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AlgebraQuestion and Answers: Page 345

Question Number 30119 Answers: 2 Comments: 1

Let N= 2^(1224) −1. S= 2^(153) +2^(77) +1. T= 2^(408) −2^(204) +1. then which of the following statment is correct? a) S and T both divide N. b) only S divides N. c) only T divides N. d) Neither S nor T divides N.

$$\mathrm{Let}\:\:\mathrm{N}=\:\mathrm{2}^{\mathrm{1224}} \:−\mathrm{1}. \\ $$$$\mathrm{S}=\:\mathrm{2}^{\mathrm{153}} +\mathrm{2}^{\mathrm{77}} +\mathrm{1}. \\ $$$$\mathrm{T}=\:\mathrm{2}^{\mathrm{408}} −\mathrm{2}^{\mathrm{204}} +\mathrm{1}. \\ $$$$\mathrm{then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statment}\:\mathrm{is} \\ $$$$\:\mathrm{correct}? \\ $$$$\left.\mathrm{a}\right)\:\mathrm{S}\:\mathrm{and}\:\mathrm{T}\:\mathrm{both}\:\mathrm{divide}\:\mathrm{N}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{only}\:\mathrm{S}\:\mathrm{divides}\:\mathrm{N}. \\ $$$$\left.\mathrm{c}\right)\:\mathrm{only}\:\mathrm{T}\:\mathrm{divides}\:\mathrm{N}. \\ $$$$\left.\mathrm{d}\right)\:\mathrm{Neither}\:\mathrm{S}\:\mathrm{nor}\:\mathrm{T}\:\mathrm{divides}\:\mathrm{N}. \\ $$

Question Number 30092 Answers: 1 Comments: 0

If x+(1/x)=3 find x^5 +(1/x^5 )

$${If}\:{x}+\frac{\mathrm{1}}{{x}}=\mathrm{3}\:{find}\:{x}^{\mathrm{5}} +\frac{\mathrm{1}}{{x}^{\mathrm{5}} } \\ $$

Question Number 29885 Answers: 1 Comments: 2

∫(1/(1+sinx))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{1}+\mathrm{sin}{x}}{dx}=? \\ $$

Question Number 29837 Answers: 0 Comments: 0

let give T_n (x)=cos(n arcosx) with x∈[−1,1] 1) prove that T_n is a polynomial and T_n ∈Z[x] 2)calculate T_1 , T_2 , T_3 ,and T_4 3) prove that T_(n+2) (x)=2x T_(n+1) (x)−T_n (x) 4)find the roots of T_n and factorize T_n (x).

$${let}\:{give}\:\:{T}_{{n}} \left({x}\right)={cos}\left({n}\:{arcosx}\right)\:{with}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{T}_{{n}} \:{is}\:{a}\:{polynomial}\:{and}\:{T}_{{n}} \in{Z}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right){calculate}\:{T}_{\mathrm{1}} ,\:{T}_{\mathrm{2}} ,\:{T}_{\mathrm{3}} ,{and}\:{T}_{\mathrm{4}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{T}_{{n}+\mathrm{2}} \left({x}\right)=\mathrm{2}{x}\:{T}_{{n}+\mathrm{1}} \left({x}\right)−{T}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{4}\right){find}\:{the}\:{roots}\:{of}\:{T}_{{n}} \:{and}\:{factorize}\:{T}_{{n}} \left({x}\right). \\ $$

Question Number 29832 Answers: 0 Comments: 0

p is a polynomial having n roots x_i with x_i ≠x_j for i≠j prove that Σ_(i=1) ^n ((p^(′′) (x_i ))/(p^′ (x_i )))=0

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{n}\:{roots}\:{x}_{{i}} \:\:{with}\:{x}_{{i}} \neq{x}_{{j}} \:{for}\:{i}\neq{j} \\ $$$${prove}\:{that}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\frac{{p}^{''} \left({x}_{{i}} \right)}{{p}^{'} \left({x}_{{i}} \right)}=\mathrm{0} \\ $$

Question Number 29820 Answers: 1 Comments: 3

Question Number 29805 Answers: 0 Comments: 1

f(x)=(x+a_1 )(x+a_2 )(x+a_3 )...(x+a_n ) find the coefficient of term x^k (0≤k≤n)

$${f}\left({x}\right)=\left({x}+{a}_{\mathrm{1}} \right)\left({x}+{a}_{\mathrm{2}} \right)\left({x}+{a}_{\mathrm{3}} \right)...\left({x}+{a}_{{n}} \right) \\ $$$${find}\:{the}\:{coefficient}\:{of}\:{term}\:{x}^{{k}} \:\left(\mathrm{0}\leqslant{k}\leqslant{n}\right) \\ $$

Question Number 29786 Answers: 0 Comments: 0

Question Number 29777 Answers: 2 Comments: 5

f(x) = (x − 1)(x − 2)(x − 3)...(x − 50) Find coefficient of x^(49)

$${f}\left({x}\right)\:=\:\left({x}\:−\:\mathrm{1}\right)\left({x}\:−\:\mathrm{2}\right)\left({x}\:−\:\mathrm{3}\right)...\left({x}\:−\:\mathrm{50}\right) \\ $$$$\mathrm{Find}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{49}} \\ $$

