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AllQuestion and Answers: Page 1975

Question Number 10787 Answers: 0 Comments: 2

Question Number 10829 Answers: 2 Comments: 0

lim_(x→1) ∫_( 1) ^( x) ((e^t^2 (dt))/(x^2 − 1)) (a) 1 (b) 0 (c) e/2 (d) e

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\int_{\:\mathrm{1}} ^{\:\mathrm{x}} \:\:\:\frac{\mathrm{e}^{\mathrm{t}^{\mathrm{2}} } \:\left(\mathrm{dt}\right)}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{1}}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{1}\:\left(\mathrm{b}\right)\:\mathrm{0}\:\left(\mathrm{c}\right)\:\mathrm{e}/\mathrm{2}\:\left(\mathrm{d}\right)\:\mathrm{e} \\ $$

Question Number 10777 Answers: 2 Comments: 2

find the range or (ranges) of value x can take for x+6>[2x+3]

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{or}\:\left(\mathrm{ranges}\right)\:\mathrm{of}\:\mathrm{value}\:\mathrm{x} \\ $$$$\mathrm{can}\:\mathrm{take}\:\mathrm{for}\:\:\mathrm{x}+\mathrm{6}>\left[\mathrm{2x}+\mathrm{3}\right] \\ $$

Question Number 10776 Answers: 1 Comments: 0

if D = determinant ((1,(3sinθ),1),((sinθ),1,(3cosθ)),(1,(sinθ),1)) the maxi mum value of D is?

$$\mathrm{if}\:\:\:\mathrm{D}\:=\begin{vmatrix}{\mathrm{1}}&{\mathrm{3sin}\theta}&{\mathrm{1}}\\{\mathrm{sin}\theta}&{\mathrm{1}}&{\mathrm{3cos}\theta}\\{\mathrm{1}}&{\mathrm{sin}\theta}&{\mathrm{1}}\end{vmatrix}\:\mathrm{the}\:\mathrm{maxi} \\ $$$$\mathrm{mum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{D}\:\mathrm{is}? \\ $$

Question Number 10774 Answers: 1 Comments: 0

Question Number 10773 Answers: 1 Comments: 0

Question Number 10768 Answers: 0 Comments: 2

Solve the following differential equations by power series. y^(//) −2y = 4x^2 e^x^2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{differential}\:\mathrm{equations}\: \\ $$$$\mathrm{by}\:\mathrm{power}\:\mathrm{series}. \\ $$$${y}^{//} −\mathrm{2}{y}\:=\:\mathrm{4}{x}^{\mathrm{2}} {e}^{{x}^{\mathrm{2}} } \\ $$

Question Number 10762 Answers: 0 Comments: 0

in eac of the following problems you are given a function on the interval −π<x<π. Sketch several periods of the corresponding periodic function of period 2π. Expand the periodic function in a sine−consine Fourier series. F(x) = { _0 ^(x+π) _(0<x<π) ^(−π<x<0)

$$\mathrm{in}\:\mathrm{eac}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{problems}\:\mathrm{you}\:{are} \\ $$$$\mathrm{given}\:\mathrm{a}\:\mathrm{function}\:\mathrm{on}\:\mathrm{the}\:\mathrm{interval}\:\:−\pi<{x}<\pi. \\ $$$$\mathrm{Sketch}\:\mathrm{several}\:\mathrm{periods}\:\mathrm{of}\:\mathrm{the}\:\mathrm{corresponding} \\ $$$$\mathrm{periodic}\:\mathrm{function}\:\mathrm{of}\:\mathrm{period}\:\mathrm{2}\pi.\:\mathrm{Expand}\:\mathrm{the} \\ $$$$\mathrm{periodic}\:\mathrm{function}\:\mathrm{in}\:\mathrm{a}\:\mathrm{sine}−\mathrm{consine}\:\mathrm{Fourier} \\ $$$$\mathrm{series}.\:\: \\ $$$$ \\ $$$${F}\left({x}\right)\:=\:\left\{\overset{{x}+\pi} {\:}_{\mathrm{0}} \:\:\:\overset{−\pi<{x}<\mathrm{0}} {\:}_{\mathrm{0}<{x}<\pi} \right. \\ $$

