Question and Answers Forum

All Questions   Topic List

OthersQuestion and Answers: Page 62

Question Number 95020    Answers: 1   Comments: 1

Question Number 94914    Answers: 1   Comments: 0

Question Number 94820    Answers: 1   Comments: 0

A train which travels at a uniform speed due to mechanical fault after traveling for an hour goes at 3/5 th of the original speed and reaches the destination 2 hours late. If the fault occured after traveling another 50 miles the train would have reached 40 minutes earlier. What is the distance between the two stations ?

$${A}\:{train}\:{which}\:{travels}\:{at}\:{a}\:{uniform}\:{speed}\:{due}\:{to}\:{mechanical}\:{fault}\:{after}\: \\ $$$${traveling}\:{for}\:{an}\:{hour}\:{goes}\:{at}\:\mathrm{3}/\mathrm{5}\:{th}\:{of}\:{the}\:{original}\:{speed}\:{and}\:{reaches}\:{the}\: \\ $$$${destination}\:\mathrm{2}\:{hours}\:{late}.\:{If}\:{the}\:{fault}\:{occured}\:{after}\:{traveling}\:{another}\:\mathrm{50} \\ $$$${miles}\:{the}\:{train}\:{would}\:{have}\:{reached}\:\mathrm{40}\:{minutes}\:{earlier}.\:{What}\:{is}\:{the}\: \\ $$$${distance}\:{between}\:{the}\:{two}\:{stations}\:? \\ $$

Question Number 94782    Answers: 1   Comments: 0

The velocity of physical quantities is given by v = (√((P + (1/n))/x)) , where P is the pressure. Find the dimention of n and x.

$$\mathrm{The}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{physical}\:\mathrm{quantities}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$$$\:\:\mathrm{v}\:\:=\:\:\sqrt{\frac{\mathrm{P}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{x}}}\:,\:\:\mathrm{where}\:\:\mathrm{P}\:\mathrm{is}\:\mathrm{the}\:\mathrm{pressure}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{dimention}\:\mathrm{of}\:\:\:\mathrm{n}\:\:\mathrm{and}\:\:\mathrm{x}. \\ $$

Question Number 94769    Answers: 2   Comments: 0

Question Number 94739    Answers: 2   Comments: 2

Question Number 94455    Answers: 1   Comments: 3

Solve (D^2 −2D+1)y=xe^x sinx pleas help me sir

$${Solve}\:\left({D}^{\mathrm{2}} −\mathrm{2}{D}+\mathrm{1}\right){y}={xe}^{{x}} {sinx}\: \\ $$$${pleas}\:{help}\:{me}\:{sir}\: \\ $$

Question Number 94419    Answers: 0   Comments: 0

Let G be a connected graph and let X be the set of vertices of G of odd degree. suppose that ∣X∣=2k, where k≥1 show that there are k edge-disjoint trail Q_1 , Q_2 ,...,Q_k in G such that E(G)=E(Q_1 )∪E(Q_2 )∪....∪E(Q_k )

$${Let}\:{G}\:{be}\:{a}\:{connected}\:{graph}\:{and}\:{let}\:{X}\:{be}\:{the}\:{set}\:{of}\:{vertices}\:{of}\:{G}\:{of}\:{odd}\:{degree}.\:{suppose}\:{that}\:\mid{X}\mid=\mathrm{2}{k},\:{where}\:{k}\geqslant\mathrm{1}\: \\ $$$${show}\:{that}\:{there}\:{are}\:{k}\:{edge}-{disjoint}\:{trail}\:{Q}_{\mathrm{1}} ,\:{Q}_{\mathrm{2}} ,...,{Q}_{{k}} \:{in}\:{G}\:{such}\:{that} \\ $$$${E}\left({G}\right)={E}\left({Q}_{\mathrm{1}} \right)\cup{E}\left({Q}_{\mathrm{2}} \right)\cup....\cup{E}\left({Q}_{{k}} \right) \\ $$

Question Number 94360    Answers: 1   Comments: 3

Question Number 94359    Answers: 0   Comments: 1

If 32 men can reap a field in 15 days .In howmany days can 20 men reap the same fied?

$${If}\:\:\mathrm{32}\:{men}\:{can}\:{reap}\:{a}\:{field}\:{in}\:\mathrm{15}\:{days}\:.{In}\:{howmany}\:{days}\:{can}\:\mathrm{20}\:{men}\:{reap}\:{the}\:{same}\:{fied}? \\ $$

