Question and Answers Forum

OthersQuestion and Answers: Page 110

Question Number 38535 Answers: 0 Comments: 0

Given the function f(x) where f(x)= { ((∫x^2 + 1 ,for {x:x D(f) 2)),((∫x^3 − 1,for y = f′(x))) :} a) Evaluate f(2) if f(a)= 2 + a^(n−1) find the value of a hence the domain of f(x).

$${Given}\:{the}\:{function} \\ $$$${f}\left({x}\right)\:{where}\: \\ $$$$ \\ $$$${f}\left({x}\right)=\:\begin{cases}{\int{x}^{\mathrm{2}} \:+\:\mathrm{1}\:,{for}\:\left\{{x}:{x}\:{D}\left({f}\right)\:\mathrm{2}\right.}\\{\int{x}^{\mathrm{3}} \:−\:\mathrm{1},{for}\:{y}\:=\:{f}'\left({x}\right)}\end{cases} \\ $$$$\left.{a}\right)\:{Evaluate}\:{f}\left(\mathrm{2}\right) \\ $$$${if}\:{f}\left({a}\right)=\:\mathrm{2}\:+\:{a}^{{n}−\mathrm{1}} \\ $$$${find}\:{the}\:{value}\:{of}\:{a} \\ $$$${hence}\:{the}\:{domain}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 38534 Answers: 0 Comments: 0

∫∫_R (2x + 3y)^2 dA=??

$$\int\underset{{R}} {\int}\left(\mathrm{2}{x}\:+\:\mathrm{3}{y}\right)^{\mathrm{2}} \:{dA}=?? \\ $$

Question Number 38517 Answers: 2 Comments: 0

simlify A= (1/((2−(√5))^4 )) + (1/((2+(√5))^4 )) B = (1/((3−(√2))^6 )) +(1/((3+(√2))^6 ))

$${simlify} \\ $$$${A}=\:\frac{\mathrm{1}}{\left(\mathrm{2}−\sqrt{\mathrm{5}}\right)^{\mathrm{4}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)^{\mathrm{4}} } \\ $$$${B}\:=\:\frac{\mathrm{1}}{\left(\mathrm{3}−\sqrt{\mathrm{2}}\right)^{\mathrm{6}} }\:+\frac{\mathrm{1}}{\left(\mathrm{3}+\sqrt{\mathrm{2}}\right)^{\mathrm{6}} } \\ $$

Question Number 38515 Answers: 1 Comments: 0

Question ; x^3 + x^3 = A) x^9 B) x^6 C) x^3 D) 1 Give a reason for your answer.

$$\:{Question}\:; \\ $$$${x}^{\mathrm{3}} \:+\:{x}^{\mathrm{3}} \:=\: \\ $$$$\left.{A}\right)\:{x}^{\mathrm{9}} \\ $$$$\left.{B}\right)\:{x}^{\mathrm{6}} \\ $$$$\left.{C}\right)\:{x}^{\mathrm{3}} \\ $$$$\left.{D}\right)\:\mathrm{1} \\ $$$${Give}\:{a}\:{reason}\:{for}\:{your}\:{answer}. \\ $$

Question Number 38495 Answers: 4 Comments: 0

prove that tan 3a tan 2a tan a = tan 3a − tan 2a − tan a

$${prove}\:{that} \\ $$$$\boldsymbol{\mathrm{tan}}\:\mathrm{3}\boldsymbol{{a}}\:\boldsymbol{\mathrm{tan}}\:\mathrm{2}\boldsymbol{{a}}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{{a}}\:=\:\:\boldsymbol{\mathrm{tan}}\:\mathrm{3}\boldsymbol{{a}}\:−\:\boldsymbol{\mathrm{tan}}\:\mathrm{2}\boldsymbol{{a}}\:−\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{{a}} \\ $$

Question Number 38488 Answers: 2 Comments: 0

find the value of x if 3^x = 9x

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{if}\: \\ $$$$\mathrm{3}^{{x}} \:=\:\mathrm{9}{x} \\ $$

Question Number 39416 Answers: 1 Comments: 0

Given that f(x) is a cubic function and f(x) = x^3 − (x^2 /4) + 5x − 7 a) find one factor of f(x) b) find (d^2 y/dx^2 ) for f(x) c) hence Evaluate y = ∫_0 ^∞ f(x).

