Question and Answers Forum

AllQuestion and Answers: Page 1677

Question Number 32331 Answers: 0 Comments: 0

p is a polynomial having n simples roots (x_i )_(1≤i≤n) prove that Σ_(k=1) ^n (1/(p^′ (x_k ))) =0

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

Question Number 32330 Answers: 0 Comments: 1

let p_n (x)=(x+1)^(6n+1) −x^(6n+1) −1 with n integr prove that ∀n (x^2 +x+1)^2 divide p_n (x).

$${let}\:{p}_{{n}} \left({x}\right)=\left({x}+\mathrm{1}\right)^{\mathrm{6}{n}+\mathrm{1}} \:−{x}^{\mathrm{6}{n}+\mathrm{1}} \:−\mathrm{1}\:{with}\:{n}\:{integr} \\ $$$${prove}\:{that}\:\forall{n}\:\:\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} \:{divide}\:{p}_{{n}} \left({x}\right). \\ $$

Question Number 32329 Answers: 2 Comments: 0

The smallest number which must be subtracted from 3400 to make it a perfect cube is _____.

$$\mathrm{The}\:\mathrm{smallest}\:\mathrm{number}\:\mathrm{which}\:\mathrm{must}\:\mathrm{be} \\ $$$$\mathrm{subtracted}\:\mathrm{from}\:\mathrm{3400}\:\mathrm{to}\:\mathrm{make}\:\mathrm{it}\:\mathrm{a}\: \\ $$$$\mathrm{perfect}\:\mathrm{cube}\:\mathrm{is}\:\_\_\_\_\_. \\ $$

Question Number 32328 Answers: 0 Comments: 0

prove that 2^(n+1) divide A_n =[(1+(√3) )^(2n+1) ].

$${prove}\:{that}\:\mathrm{2}^{{n}+\mathrm{1}} \:{divide}\:\:{A}_{{n}} =\left[\left(\mathrm{1}+\sqrt{\mathrm{3}}\:\right)^{\mathrm{2}{n}+\mathrm{1}} \right]. \\ $$

Question Number 32326 Answers: 0 Comments: 0

simplify Σ_(k=p) ^(2p) (C_k ^p /2^k ) .

$${simplify}\:\:\sum_{{k}={p}} ^{\mathrm{2}{p}} \:\:\:\:\:\frac{{C}_{{k}} ^{{p}} }{\mathrm{2}^{{k}} }\:. \\ $$

Question Number 32323 Answers: 1 Comments: 0

Given f(x) = (3/(16))(∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Find f(x)

$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} {f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Find}\:{f}\left({x}\right) \\ $$

Question Number 32316 Answers: 0 Comments: 4

A gas expands according to the law pv=constant,where p is the pressure and v is the volume of the gas.Initially,v=1000m^3 and p=40N/m^2 .If the pressure is decreased at the rate of 5Nm^(−2) min^(−1) find the rate at which the gas is expanding when its volume is 2000m^(3.)

$${A}\:{gas}\:{expands}\:{according}\:{to}\:{the} \\ $$$${law}\:{pv}={constant},{where}\:{p}\:{is}\:{the} \\ $$$${pressure}\:{and}\:{v}\:{is}\:{the}\:{volume}\:{of} \\ $$$${the}\:{gas}.{Initially},{v}=\mathrm{1000}{m}^{\mathrm{3}} \:{and} \\ $$$${p}=\mathrm{40}{N}/{m}^{\mathrm{2}} .{If}\:{the}\:{pressure}\:{is} \\ $$$${decreased}\:{at}\:{the}\:{rate}\:{of}\:\mathrm{5}{Nm}^{−\mathrm{2}} {min}^{−\mathrm{1}} \\ $$$${find}\:{the}\:{rate}\:{at}\:{which}\:{the}\:{gas}\:{is} \\ $$$${expanding}\:{when}\:{its}\:{volume}\:{is} \\ $$$$\mathrm{2000}{m}^{\mathrm{3}.} \\ $$

Question Number 32313 Answers: 0 Comments: 0

If y=sinx then show that (d^n y/dx^n )=sin(x+((nπ)/2))

$${If}\:{y}={sinx}\:{then}\:{show}\:{that} \\ $$$$\frac{{d}^{{n}} {y}}{{dx}^{{n}} }={sin}\left({x}+\frac{{n}\pi}{\mathrm{2}}\right) \\ $$

Question Number 32312 Answers: 1 Comments: 1

A small ball is dropped from a height of 1m into a horizontal floor.Each time it rebounces to 3/5 of the height it has fallen. a)show that when the ball strikes the ground for the third time ,it has travelled a distance of 2.92m b)Show that the total distance travelled by the ball cant exceed 4m.

$${A}\:{small}\:{ball}\:{is}\:{dropped}\:{from}\:{a}\: \\ $$$${height}\:{of}\:\mathrm{1}{m}\:{into}\:{a}\:{horizontal} \\ $$$${floor}.{Each}\:{time}\:{it}\:{rebounces}\:{to} \\ $$$$\mathrm{3}/\mathrm{5}\:{of}\:{the}\:{height}\:{it}\:{has}\:{fallen}. \\ $$$$\left.{a}\right){show}\:{that}\:{when}\:{the}\:{ball}\:{strikes} \\ $$$${the}\:{ground}\:{for}\:{the}\:{third}\:{time}\:,{it} \\ $$$${has}\:{travelled}\:{a}\:{distance}\:{of}\:\mathrm{2}.\mathrm{92}{m} \\ $$$$\left.{b}\right){Show}\:{that}\:{the}\:{total}\:{distance} \\ $$$${travelled}\:{by}\:{the}\:{ball}\:{cant}\:{exceed} \\ $$$$\mathrm{4}{m}. \\ $$

Question Number 32309 Answers: 0 Comments: 0