Question Number 94319    Answers: 0   Comments: 4

Question Number 94081    Answers: 0   Comments: 0

P is the point representing the complex number z = r( cos θ + i sin θ) in an argand diagram such that ∣z−a∣∣z + a∣ = a^2 . Show that P moves on the curve whose equation is r^2 =2a^2 cos2θ. sketch the curve r^2 = 2a^2 cos 2θ , showing clearly the tangents at the pole.

$${P}\:\mathrm{is}\:\mathrm{the}\:\mathrm{point}\:\mathrm{representing}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{number} \\ $$$$\:{z}\:=\:{r}\left(\:\mathrm{cos}\:\theta\:+\:{i}\:\mathrm{sin}\:\theta\right)\:\mathrm{in}\:\mathrm{an}\:\mathrm{argand}\:\mathrm{diagram}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mid{z}−{a}\mid\mid{z}\:+\:{a}\mid\:=\:{a}^{\mathrm{2}} .\:\mathrm{Show}\:\mathrm{that}\:{P}\:\mathrm{moves}\:\mathrm{on}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\mathrm{whose}\:\mathrm{equation}\:\mathrm{is}\:{r}^{\mathrm{2}} \:=\mathrm{2}{a}^{\mathrm{2}} \:\mathrm{cos2}\theta.\:\mathrm{sketch}\:\mathrm{the}\:\mathrm{curve}\: \\ $$$${r}^{\mathrm{2}} \:=\:\mathrm{2}{a}^{\mathrm{2}} \:\mathrm{cos}\:\mathrm{2}\theta\:,\:\mathrm{showing}\:\mathrm{clearly}\:\mathrm{the}\:\mathrm{tangents}\:\mathrm{at}\:\mathrm{the}\:\mathrm{pole}. \\ $$

Question Number 94079    Answers: 3   Comments: 0

∫_2 ^4 ((3x−2)/(x^2 −4)) dx = ?

$$\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\frac{\mathrm{3}{x}−\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{4}}\:{dx}\:=\:? \\ $$

Question Number 94078    Answers: 0   Comments: 0

Given the function f defined by f(x) = ((∣x−2∣)/(1−∣x∣)) (i) state the domain of f. (ii) show that f(x) = { ((((2−x)/(1+x)) , x < 0)),((((2−x)/(1−x)), 0 ≤ x < 2)),((((x−2)/(1−x)) , x ≥ 2)) :} (iii) Investigate the continuity of f at x = 2.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:{f}\:\mathrm{defined}\:\mathrm{by}\:{f}\left({x}\right)\:=\:\frac{\mid{x}−\mathrm{2}\mid}{\mathrm{1}−\mid{x}\mid} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{state}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:{f}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:{f}\left({x}\right)\:=\:\begin{cases}{\frac{\mathrm{2}−{x}}{\mathrm{1}+{x}}\:,\:{x}\:<\:\mathrm{0}}\\{\frac{\mathrm{2}−{x}}{\mathrm{1}−{x}},\:\mathrm{0}\:\leqslant\:{x}\:<\:\mathrm{2}}\\{\frac{{x}−\mathrm{2}}{\mathrm{1}−{x}}\:,\:{x}\:\geqslant\:\mathrm{2}}\end{cases} \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{Investigate}\:\mathrm{the}\:\mathrm{continuity}\:\mathrm{of}\:{f}\:\mathrm{at}\:{x}\:=\:\mathrm{2}. \\ $$

Question Number 93787    Answers: 0   Comments: 4

Question Number 93730    Answers: 0   Comments: 3

what are the reasons for not using x = ((2c)/(−b ±(√(b^2 −4ac)))) as the quadratic formula? i proved it.

$$\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:\mathrm{reasons}\:\mathrm{for}\:\mathrm{not}\:\mathrm{using}\: \\ $$$$\:{x}\:=\:\frac{\mathrm{2}{c}}{−{b}\:\pm\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}\:\:\mathrm{as}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{formula}?\: \\ $$$$\mathrm{i}\:\mathrm{proved}\:\mathrm{it}. \\ $$

Question Number 93691    Answers: 1   Comments: 3

If Σ_(i = 1) ^(10) (x_i +4) = 60 then find the value of x^ .