Question Number 29647 Answers: 0 Comments: 1

Question Number 29520 Answers: 1 Comments: 0

to make an open fish tank a glass sheet of 2mm gauge is used .the outer length ,breadth and height are 60.4 , 40.4, 40.2 respectively .how much maximum volume of water will be contained in it ?

$$\mathrm{to}\:\mathrm{make}\:\mathrm{an}\:\mathrm{open}\:\mathrm{fish}\:\mathrm{tank}\:\mathrm{a}\:\mathrm{glass}\:\mathrm{sheet}\:\mathrm{of}\: \\ $$$$\mathrm{2mm}\:\mathrm{gauge}\:\mathrm{is}\:\mathrm{used}\:.\mathrm{the}\:\mathrm{outer}\:\mathrm{length} \\ $$$$,\mathrm{breadth}\:\mathrm{and}\:\mathrm{height}\:\mathrm{are}\:\mathrm{60}.\mathrm{4}\:,\:\mathrm{40}.\mathrm{4},\: \\ $$$$\mathrm{40}.\mathrm{2}\:\mathrm{respectively}\:.\mathrm{how}\:\mathrm{much}\:\mathrm{maximum} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{water}\:\mathrm{will}\:\mathrm{be}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{it}\:? \\ $$$$ \\ $$

Question Number 29433 Answers: 1 Comments: 0

4(2x^2 )=8^x

$$\mathrm{4}\left(\mathrm{2x}^{\mathrm{2}} \right)=\mathrm{8}^{\mathrm{x}} \\ $$

Question Number 29424 Answers: 0 Comments: 8

Question Number 29362 Answers: 1 Comments: 2

Question Number 29345 Answers: 1 Comments: 0

solve simultanrodly x+y=5.....(1) x^y +y^x =17.....(2)

$$\mathrm{solve}\:\mathrm{simultanrodly} \\ $$$$ \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{5}.....\left(\mathrm{1}\right) \\ $$$$\mathrm{x}^{\mathrm{y}} +\mathrm{y}^{\mathrm{x}} =\mathrm{17}.....\left(\mathrm{2}\right) \\ $$

Question Number 29264 Answers: 1 Comments: 0

there are 25 persons in a conical tent every person needs an area of 4 sq m on the ground under the tent. if height if the tent is 18m.find the volume of the tent.

$$\mathrm{there}\:\mathrm{are}\:\mathrm{25}\:\mathrm{persons}\:\mathrm{in}\:\mathrm{a}\:\mathrm{conical}\:\mathrm{tent} \\ $$$$\mathrm{every}\:\mathrm{person}\:\mathrm{needs}\:\mathrm{an}\:\mathrm{area}\:\mathrm{of}\:\mathrm{4}\:\mathrm{sq}\:\mathrm{m}\: \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{under}\:\mathrm{the}\:\mathrm{tent}.\:\mathrm{if} \\ $$$$\mathrm{height}\:\mathrm{if}\:\mathrm{the}\:\mathrm{tent}\:\mathrm{is}\:\mathrm{18m}.\mathrm{find}\:\mathrm{the}\:\mathrm{volume} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{tent}. \\ $$

Question Number 29259 Answers: 1 Comments: 0

solve the equation 2x+3y=5 3x+4y=4

$${solve}\:{the}\:{equation} \\ $$$$\mathrm{2}{x}+\mathrm{3}{y}=\mathrm{5} \\ $$$$\mathrm{3}{x}+\mathrm{4}{y}=\mathrm{4}\: \\ $$

Question Number 29196 Answers: 0 Comments: 0

Let s = n_c_1 − (1+(1/2))n_c_2 +(1+(1/2)+(1/3))n_c_3 +.......+(−1)^(n−1) (1+(1/2)+(1/3)+....+(1/n))n_c_n then prove that s×n =1.

$$\mathrm{Let}\:\mathrm{s}\:=\:\mathrm{n}_{\mathrm{c}_{\mathrm{1}} } \:−\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right)\mathrm{n}_{\mathrm{c}_{\mathrm{2}} } \:+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}\right)\mathrm{n}_{\mathrm{c}_{\mathrm{3}} } \\ $$$$+.......+\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+....+\frac{\mathrm{1}}{\mathrm{n}}\right)\mathrm{n}_{\mathrm{c}_{\mathrm{n}} } \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{s}×\mathrm{n}\:=\mathrm{1}. \\ $$

Question Number 29167 Answers: 0 Comments: 1

let give S_n = Σ_(k=1) ^(n−1) sin(((kπ)/n)) find lim_(n→+∞) (S_n /n) .

$${let}\:{give}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} {sin}\left(\frac{{k}\pi}{{n}}\right)\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{S}_{{n}} }{{n}}\:\:. \\ $$

Question Number 29166 Answers: 0 Comments: 1

simlify S_n = Σ_(k=1) ^n k(1+i)^(k−1) .