Question and Answers Forum

AlgebraQuestion and Answers: Page 312

Question Number 55576 Answers: 0 Comments: 0

An aeroplane has an air speed of 120kmh^(−1) and flies on a course of bearing S60°E. A wind is blowing steadily at 30kmh^(−1) from a bearing of N60°E. Find; i. the ground speed of the aeroplane ii. the path of the aeroplane

$$\mathrm{An}\:\mathrm{aeroplane}\:\mathrm{has}\:\mathrm{an}\:\mathrm{air}\:\mathrm{speed}\:\mathrm{of}\: \\ $$$$\mathrm{120kmh}^{−\mathrm{1}} \:\mathrm{and}\:\mathrm{flies}\:\mathrm{on}\:\mathrm{a}\:\mathrm{course}\:\mathrm{of} \\ $$$$\mathrm{bearing}\:\mathrm{S60}°\mathrm{E}.\:\mathrm{A}\:\mathrm{wind}\:\mathrm{is}\:\mathrm{blowing}\:\mathrm{steadily} \\ $$$$\mathrm{at}\:\mathrm{30kmh}^{−\mathrm{1}} \:\mathrm{from}\:\mathrm{a}\:\mathrm{bearing}\:\mathrm{of}\:\mathrm{N60}°\mathrm{E}. \\ $$$$\mathrm{Find}; \\ $$$$\mathrm{i}.\:\mathrm{the}\:\mathrm{ground}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{aeroplane} \\ $$$$\mathrm{ii}.\:\mathrm{the}\:\mathrm{path}\:\mathrm{of}\:\mathrm{the}\:\mathrm{aeroplane} \\ $$

Question Number 55501 Answers: 0 Comments: 5

Roots of x^3 +px+q=0 are x = u+v u^3 , v^3 are roots of z^2 −α^3 z+β^6 =0 ((d^2 (y/x))/dx^2 )∣_(x=α) =0 , (dy/dx)∣_(x=β) =0 . I have noticed long way back, please hunt why (how come) ?

$${Roots}\:{of}\:{x}^{\mathrm{3}} +{px}+{q}=\mathrm{0} \\ $$$${are}\:\:{x}\:=\:{u}+{v} \\ $$$${u}^{\mathrm{3}} ,\:{v}^{\mathrm{3}} \:{are}\:{roots}\:{of} \\ $$$$\:\:\:\:\:\:\:{z}^{\mathrm{2}} −\alpha^{\mathrm{3}} {z}+\beta^{\mathrm{6}} =\mathrm{0} \\ $$$$\:\:\:\frac{{d}^{\mathrm{2}} \left({y}/{x}\right)}{{dx}^{\mathrm{2}} }\mid_{{x}=\alpha} =\mathrm{0}\:\:,\:\frac{{dy}}{{dx}}\mid_{{x}=\beta} =\mathrm{0}\:. \\ $$$${I}\:{have}\:{noticed}\:{long}\:{way}\:{back}, \\ $$$${please}\:{hunt}\:{why}\:\left({how}\:{come}\right)\:? \\ $$

Question Number 55482 Answers: 1 Comments: 0

A group of n students are numbered continously from first sdtudent as 1,2,3,...... If 1101 digits had to be used in all,what is the number of students in the group

$${A}\:{group}\:{of}\:{n}\:{students}\:{are}\:{numbered} \\ $$$${continously}\:{from}\:{first}\:{sdtudent}\:{as}\:\mathrm{1},\mathrm{2},\mathrm{3},...... \\ $$$${If}\:\mathrm{1101}\:{digits}\:{had}\:{to}\:{be}\:{used}\:{in}\:{all},{what}\:{is}\:{the}\: \\ $$$${number}\:{of}\:{students}\:{in}\:{the}\:{group} \\ $$

Question Number 55415 Answers: 2 Comments: 0

a, b, d are gp. such that a, b, c are real. if a + b + c = 26 and a^2 + b^2 + c^2 = 364, find b ?