Question Number 10761 Answers: 0 Comments: 0

Express the following Γ function using tables and evaluate them using a table or Γ functions. ∫_0 ^∞ x^(−1/3) e^(−8x ) dx

$$\mathrm{Express}\:\mathrm{the}\:\mathrm{following}\:\Gamma\:\mathrm{function}\:\mathrm{using}\:\mathrm{tables} \\ $$$$\mathrm{and}\:\mathrm{evaluate}\:\mathrm{them}\:\mathrm{using}\:\mathrm{a}\:\mathrm{table}\:\mathrm{or}\:\Gamma\:\mathrm{functions}. \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{−\mathrm{1}/\mathrm{3}} {e}^{−\mathrm{8}{x}\:} {dx} \\ $$

Question Number 10760 Answers: 1 Comments: 1

A block of mass 2.0kg resting on a smooth horizontal plane is acted upon simultaneously by two forces 10N due north and 10N due east. The magnitude of the acceleration produce by the force on the block.

$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}.\mathrm{0kg}\:\mathrm{resting}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth}\:\mathrm{horizontal}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{acted} \\ $$$$\mathrm{upon}\:\mathrm{simultaneously}\:\mathrm{by}\:\mathrm{two}\:\mathrm{forces}\:\mathrm{10N}\:\mathrm{due}\:\mathrm{north}\:\mathrm{and}\:\mathrm{10N}\:\mathrm{due}\:\mathrm{east}. \\ $$$$\mathrm{The}\:\mathrm{magnitude}\:\mathrm{of}\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{produce}\:\mathrm{by}\:\mathrm{the}\:\mathrm{force}\:\mathrm{on}\:\mathrm{the}\:\mathrm{block}. \\ $$

Question Number 10755 Answers: 1 Comments: 0

x^2 + y^2 + xy + 2(x − y) = 9 How many solution that fulfilled the equation above ? x, y ∈ N

$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{xy}\:+\:\mathrm{2}\left({x}\:−\:{y}\right)\:=\:\mathrm{9} \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{solution}\:\mathrm{that}\:\mathrm{fulfilled}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{above}\:? \\ $$$${x},\:{y}\:\in\:\mathbb{N} \\ $$

Question Number 10750 Answers: 1 Comments: 0

Exact value of sin 9° = ...

$$\mathrm{Exact}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{sin}\:\mathrm{9}°\:=\:... \\ $$

Question Number 10746 Answers: 1 Comments: 0

i)express the function f(θ)=sinθ + cosθ in the form rsin(θ+α), r>0 and 0≤θ≤≤(π/2) ii)hence find the maximum value of f and the smallest non−negative value of θ at which it occurs.

$$\left.{i}\right){express}\:{the}\:{function}\:{f}\left(\theta\right)={sin}\theta\:+\:{cos}\theta\:{in}\:{the}\:{form}\:{rsin}\left(\theta+\alpha\right),\:{r}>\mathrm{0}\:{and}\:\mathrm{0}\leqslant\theta\leqslant\leqslant\frac{\pi}{\mathrm{2}} \\ $$$$\left.{ii}\right){hence}\:{find}\:{the}\:{maximum}\:{value}\:{of}\:{f}\:{and} \\ $$$${the}\:{smallest}\:{non}−{negative}\:{value}\:{of}\:\theta\:{at}\:{which}\:{it}\:{occurs}. \\ $$

Question Number 10744 Answers: 1 Comments: 0

hence or otherwise,solve the equation ((cosec θ)/(cosec θ−sin θ))=(4/3) for 0≤θ≤2Π

$${hence}\:{or}\:{otherwise},{solve}\:{the}\:{equation}\:\frac{\mathrm{cosec}\:\theta}{\mathrm{cosec}\:\theta−\mathrm{sin}\:\theta}=\frac{\mathrm{4}}{\mathrm{3}}\:{for}\:\mathrm{0}\leqslant\theta\leqslant\mathrm{2}\Pi \\ $$

Question Number 10743 Answers: 1 Comments: 2

show that sec^2 θ=((cosec θ)/(cosec θ−sin ))

$${show}\:{that}\:\mathrm{sec}\:^{\mathrm{2}} \theta=\frac{\mathrm{cosec}\:\theta}{\mathrm{cosec}\:\theta−\mathrm{sin}\:} \\ $$

