ $$$$\mathrm{novels}\:\mathrm{are}\:\mathrm{selected}\:\mathrm{and}\:\mathrm{arranged}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{row}\:\mathrm{such}\:\mathrm{that}\:\mathrm{they}\:\mathrm{are}\:\mathrm{alternate}\:\mathrm{then} \\ $$$$\mathrm{no}.\:\mathrm{of}\:\mathrm{ways}\:\mathrm{is} \\ $$

Question Number 21925 Answers: 1 Comments: 0

In an xy plane the graph of the equation (x−6)^2 +(y+5)^2 =16 is a circle. P(10,−5) is on the circle. If PQ is a diameter of the circle, what is the co-ordinate at Q?

$$\mathrm{In}\:\mathrm{an}\:\mathrm{xy}\:\mathrm{plane}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left(\mathrm{x}−\mathrm{6}\right)^{\mathrm{2}} +\left(\mathrm{y}+\mathrm{5}\right)^{\mathrm{2}} =\mathrm{16}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}.\:\mathrm{P}\left(\mathrm{10},−\mathrm{5}\right) \\ $$$$\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}.\:\mathrm{If}\:\mathrm{PQ}\:\mathrm{is}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{circle},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{co}-\mathrm{ordinate}\:\mathrm{at}\:\mathrm{Q}? \\ $$

Question Number 21917 Answers: 0 Comments: 4

How many seven letter words can be formed by using the letters of the word SUCCESS so that neither two C nor two S are together?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{seven}\:\mathrm{letter}\:\mathrm{words}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{using}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word} \\ $$$$\mathrm{SUCCESS}\:\mathrm{so}\:\mathrm{that}\:\mathrm{neither}\:\mathrm{two}\:\mathrm{C}\:\mathrm{nor} \\ $$$$\mathrm{two}\:\mathrm{S}\:\mathrm{are}\:\mathrm{together}? \\ $$

Question Number 21913 Answers: 0 Comments: 0

Let a_n denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_n = the number of such n-digit integers ending with digit 1 and c_n = the number of such n-digit integers ending with digit 0. 1. Which of the following is correct? (1) a_(17) = a_(16) + a_(15) (2) c_(17) ≠ c_(16) + c_(15) (3) b_(17) ≠ b_(16) + c_(16) (4) a_(17) = c_(17) + b_(16) 2. The value of b_6 is

$$\mathrm{Let}\:{a}_{{n}} \:\mathrm{denote}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{all}\:{n}-\mathrm{digit} \\ $$$$\mathrm{positive}\:\mathrm{integers}\:\mathrm{formed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{0},\:\mathrm{1}\:\mathrm{or}\:\mathrm{both}\:\mathrm{such}\:\mathrm{that}\:\mathrm{no}\:\mathrm{consecutive} \\ $$$$\mathrm{digits}\:\mathrm{in}\:\mathrm{them}\:\mathrm{are}\:\mathrm{0}.\:\mathrm{Let}\:{b}_{{n}} \:=\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{such}\:{n}-\mathrm{digit}\:\mathrm{integers}\:\mathrm{ending} \\ $$$$\mathrm{with}\:\mathrm{digit}\:\mathrm{1}\:\mathrm{and}\:{c}_{{n}} \:=\:\mathrm{the}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{such}\:{n}-\mathrm{digit}\:\mathrm{integers}\:\mathrm{ending}\:\mathrm{with} \\ $$$$\mathrm{digit}\:\mathrm{0}. \\ $$$$\mathrm{1}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{correct}? \\ $$$$\left(\mathrm{1}\right)\:{a}_{\mathrm{17}} \:=\:{a}_{\mathrm{16}} \:+\:{a}_{\mathrm{15}} \\ $$$$\left(\mathrm{2}\right)\:{c}_{\mathrm{17}} \:\neq\:{c}_{\mathrm{16}} \:+\:{c}_{\mathrm{15}} \\ $$$$\left(\mathrm{3}\right)\:{b}_{\mathrm{17}} \:\neq\:{b}_{\mathrm{16}} \:+\:{c}_{\mathrm{16}} \\ $$$$\left(\mathrm{4}\right)\:{a}_{\mathrm{17}} \:=\:{c}_{\mathrm{17}} \:+\:{b}_{\mathrm{16}} \\ $$$$\mathrm{2}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{b}_{\mathrm{6}} \:\mathrm{is} \\ $$

Question Number 21904 Answers: 0 Comments: 3

Compute the area of a loop of the curve ((x^2 /a^2 )+(y^2 /b^2 ))=((2xy)/c^2 ) .

$${Compute}\:{the}\:{area}\:{of}\:{a}\:{loop}\:{of}\:{the} \\ $$$${curve}\:\left(\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\right)=\frac{\mathrm{2}{xy}}{{c}^{\mathrm{2}} }\:. \\ $$

Question Number 21903 Answers: 0 Comments: 0

ΔABC with sides a, b, c ∈ Z and ∠A = 3∠B If its circumference is minimum, then find a, b, c

Question Number 21889 Answers: 1 Comments: 0

If a,b, and A of a triangle are fixed and two possible values of the third side be c_1 and c_2 such that c_1 ^2 +c_1 c_2 +c_2 ^2 =a^2 , then find angle A.