$$\boldsymbol{\mathrm{If}}\:\underset{\boldsymbol{{i}}\:=\:\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left(\boldsymbol{\mathrm{x}}_{\mathrm{i}} +\mathrm{4}\right)\:=\:\mathrm{60}\:\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\bar {\boldsymbol{{x}}}. \\ $$

Question Number 93483    Answers: 0   Comments: 0

prove that the equation of the normal to the rectangular hyperbola xy = c^2 at the point P(ct, c/t) is t^3 x −ty = c(t^4 −1). the normal to P on the hyperbola meets the x−axis at Q and the tangent to P meets the yaxis at R. show that the locus of the midpoint oc QR, as P varies is 2c^2 xy + y^4 = c^4 .

$$\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{normal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{rectangular} \\ $$$$\mathrm{hyperbola}\:{xy}\:=\:{c}^{\mathrm{2}} \:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:{P}\left({ct},\:{c}/{t}\right)\:\mathrm{is}\:{t}^{\mathrm{3}} {x}\:−{ty}\:=\:{c}\left({t}^{\mathrm{4}} −\mathrm{1}\right). \\ $$$$\mathrm{the}\:\mathrm{normal}\:\mathrm{to}\:{P}\:\:\mathrm{on}\:\mathrm{the}\:\mathrm{hyperbola}\:\mathrm{meets}\:\mathrm{the}\:\mathrm{x}−\mathrm{axis}\:\mathrm{at}\:{Q}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{tangent}\:\mathrm{to}\:{P}\:\mathrm{meets}\:\mathrm{the}\:\mathrm{yaxis}\:\mathrm{at}\:{R}.\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{midpoint}\:\:\mathrm{oc}\:{QR},\:\mathrm{as}\:{P}\:\mathrm{varies}\:\mathrm{is}\:\mathrm{2}{c}^{\mathrm{2}} {xy}\:+\:{y}^{\mathrm{4}} \:=\:{c}^{\mathrm{4}} . \\ $$

Question Number 93474    Answers: 2   Comments: 0

Q. Prove by mathematical induction that Σ_(r=1) ^n (4r + 5) = 2n^2 + 7n

$$\mathrm{Q}.\:\mathrm{Prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\mathrm{4}{r}\:+\:\mathrm{5}\right)\:=\:\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{7}{n}\: \\ $$

Question Number 93397    Answers: 0   Comments: 2

prove that ((1−tan^3 θ)/(1 + tan^3 θ)) = 1−2sin^3 θ

$$\mathrm{prove}\:\mathrm{that}\:\frac{\mathrm{1}−\mathrm{tan}^{\mathrm{3}} \theta}{\mathrm{1}\:+\:\mathrm{tan}^{\mathrm{3}} \theta}\:=\:\mathrm{1}−\mathrm{2sin}^{\mathrm{3}} \theta\: \\ $$

Question Number 93368    Answers: 1   Comments: 0

given that f(r)= sin (1 + 2r)θ show that f(r)−f(r−1) = 2 cos 2r θ sin θ hence show that Σ_(r=1) ^n cos 2r θ sin θ = cos (n +1)θ sin nθ

$$\mathrm{given}\:\mathrm{that}\:{f}\left({r}\right)=\:\:\mathrm{sin}\:\left(\mathrm{1}\:+\:\mathrm{2}{r}\right)\theta \\ $$$$\mathrm{show}\:\mathrm{that}\:{f}\left({r}\right)−{f}\left({r}−\mathrm{1}\right)\:=\:\mathrm{2}\:\mathrm{cos}\:\mathrm{2}{r}\:\theta\:\mathrm{sin}\:\theta \\ $$$$\mathrm{hence}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:\mathrm{2}{r}\:\theta\:\mathrm{sin}\:\theta\:\:=\:\mathrm{cos}\:\left({n}\:+\mathrm{1}\right)\theta\:\mathrm{sin}\:{n}\theta \\ $$

Question Number 93367    Answers: 0   Comments: 2

show using demoives theorem that sin^2 θ cos^2 4θ =(1/8)(1−cos4θ)

$$\mathrm{show}\:\mathrm{using}\:\mathrm{demoives}\:\mathrm{theorem}\:\mathrm{that}\: \\ $$$$\mathrm{sin}^{\mathrm{2}} \theta\:\mathrm{cos}^{\mathrm{2}} \mathrm{4}\theta\:=\frac{\mathrm{1}}{\mathrm{8}}\left(\mathrm{1}−\mathrm{cos4}\theta\right) \\ $$