$${Given}\:{that}\:{f}\left({x}\right)\:{is}\:{a}\:{cubic}\:{function} \\ $$$${and}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:\:−\:\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:+\:\mathrm{5}{x}\:−\:\mathrm{7} \\ $$$$\left.{a}\right)\:{find}\:{one}\:{factor}\:{of}\:{f}\left({x}\right) \\ $$$$\left.{b}\right)\:{find}\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:{for}\:{f}\left({x}\right) \\ $$$$\left.{c}\right)\:{hence}\:{Evaluate}\:\:{y}\:=\:\int_{\mathrm{0}} ^{\infty} {f}\left({x}\right). \\ $$

Question Number 38391 Answers: 1 Comments: 0

a+2b+3c=12 2ab+3ac+6bc=48 a+b+c=...

$${a}+\mathrm{2}{b}+\mathrm{3}{c}=\mathrm{12} \\ $$$$\mathrm{2}{ab}+\mathrm{3}{ac}+\mathrm{6}{bc}=\mathrm{48} \\ $$$${a}+{b}+{c}=... \\ $$

Question Number 38420 Answers: 0 Comments: 0

In the figure below,a particle A of mass 2kg is lying on a rough wooden block.The particle A is connected by a light inextensible horizontal string passing over a smooth light fixed pulley at the edge of the block,to a particle B of mass 3kg which hangs freely. The coefficent of friction between the particle A and the surface of the block is μ. Given that the string is taut and the system is released from rest such that the particle move with an acceleration of 4ms^(−1) . Find a) the tension b) the value of μ.

$${In}\:{the}\:{figure}\:{below},{a}\:{particle}\:{A}\:{of} \\ $$$${mass}\:\mathrm{2}{kg}\:{is}\:{lying}\:{on}\:{a}\:{rough}\:{wooden} \\ $$$${block}.{The}\:{particle}\:{A}\:{is}\:{connected}\:{by} \\ $$$${a}\:{light}\:{inextensible}\:{horizontal}\:{string} \\ $$$${passing}\:{over}\:{a}\:{smooth}\:{light}\:{fixed} \\ $$$${pulley}\:{at}\:{the}\:{edge}\:{of}\:{the}\:{block},{to}\:{a} \\ $$$${particle}\:{B}\:{of}\:{mass}\:\mathrm{3}{kg}\:{which}\:{hangs} \\ $$$${freely}.\:{The}\:{coefficent}\:{of}\:{friction} \\ $$$${between}\:{the}\:{particle}\:{A}\:{and}\:{the}\:{surface} \\ $$$${of}\:{the}\:{block}\:{is}\:\mu. \\ $$$${Given}\:{that}\:{the}\:{string}\:{is}\:{taut}\:{and}\:{the}\:{system}\:{is}\:{released}\:{from}\:{rest}\:\:{such}\:{that}\:{the}\:{particle}\:{move}\:{with}\:{an} \\ $$$${acceleration}\:{of}\:\mathrm{4}{ms}^{−\mathrm{1}} .\:{Find} \\ $$$$\left.{a}\right)\:{the}\:{tension} \\ $$$$\left.{b}\right)\:{the}\:{value}\:{of}\:\mu. \\ $$

Question Number 38419 Answers: 1 Comments: 0

Question Number 38373 Answers: 1 Comments: 7

Question Number 38366 Answers: 1 Comments: 0

At time t,the force acting on a particle P of mass 2kg is (2ti + 4j)N.P is initially at rest at the point with position vector (i + 2j). Find: a) the velocity of P when t = 2. b) the position vector when t = 2.

$${At}\:{time}\:{t},{the}\:{force}\:{acting}\:{on}\:{a}\:{particle} \\ $$$${P}\:{of}\:{mass}\:\mathrm{2}{kg}\:{is}\:\left(\mathrm{2}\boldsymbol{{ti}}\:+\:\mathrm{4}\boldsymbol{{j}}\right){N}.{P} \\ $$$${is}\:{initially}\:{at}\:{rest}\:{at}\:{the}\:{point}\:{with} \\ $$$${position}\:{vector}\:\left(\boldsymbol{{i}}\:+\:\mathrm{2}\boldsymbol{{j}}\right). \\ $$$${Find}: \\ $$$$\left.{a}\right)\:{the}\:{velocity}\:{of}\:{P}\:{when}\:{t}\:=\:\mathrm{2}. \\ $$$$\left.{b}\right)\:{the}\:{position}\:{vector}\:{when}\:{t}\:=\:\mathrm{2}. \\ $$

Question Number 38362 Answers: 0 Comments: 0

Question Number 38365 Answers: 1 Comments: 0

A particle P moves on a straightline from a fixed point O and the distance x from O after t seconds is given as x = (1/(4 )) t^4 − (3/2) t^2 + 2t. Find: a) the velocity of P when t = 2, b) the acceleration of P when t = 2, c) the time at which the speed P is Minimum.