$$\mathrm{a},\:\mathrm{b},\:\mathrm{d}\:\:\mathrm{are}\:\mathrm{gp}.\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{a},\:\mathrm{b},\:\mathrm{c}\:\mathrm{are}\:\mathrm{real}.\:\:\mathrm{if}\:\: \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\:=\:\:\mathrm{26}\:\:\:\mathrm{and}\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:\:=\:\:\mathrm{364},\:\:\:\:\:\mathrm{find}\:\:\mathrm{b}\:\:? \\ $$

Question Number 55412 Answers: 0 Comments: 0

Solve for x and y. 2^x + 2y = 1 ....... equation (i) 3^(2x) + y = 27 ..... equation (ii)

$$\mathrm{Solve}\:\mathrm{for}\:\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}. \\ $$$$\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{x}} \:+\:\mathrm{2y}\:\:=\:\:\mathrm{1}\:\:\:\:\:\:\:.......\:\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{3}^{\mathrm{2x}} \:+\:\mathrm{y}\:\:=\:\:\mathrm{27}\:\:\:\:\:\:\:\:.....\:\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 55359 Answers: 1 Comments: 0

Find a formula for the general term of the squence 1, 2, 2, 3, 3, 3, 4, 4, 4,4, ...

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{formula}\:\mathrm{for}\:\mathrm{the}\:\mathrm{general}\: \\ $$$$\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{squence} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{2},\:\mathrm{3},\:\mathrm{3},\:\mathrm{3},\:\mathrm{4},\:\mathrm{4},\:\mathrm{4},\mathrm{4},\:... \\ $$

Question Number 55358 Answers: 0 Comments: 0

Determine all functions f : N → N satisfying xf(y)+yf(x)=(x+y)f(x^2 +y^2 ) for all positive integers x and y

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{functions}\:{f}\::\:\mathbb{N}\:\rightarrow\:\mathbb{N}\: \\ $$$$\mathrm{satisfying} \\ $$$${xf}\left({y}\right)+{yf}\left({x}\right)=\left({x}+{y}\right){f}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:{x}\:\mathrm{and}\:{y} \\ $$

Question Number 55326 Answers: 1 Comments: 0

find the minimum value (a/(√(a^2 +8bc)))+(b/(√(b^2 +8ac)))+(c/(√(c^2 +8ab)))

$${find}\:\:{the}\:{minimum}\:{value} \\ $$$$\frac{{a}}{\sqrt{{a}^{\mathrm{2}} +\mathrm{8}{bc}}}+\frac{{b}}{\sqrt{{b}^{\mathrm{2}} +\mathrm{8}{ac}}}+\frac{{c}}{\sqrt{{c}^{\mathrm{2}} +\mathrm{8}{ab}}} \\ $$

Question Number 55312 Answers: 1 Comments: 0

Consider the system { ((x^2 + y^2 = z)),((2x + 2y + z = k)) :} The value of xy + zk for which the system has a unique solution is ...

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{system} \\ $$$$\begin{cases}{{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:{z}}\\{\mathrm{2}{x}\:+\:\mathrm{2}{y}\:+\:{z}\:=\:{k}}\end{cases} \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{xy}\:+\:{zk}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{system} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{is}\:... \\ $$

Question Number 55300 Answers: 1 Comments: 0

Question Number 55284 Answers: 0 Comments: 1

5^(x−2) −5^x +5^(x+1) =505

$$\mathrm{5}^{{x}−\mathrm{2}} −\mathrm{5}^{{x}} +\mathrm{5}^{{x}+\mathrm{1}} =\mathrm{505} \\ $$

Question Number 55258 Answers: 0 Comments: 0

3x+5y=?_

$$\mathrm{3}{x}+\mathrm{5}{y}=?_{} \\ $$

Question Number 55198 Answers: 2 Comments: 1

x^5 −4x^4 +6x^3 +8x−32=0 Find at least one root.