$${If}\:{a},{b},\:{and}\:{A}\:{of}\:{a}\:{triangle}\:\:{are}\: \\ $$$${fixed}\:{and}\:{two}\:{possible}\:{values}\:{of}\:{the}\: \\ $$$${third}\:{side}\:{be}\:{c}_{\mathrm{1}} {and}\:{c}_{\mathrm{2}} {such}\:{that} \\ $$$$\boldsymbol{{c}}_{\mathrm{1}} ^{\mathrm{2}} +\boldsymbol{{c}}_{\mathrm{1}} \boldsymbol{{c}}_{\mathrm{2}} +\boldsymbol{{c}}_{\mathrm{2}} ^{\mathrm{2}} =\boldsymbol{{a}}^{\mathrm{2}} ,\:{then}\:{find}\:{angle}\:{A}. \\ $$

Question Number 21877 Answers: 2 Comments: 3

x=^3 (√(7+5(√2)))+^3 (√(7−5(√2))) 1. According to a video, x=2 2. According to WolframAlpha, x≈0.2071+0.3587i for “principal root” and x=2(√2) for “real-valued root” 3. According to google, x≈2.8284 Please help and explain! :)

$${x}=^{\mathrm{3}} \sqrt{\mathrm{7}+\mathrm{5}\sqrt{\mathrm{2}}}+^{\mathrm{3}} \sqrt{\mathrm{7}−\mathrm{5}\sqrt{\mathrm{2}}} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{a}\:\mathrm{video},\:{x}=\mathrm{2} \\ $$$$\: \\ $$$$\mathrm{2}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{WolframAlpha}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{x}\approx\mathrm{0}.\mathrm{2071}+\mathrm{0}.\mathrm{3587}{i}\:\:\mathrm{for}\:``\mathrm{principal}\:\mathrm{root}'' \\ $$$$\mathrm{and}\:\:\:{x}=\mathrm{2}\sqrt{\mathrm{2}}\:\:\mathrm{for}\:``\mathrm{real}-\mathrm{valued}\:\mathrm{root}'' \\ $$$$\: \\ $$$$\mathrm{3}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{google},\:{x}\approx\mathrm{2}.\mathrm{8284} \\ $$$$\: \\ $$$$\left.\mathrm{Please}\:\mathrm{help}\:\mathrm{and}\:\mathrm{explain}!\:\:\::\right) \\ $$

Question Number 21874 Answers: 2 Comments: 0

Find the remainder if 2^(2006) is divided by 17

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:\:\:\mathrm{2}^{\mathrm{2006}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{17} \\ $$

Question Number 21870 Answers: 1 Comments: 0

A wire of mass 9.8 × 10^(−3) kg per meter passes over a frictionless pulley fixed on the top of an inclined frictionless plane which makes an angle of 30° with the horizontal. Masses M_1 and M_2 are tied at the two ends of the wire. The mass M_1 rests on the plane and the mass M_2 hangs freely vertically downward. The whole system is in equilibrium. Now a transverse wave propagates along the wire with a velocity of 100 m/s. Find M_1 and M_2 (g = 9.8 m/s^2 ).

$$\mathrm{A}\:\mathrm{wire}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{9}.\mathrm{8}\:×\:\mathrm{10}^{−\mathrm{3}} \:\mathrm{kg}\:\mathrm{per}\:\mathrm{meter} \\ $$$$\mathrm{passes}\:\mathrm{over}\:\mathrm{a}\:\mathrm{frictionless}\:\mathrm{pulley}\:\mathrm{fixed} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{an}\:\mathrm{inclined}\:\mathrm{frictionless} \\ $$$$\mathrm{plane}\:\mathrm{which}\:\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{30}°\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{Masses}\:{M}_{\mathrm{1}} \:\mathrm{and}\:{M}_{\mathrm{2}} \:\mathrm{are} \\ $$$$\mathrm{tied}\:\mathrm{at}\:\mathrm{the}\:\mathrm{two}\:\mathrm{ends}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire}.\:\mathrm{The} \\ $$$$\mathrm{mass}\:{M}_{\mathrm{1}} \:\mathrm{rests}\:\mathrm{on}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{mass}\:{M}_{\mathrm{2}} \:\mathrm{hangs}\:\mathrm{freely}\:\mathrm{vertically} \\ $$$$\mathrm{downward}.\:\mathrm{The}\:\mathrm{whole}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}.\:\mathrm{Now}\:\mathrm{a}\:\mathrm{transverse}\:\mathrm{wave} \\ $$$$\mathrm{propagates}\:\mathrm{along}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{100}\:\mathrm{m}/\mathrm{s}.\:\mathrm{Find}\:{M}_{\mathrm{1}} \:\mathrm{and}\:{M}_{\mathrm{2}} \\ $$$$\left({g}\:=\:\mathrm{9}.\mathrm{8}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right). \\ $$

Question Number 21840 Answers: 1 Comments: 0

A cone is placed inside a sphere. If volume of the cone is maximum, find the ratio of radius from the cone and sphere

$$\mathrm{A}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{placed}\:\mathrm{inside}\:\mathrm{a}\:\mathrm{sphere}. \\ $$$$\mathrm{If}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{maximum}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{from}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{and}\:\mathrm{sphere} \\ $$

Question Number 21837 Answers: 0 Comments: 4

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