$${A}\:{particle}\:{P}\:{moves}\:{on}\:{a}\:{straightline} \\ $$$${from}\:{a}\:{fixed}\:{point}\:{O}\:{and}\:{the}\:{distance} \\ $$$${x}\:{from}\:{O}\:{after}\:{t}\:{seconds}\:{is}\:{given}\:{as} \\ $$$$\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{4}\:}\:{t}^{\mathrm{4}} \:−\:\frac{\mathrm{3}}{\mathrm{2}}\:{t}^{\mathrm{2}} \:+\:\mathrm{2}{t}. \\ $$$${Find}: \\ $$$$\left.{a}\right)\:{the}\:{velocity}\:{of}\:{P}\:{when}\:{t}\:=\:\mathrm{2}, \\ $$$$\left.{b}\right)\:{the}\:{acceleration}\:{of}\:{P}\:{when}\:{t}\:=\:\mathrm{2}, \\ $$$$\left.{c}\right)\:{the}\:{time}\:{at}\:{which}\:{the}\:{speed}\:{P}\: \\ $$$${is}\:{Minimum}. \\ $$

Question Number 38296 Answers: 0 Comments: 7

i have a suggestion pls comment...we are all virtual friends common bond is mathematics so may know each other by posting our self photo...if administator give permission..

$${i}\:{have}\:{a}\:{suggestion}\:{pls}\:{comment}...{we}\:{are}\:{all} \\ $$$${virtual}\:{friends}\:{common}\:{bond}\:{is}\:{mathematics} \\ $$$${so}\:{may}\:{know}\:{each}\:{other}\:{by}\:{posting}\:{our}\:{self} \\ $$$${photo}...{if}\:{administator}\:{give}\:{permission}.. \\ $$

Question Number 38252 Answers: 0 Comments: 2

if x^2 + 3xy − y^2 = 3 find (dy/dx) at point (1,1) hence differentiate ((sin x)/(1 + x)) with respect to x.

$${if}\:{x}^{\mathrm{2}} \:+\:\mathrm{3}{xy}\:−\:{y}^{\mathrm{2}} \:=\:\mathrm{3}\:{find}\: \\ $$$$\frac{{dy}}{{dx}}\:{at}\:{point}\:\left(\mathrm{1},\mathrm{1}\right)\:{hence} \\ $$$${differentiate}\:\frac{{sin}\:{x}}{\mathrm{1}\:+\:{x}}\:{with}\:{respect} \\ $$$${to}\:{x}. \\ $$

Question Number 38250 Answers: 0 Comments: 0

It is given that the first term of a GP is the last term of an AP. the second term of the AP is the third term of the GP..detemine the Geometric mean of the GP is the fourth term of the GP is 16.

$$\:\:{It}\:{is}\:{given}\:{that}\:{the}\:{first}\:{term}\:{of} \\ $$$${a}\:{GP}\:\:{is}\:{the}\:{last}\:{term}\:{of}\:{an}\:{AP}. \\ $$$${the}\:{second}\:{term}\:{of}\:{the}\:{AP}\:{is}\:{the} \\ $$$${third}\:{term}\:{of}\:{the}\:{GP}..{detemine} \\ $$$${the}\:{Geometric}\:{mean}\:{of}\:{the}\:{GP}\:{is}\: \\ $$$$\:{the}\:{fourth}\:{term}\:{of}\:{the}\:{GP}\:{is}\:\mathrm{16}. \\ $$

Question Number 38059 Answers: 1 Comments: 0

Prove that Σ(x_i −x^− )=0

$${Prove}\:{that}\:\Sigma\left({x}_{{i}} −\overset{−} {{x}}\right)=\mathrm{0} \\ $$

Question Number 38051 Answers: 0 Comments: 0

A manufactual plans to build two types of tables. For table A,the cost of material is 20 000 bucks, the number of man hours needed to complete it is 10, and the profit is 15 000 bucks. Table B requires materials costing 12000 bucks,15 man hour of labour and makes same profit as table A. The total money avialable for materials is 500 000 bucks and labour avialable is 330 man hours.Find the maximum profit that can be made and the number of each type of table that should be made to produces it.

