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AllQuestion and Answers: Page 1907

Question Number 10820 Answers: 1 Comments: 0

Question Number 10836 Answers: 1 Comments: 0

solve cos2θ−3cosθ=1 for o≤θ≤2π

$${solve}\:{cos}\mathrm{2}\theta−\mathrm{3}{cos}\theta=\mathrm{1} \\ $$$${for}\:{o}\leqslant\theta\leqslant\mathrm{2}\pi \\ $$

Question Number 10815 Answers: 0 Comments: 0

Evalute ∫(x^(2 ) + 9)^9 dx .

$${Evalute}\:\:\int\left({x}^{\mathrm{2}\:} \:\:+\:\mathrm{9}\right)^{\mathrm{9}} \:{dx}\:. \\ $$

Question Number 10807 Answers: 2 Comments: 2

Question Number 10795 Answers: 1 Comments: 1

by use sketching determine the range or(ranges) of the value x can take for each of the following inqualities (i) 3x^2 −19x−6≤0 (ii)2x^2 −5x−3≥0

$$\mathrm{by}\:\mathrm{use}\:\mathrm{sketching}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{range} \\ $$$$\mathrm{or}\left(\mathrm{ranges}\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{value}\:\mathrm{x}\:\mathrm{can}\:\mathrm{take}\:\mathrm{for} \\ $$$$\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inqualities} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{3x}^{\mathrm{2}} −\mathrm{19x}−\mathrm{6}\leqslant\mathrm{0} \\ $$$$\left(\mathrm{ii}\right)\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}−\mathrm{3}\geqslant\mathrm{0} \\ $$

Question Number 10794 Answers: 0 Comments: 1

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

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

Question Number 10793 Answers: 1 Comments: 0

let A = determinant ((4,(4k),k),(0,k,(4k)),(0,0,4)) if det(A^2 )=16 then ∣k∣ is?

$$\mathrm{let}\:\:\mathrm{A}\:=\begin{vmatrix}{\mathrm{4}}&{\mathrm{4k}}&{\mathrm{k}}\\{\mathrm{0}}&{\mathrm{k}}&{\mathrm{4k}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{4}}\end{vmatrix}\:\mathrm{if}\:\mathrm{det}\left(\mathrm{A}^{\mathrm{2}} \right)=\mathrm{16} \\ $$$$\mathrm{then}\:\mid\mathrm{k}\mid\:\mathrm{is}? \\ $$

Question Number 10790 Answers: 1 Comments: 0

∫_0 ^(π/2) (√(sin x)) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{sin}\:{x}}\:{dx} \\ $$

Question Number 10789 Answers: 1 Comments: 0

(2+3i)x^2 −(3−2i)y=2x−3y+5i

$$\left(\mathrm{2}+\mathrm{3}{i}\right){x}^{\mathrm{2}} −\left(\mathrm{3}−\mathrm{2}{i}\right){y}=\mathrm{2}{x}−\mathrm{3}{y}+\mathrm{5}{i} \\ $$

Question Number 10788 Answers: 1 Comments: 0

factorise the expression sin4x−sinx

$${factorise}\:{the}\:{expression}\:{sin}\mathrm{4}{x}−{sinx} \\ $$

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 )

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

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