Question Number 93366    Answers: 1   Comments: 0

solve the equation (z−2)^3 = (1/2)−i((√3)/2)

$$\:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\left({z}−\mathrm{2}\right)^{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}−{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$

Question Number 93365    Answers: 0   Comments: 3

given that α is a real number, use mathematical induction or otherwise to show that cos ((α/2))cos((α/2^2 ))cos((α/2^3 )) ...cos((α/2^n )) = ((sin α)/(2^n sin((α/2^n )))) hence find the lim_(n→∞) cos((α/2))cos((α/2^2 ))cos((α/2^3 )) ... cos((α/2^n ))

$$\mathrm{given}\:\mathrm{that}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number},\:\mathrm{use}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{or} \\ $$$$\mathrm{otherwise}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{2}} }\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{3}} }\right)\:...\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{{n}} }\right)\:=\:\frac{\mathrm{sin}\:\alpha}{\mathrm{2}^{{n}} \:\mathrm{sin}\left(\frac{\alpha}{\mathrm{2}^{{n}} }\right)} \\ $$$$\mathrm{hence}\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{2}} }\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{3}} }\right)\:...\:\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{{n}} }\right) \\ $$

Question Number 93220    Answers: 0   Comments: 5

Please in an arithmetic mean a, A_1 , A_2 , A_3 , ... , A_n , b where A_1 , A_2 , A_3 , ... , A_n are nth arithmetic mean why is b = (n + 2)th term: like T_(n + 2) Please

$$\mathrm{Please}\:\mathrm{in}\:\mathrm{an}\:\mathrm{arithmetic}\:\mathrm{mean} \\ $$$$\:\:\:\:\:\:\:\mathrm{a},\:\:\mathrm{A}_{\mathrm{1}} ,\:\mathrm{A}_{\mathrm{2}} ,\:\mathrm{A}_{\mathrm{3}} ,\:...\:,\:\mathrm{A}_{\mathrm{n}} ,\:\mathrm{b} \\ $$$$\mathrm{where}\:\:\:\mathrm{A}_{\mathrm{1}} ,\:\mathrm{A}_{\mathrm{2}} ,\:\mathrm{A}_{\mathrm{3}} ,\:...\:,\:\mathrm{A}_{\mathrm{n}} \:\:\mathrm{are}\:\mathrm{nth}\:\mathrm{arithmetic}\:\mathrm{mean} \\ $$$$\mathrm{why}\:\mathrm{is}\:\:\mathrm{b}\:\:=\:\:\left(\mathrm{n}\:\:+\:\:\mathrm{2}\right)\mathrm{th}\:\:\mathrm{term}:\:\:\mathrm{like}\:\:\mathrm{T}_{\mathrm{n}\:\:+\:\:\mathrm{2}} \\ $$$$\mathrm{Please} \\ $$

Question Number 93184    Answers: 0   Comments: 0

in solving the linear congruence ax ≡ b (mod n) ⇒ n∣(ax − b) ⇒ ax −b = kn ⇔ ax −kn = b ⇒ solving the linear diophantine equation ax −kn = b what are the general solution to the equation ax−kn = b

$$\mathrm{in}\:\mathrm{solving}\:\mathrm{the}\:\mathrm{linear}\:\mathrm{congruence} \\ $$$${ax}\:\equiv\:{b}\:\left(\mathrm{mod}\:{n}\right)\:\Rightarrow\:{n}\mid\left({ax}\:−\:{b}\right)\:\Rightarrow\:{ax}\:−{b}\:=\:{kn}\:\Leftrightarrow\:{ax}\:−{kn}\:=\:{b} \\ $$$$\Rightarrow\:\mathrm{solving}\:\mathrm{the}\:\mathrm{linear}\:\mathrm{diophantine}\:\mathrm{equation}\:{ax}\:−{kn}\:=\:{b} \\ $$$$\:\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:{ax}−{kn}\:=\:{b} \\ $$$$\: \\ $$$$ \\ $$

  Pg 57      Pg 58      Pg 59      Pg 60      Pg 61      Pg 62      Pg 63      Pg 64      Pg 65      Pg 66   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com