$${A}\:{manufactual}\:{plans}\:{to}\:{build}\:{two} \\ $$$${types}\:{of}\:{tables}.\:{For}\:{table}\:{A},{the}\:{cost} \\ $$$${of}\:{material}\:{is}\:\mathrm{20}\:\mathrm{000}\:{bucks},\:{the}\: \\ $$$${number}\:{of}\:{man}\:{hours}\:{needed}\:{to}\: \\ $$$${complete}\:{it}\:{is}\:\mathrm{10},\:{and}\:{the}\:{profit}\: \\ $$$${is}\:\mathrm{15}\:\mathrm{000}\:{bucks}.\:{Table}\:{B}\:{requires} \\ $$$${materials}\:{costing}\:\mathrm{12000}\:{bucks},\mathrm{15} \\ $$$${man}\:{hour}\:{of}\:{labour}\:{and}\:{makes} \\ $$$${same}\:{profit}\:{as}\:{table}\:{A}. \\ $$$$\:\:{The}\:{total}\:{money}\:{avialable}\:{for}\:{materials} \\ $$$${is}\:\mathrm{500}\:\mathrm{000}\:{bucks}\:{and}\:{labour}\:{avialable} \\ $$$${is}\:\mathrm{330}\:{man}\:{hours}.{Find}\:{the}\:{maximum} \\ $$$${profit}\:{that}\:{can}\:{be}\:{made}\:{and}\:{the}\:{number}\:{of}\:{each} \\ $$$${type}\:{of}\:{table}\:{that}\:{should}\:{be}\:{made}\:{to}\:{produces} \\ $$$${it}. \\ $$

Question Number 38049 Answers: 1 Comments: 0

Find the equation of the two lines throught (2,−3) which makes 45° with the line 2x − y = 2..hence find the cosine of the acute between the lines l_1 : y− 2x + 5=0 and l_2 : y − x + 6 (leave your answer in surd form)

$${Find}\:{the}\:{equation}\:{of}\:{the}\:{two}\:{lines} \\ $$$${throught}\:\left(\mathrm{2},−\mathrm{3}\right)\:{which}\:{makes}\:\mathrm{45}° \\ $$$${with}\:{the}\:{line}\:\mathrm{2}{x}\:−\:{y}\:=\:\mathrm{2}..{hence} \\ $$$${find}\:{the}\:{cosine}\:{of}\:{the}\:{acute}\:{between} \\ $$$${the}\:{lines}\:{l}_{\mathrm{1}} :\:{y}−\:\mathrm{2}{x}\:+\:\mathrm{5}=\mathrm{0}\:{and}\: \\ $$$${l}_{\mathrm{2}} :\:{y}\:−\:{x}\:+\:\mathrm{6}\:\left({leave}\:{your}\:{answer}\:{in}\right. \\ $$$$\left.{surd}\:{form}\right) \\ $$

Question Number 38012 Answers: 3 Comments: 1

The roots of the equation 2x^2 − x + 3 = 0 are α and β if the roots of 3x^2 + px + q=0 are α + (1/α) and β + (1/(β )) find the value of p and q.

$${The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:−\:{x}\:+\:\mathrm{3}\:=\:\mathrm{0}\:{are}\:\alpha\:{and}\:\beta \\ $$$${if}\:{the}\:{roots}\:{of}\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{px}\:+\:{q}=\mathrm{0}\: \\ $$$${are}\:\alpha\:+\:\frac{\mathrm{1}}{\alpha}\:{and}\:\beta\:+\:\frac{\mathrm{1}}{\beta\:}\:{find}\:{the}\:{value} \\ $$$${of}\:{p}\:{and}\:{q}. \\ $$$$\: \\ $$

Question Number 38011 Answers: 2 Comments: 0

Show that ((sin2A)/(1+ cos2A)) = TanA

$${Show}\:{that}\: \\ $$$$\:\:\frac{{sin}\mathrm{2}{A}}{\mathrm{1}+\:{cos}\mathrm{2}{A}}\:=\:{TanA} \\ $$

Question Number 37972 Answers: 1 Comments: 0

The distance S metres is given as a funtion f(t) where is time taken... if S = t^3 + t^2 + 4 find the velocity and acceleration

$$\:{The}\:{distance}\:{S}\:{metres}\:{is}\: \\ $$$${given}\:{as}\:{a}\:{funtion}\: \\ $$$${f}\left({t}\right)\:{where}\:{is}\:{time}\:{taken}... \\ $$$${if}\:{S}\:=\:{t}^{\mathrm{3}} \:+\:{t}^{\mathrm{2}} \:+\:\mathrm{4} \\ $$$${find}\:{the}\:{velocity}\:{and}\:{acceleration} \\ $$

Question Number 37948 Answers: 0 Comments: 1

If x ∈R show that (2+i)e^((1+3i)) +(2−i)e^((1−3i)) is also real.