$${x}^{\mathrm{5}} −\mathrm{4}{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{3}} +\mathrm{8}{x}−\mathrm{32}=\mathrm{0} \\ $$$${Find}\:{at}\:{least}\:{one}\:{root}. \\ $$

Question Number 55195 Answers: 1 Comments: 0

x^2 +ax+(√2)b=0,has 2 roots:c and d,also x^2 +cx+(√2)d=0,has 2 roots:a and b.such that:a, b, c, d,are defferent non zero numbers. find possible value(s) for:a^2 +b^2 +c^2 +d^2 .

Question Number 55108 Answers: 0 Comments: 0

Question Number 55088 Answers: 0 Comments: 0

Question Number 55076 Answers: 1 Comments: 0

Known polynom P(z)=a_0 z^n +a_1 z^(n−1) +…+a_(n ) With explain real number. If z_0 =3−4i form root is from polynom, then one other root defonitely appeared is..

$$\mathrm{Known}\:\mathrm{polynom} \\ $$$$\mathrm{P}\left({z}\right)={a}_{\mathrm{0}} {z}^{{n}} +{a}_{\mathrm{1}} {z}^{{n}−\mathrm{1}} +\ldots+{a}_{{n}\:} \mathrm{With} \\ $$$$\mathrm{explain}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{If}\:{z}_{\mathrm{0}} =\mathrm{3}−\mathrm{4}{i} \\ $$$$\mathrm{form}\:\mathrm{root}\:\mathrm{is}\:\mathrm{from}\:\mathrm{polynom}, \\ $$$$\mathrm{then}\:\mathrm{one}\:\mathrm{other}\:\mathrm{root}\:\mathrm{defonitely}\:\mathrm{appeared} \\ $$$$\mathrm{is}.. \\ $$

Question Number 55070 Answers: 0 Comments: 1

Factorised the polynom z^4 +1 be polynom with lower degree, but have real coefficient

$$\mathrm{Factorised}\:\mathrm{the}\:\mathrm{polynom}\:{z}^{\mathrm{4}} +\mathrm{1}\: \\ $$$$\mathrm{be}\:\mathrm{polynom}\:\mathrm{with}\:\mathrm{lower}\:\mathrm{degree}, \\ $$$$\mathrm{but}\:\mathrm{have}\:\mathrm{real}\:\mathrm{coefficient} \\ $$

Question Number 55069 Answers: 1 Comments: 3

Known analytic function f(z)=((2(z−2))/(z(z−4))) and written as f(z)=Σ_(n=0) ^(∝) a_n (z−1)^n The value of a_(100) is...

$$\mathrm{Known}\:\mathrm{analytic}\:\mathrm{function} \\ $$$${f}\left({z}\right)=\frac{\mathrm{2}\left({z}−\mathrm{2}\right)}{{z}\left({z}−\mathrm{4}\right)} \\ $$$$\mathrm{and}\:\mathrm{written}\:\mathrm{as}\:{f}\left({z}\right)=\underset{{n}=\mathrm{0}} {\overset{\propto} {\Sigma}}\:{a}_{{n}} \left({z}−\mathrm{1}\right)^{{n}} \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{100}} \:\mathrm{is}... \\ $$

Question Number 55066 Answers: 0 Comments: 0

Find radius convergence for series 1−z^2 +z^4 −z^6 +...

$$\mathrm{Find}\:\mathrm{radius}\:\mathrm{convergence}\:\mathrm{for}\:\mathrm{series} \\ $$$$\mathrm{1}−{z}^{\mathrm{2}} +{z}^{\mathrm{4}} −{z}^{\mathrm{6}} +... \\ $$

Question Number 55039 Answers: 1 Comments: 3

α and β,are 2 roots of eq: ax^2 +bx+c=0 with conditions: { ((α^2 =β+b)),((β^2 =α+a)) :} find: c in terms of: a and b.