Question Number 10742 Answers: 2 Comments: 0

let the roots of the equation2x^3 −5x^2 +4x+6=0 be α,β and γ. i)state the values of α+β+γ, αβ+αγ+βγ and αβγ. ii)hence or otherwise determine an equation with integer coefficients which as roots (1/α^(2 ) ), (1/β^2 ) , and (1/γ^2 )

$${let}\:{the}\:{roots}\:{of}\:{the}\:{equation}\mathrm{2}{x}^{\mathrm{3}} −\mathrm{5}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{6}=\mathrm{0} \\ $$$${be}\:\alpha,\beta\:{and}\:\gamma. \\ $$$$\left.{i}\right){state}\:{the}\:{values}\:{of}\:\alpha+\beta+\gamma,\:\alpha\beta+\alpha\gamma+\beta\gamma\:{and}\:\alpha\beta\gamma. \\ $$$$\left.{ii}\right){hence}\:{or}\:{otherwise}\:{determine}\:{an}\:{equation}\:{with}\:{integer}\:{coefficients}\:{which}\:{as}\:{roots}\:\frac{\mathrm{1}}{\alpha^{\mathrm{2}\:} },\:\frac{\mathrm{1}}{\beta^{\mathrm{2}} }\:,\:{and}\:\frac{\mathrm{1}}{\gamma^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 10741 Answers: 1 Comments: 0

a function f is defined by f(x)= ((x+3)/(x−1)), x not equal to 1.determine whether f is bijective,that is,both one to one and onto

$${a}\:{function}\:{f}\:{is}\:{defined}\:{by}\:{f}\left({x}\right)=\:\frac{{x}+\mathrm{3}}{{x}−\mathrm{1}},\:{x}\:{not}\:{equal}\:{to}\:\mathrm{1}.{determine}\:{whether}\:{f}\:{is}\:{bijective},{that}\:{is},{both}\:{one}\:{to}\:{one}\:{and}\:{onto} \\ $$$$ \\ $$

Question Number 10736 Answers: 1 Comments: 0

A tennis ball is thrown vertically upward with an initial velocity of 50m/s, when the ball return to the point of projection it renounce with the velocity of (2/3) of the velocity. Calculate the heigth after renounce.

$$\mathrm{A}\:\mathrm{tennis}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{vertically}\:\mathrm{upward}\:\mathrm{with}\:\mathrm{an}\:\mathrm{initial}\:\mathrm{velocity}\:\mathrm{of} \\ $$$$\mathrm{50m}/\mathrm{s},\:\mathrm{when}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{return}\:\mathrm{to}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{projection}\:\mathrm{it}\:\mathrm{renounce}\: \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:\:\frac{\mathrm{2}}{\mathrm{3}}\:\mathrm{of}\:\mathrm{the}\:\mathrm{velocity}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{heigth}\:\mathrm{after}\:\mathrm{renounce}. \\ $$

Question Number 10734 Answers: 2 Comments: 0

f(x−1)+f(x+1)=2x^2 +6 ⇒f(x)=?

$${f}\left({x}−\mathrm{1}\right)+{f}\left({x}+\mathrm{1}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{6}\:\Rightarrow{f}\left({x}\right)=? \\ $$

Question Number 10728 Answers: 1 Comments: 0

A pipe closed at one end emits sound wave of frequency 440 Hz. (i) Determine the length of the air column (ii) Frequencies of the second harmonic and the third harmonic Velocity of sound is 330m/s Refractive index of glass is 1.50, that of air is 1.00, and that of water is 1.33

$$\mathrm{A}\:\mathrm{pipe}\:\mathrm{closed}\:\mathrm{at}\:\mathrm{one}\:\mathrm{end}\:\mathrm{emits}\:\mathrm{sound}\:\mathrm{wave}\:\mathrm{of}\:\mathrm{frequency}\:\mathrm{440}\:\mathrm{Hz}.\: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{air}\:\mathrm{column} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Frequencies}\:\mathrm{of}\:\mathrm{the}\:\mathrm{second}\:\mathrm{harmonic}\:\mathrm{and}\:\mathrm{the}\:\mathrm{third}\:\mathrm{harmonic} \\ $$$$\mathrm{Velocity}\:\mathrm{of}\:\mathrm{sound}\:\mathrm{is}\:\mathrm{330m}/\mathrm{s} \\ $$$$\mathrm{Refractive}\:\mathrm{index}\:\mathrm{of}\:\mathrm{glass}\:\mathrm{is}\:\mathrm{1}.\mathrm{50},\:\mathrm{that}\:\mathrm{of}\:\:\mathrm{air}\:\mathrm{is}\:\mathrm{1}.\mathrm{00},\: \\ $$$$\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{water}\:\mathrm{is}\:\mathrm{1}.\mathrm{33} \\ $$

Question Number 10721 Answers: 1 Comments: 0

Q.1 x^2 −y^2 −i(2x+y)=2i Q.2 (2+3i)x^2 −(3−x)y=2x−3y

$${Q}.\mathrm{1}\:\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{i}\left(\mathrm{2}{x}+{y}\right)=\mathrm{2}{i} \\ $$$${Q}.\mathrm{2}\:\:\:\left(\mathrm{2}+\mathrm{3}{i}\right){x}^{\mathrm{2}} −\left(\mathrm{3}−{x}\right){y}=\mathrm{2}{x}−\mathrm{3}{y} \\ $$

Question Number 10714 Answers: 0 Comments: 0

Given the following information regarding the follwing distribution : n = 5, x^− = 10, y^− = 20, Σ(x − 4)^2 = 100, Σ(y − 10)^2 = 160, Σ(x − 4)(y − 10) = 80 (i) Find the two regression coefficient and (ii) Calculate correlation coefficient

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{following}\:\mathrm{information}\:\mathrm{regarding}\:\mathrm{the}\:\mathrm{follwing}\: \\ $$$$\mathrm{distribution}\::\:\: \\ $$$$\mathrm{n}\:=\:\mathrm{5},\:\:\:\:\overset{−} {\mathrm{x}}\:=\:\mathrm{10},\:\:\:\overset{−} {\mathrm{y}}\:=\:\mathrm{20},\:\:\:\Sigma\left(\mathrm{x}\:−\:\mathrm{4}\right)^{\mathrm{2}} \:=\:\mathrm{100},\:\:\Sigma\left(\mathrm{y}\:−\:\mathrm{10}\right)^{\mathrm{2}} \:=\:\mathrm{160}, \\ $$$$\Sigma\left(\mathrm{x}\:−\:\mathrm{4}\right)\left(\mathrm{y}\:−\:\mathrm{10}\right)\:=\:\mathrm{80} \\ $$$$\left(\mathrm{i}\right) \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{two}\:\mathrm{regression}\:\mathrm{coefficient}\:\mathrm{and} \\ $$$$\left(\mathrm{ii}\right) \\ $$$$\mathrm{Calculate}\:\mathrm{correlation}\:\mathrm{coefficient} \\ $$

Question Number 10713 Answers: 2 Comments: 0

y = 5sin(2000πt − 0.4x), calculate the wavelength

$$\mathrm{y}\:=\:\mathrm{5sin}\left(\mathrm{2000}\pi\mathrm{t}\:−\:\mathrm{0}.\mathrm{4x}\right),\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{wavelength} \\ $$

Question Number 10712 Answers: 2 Comments: 0

If y = x^x , find (dy/dx)

$$\mathrm{If}\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{x}} ,\:\:\mathrm{find}\:\:\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$

Question Number 10711 Answers: 1 Comments: 0

If y = x^x^x , find (dy/dx)

$$\mathrm{If}\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \:,\:\:\mathrm{find}\:\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$

Question Number 10705 Answers: 0 Comments: 0

A pipe closed at one end emits sound waves of frequency 440Hz. Determine (i) The lenght of air column (ii) The frequencies of the first and second overtones.

$$\mathrm{A}\:\mathrm{pipe}\:\mathrm{closed}\:\mathrm{at}\:\mathrm{one}\:\mathrm{end}\:\mathrm{emits}\:\mathrm{sound}\:\mathrm{waves}\:\mathrm{of}\:\mathrm{frequency}\:\mathrm{440Hz}.\: \\ $$$$\mathrm{Determine}\:\left(\mathrm{i}\right)\:\mathrm{The}\:\mathrm{lenght}\:\mathrm{of}\:\mathrm{air}\:\mathrm{column}\:\: \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{The}\:\mathrm{frequencies}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{and}\:\mathrm{second}\:\mathrm{overtones}. \\ $$

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