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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.

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

Question Number 10704 Answers: 1 Comments: 0

If the bulk modulus for water is 2.28 × 10^9 Nm^(−2) , determine the time it takes for a sound to travel a distant of 110m in water of density 1000kgm^(−3) .

$$\mathrm{If}\:\mathrm{the}\:\mathrm{bulk}\:\mathrm{modulus}\:\mathrm{for}\:\mathrm{water}\:\mathrm{is}\:\mathrm{2}.\mathrm{28}\:×\:\mathrm{10}^{\mathrm{9}} \mathrm{Nm}^{−\mathrm{2}} \:,\:\mathrm{determine}\:\mathrm{the}\: \\ $$$$\mathrm{time}\:\mathrm{it}\:\mathrm{takes}\:\mathrm{for}\:\mathrm{a}\:\mathrm{sound}\:\mathrm{to}\:\mathrm{travel}\:\mathrm{a}\:\mathrm{distant}\:\mathrm{of}\:\mathrm{110m}\:\mathrm{in}\:\mathrm{water}\:\mathrm{of}\: \\ $$$$\mathrm{density}\:\mathrm{1000kgm}^{−\mathrm{3}} . \\ $$

Question Number 10696 Answers: 1 Comments: 0

given that tan2x=(1/4)and that angle x is acute, calculate,without using a calculator the value of these. (a) cos2x (b) sinx

$${given}\:{that}\:{tan}\mathrm{2}{x}=\frac{\mathrm{1}}{\mathrm{4}}{and}\:{that}\:{angle}\:{x}\:{is}\:{acute},\:{calculate},{without}\:{using}\:{a}\:{calculator}\:{the}\:{value}\:{of}\:{these}. \\ $$$$\left({a}\right)\:{cos}\mathrm{2}{x} \\ $$$$\left({b}\right)\:{sinx} \\ $$$$ \\ $$

Question Number 10695 Answers: 2 Comments: 0

prove that tanx−cotx=−2cot2x

$${prove}\:{that}\:{tanx}−{cotx}=−\mathrm{2}{cot}\mathrm{2}{x} \\ $$

Question Number 10694 Answers: 1 Comments: 0

Question Number 10693 Answers: 1 Comments: 0

f(x)=((5x−1)/4) , f(a)+f(b)=(9/2) ⇒a+b=?

$${f}\left({x}\right)=\frac{\mathrm{5}{x}−\mathrm{1}}{\mathrm{4}}\:\:,\:{f}\left({a}\right)+{f}\left({b}\right)=\frac{\mathrm{9}}{\mathrm{2}} \\ $$$$\Rightarrow{a}+{b}=? \\ $$

Question Number 10692 Answers: 1 Comments: 0

f(x+y)=f(x)×f(y) f(16)=8 ⇒f(64)

$${f}\left({x}+{y}\right)={f}\left({x}\right)×{f}\left({y}\right) \\ $$$${f}\left(\mathrm{16}\right)=\mathrm{8}\:\Rightarrow{f}\left(\mathrm{64}\right) \\ $$

Question Number 10690 Answers: 1 Comments: 0

Question Number 10687 Answers: 1 Comments: 0

Abbey and mary carry a uniform log of length 10m and weight 100N. Abbey is 1m from one end and mary is 2m from the other end. what weight does abbey and mary support.

$$\mathrm{Abbey}\:\mathrm{and}\:\mathrm{mary}\:\mathrm{carry}\:\mathrm{a}\:\mathrm{uniform}\:\mathrm{log}\:\mathrm{of}\:\mathrm{length}\:\mathrm{10m}\:\mathrm{and}\:\mathrm{weight}\:\mathrm{100N}. \\ $$$$\mathrm{Abbey}\:\mathrm{is}\:\mathrm{1m}\:\mathrm{from}\:\mathrm{one}\:\mathrm{end}\:\mathrm{and}\:\mathrm{mary}\:\mathrm{is}\:\mathrm{2m}\:\mathrm{from}\:\mathrm{the}\:\mathrm{other}\:\mathrm{end}.\:\mathrm{what} \\ $$$$\mathrm{weight}\:\mathrm{does}\:\mathrm{abbey}\:\mathrm{and}\:\mathrm{mary}\:\mathrm{support}. \\ $$

Question Number 10675 Answers: 1 Comments: 1

given that x,y∈R solve. (1) (x+2y)+i(2x−3y)=5−4i (2) (x+iy)×(7−5i)=9+4i

$${given}\:{that}\:{x},{y}\in{R}\:{solve}. \\ $$$$\left(\mathrm{1}\right)\:\left({x}+\mathrm{2}{y}\right)+{i}\left(\mathrm{2}{x}−\mathrm{3}{y}\right)=\mathrm{5}−\mathrm{4}{i} \\ $$$$\left(\mathrm{2}\right)\:\left({x}+{iy}\right)×\left(\mathrm{7}−\mathrm{5}{i}\right)=\mathrm{9}+\mathrm{4}{i} \\ $$

Question Number 10671 Answers: 2 Comments: 0

Σ_(n = 1) ^∞ 5((1/4))^(n − 1)

$$\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{5}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{n}\:−\:\mathrm{1}} \\ $$

Question Number 10670 Answers: 0 Comments: 0

Prove that: ζ(s)=Σ_(n=1) ^∞ n^(−s) =Π_(p∈P) ^∞ (1−p^(−s) )^(−1)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\zeta\left({s}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{n}^{−{s}} =\underset{{p}\in\mathbb{P}} {\overset{\infty} {\prod}}\left(\mathrm{1}−{p}^{−{s}} \right)^{−\mathrm{1}} \\ $$

Question Number 10669 Answers: 0 Comments: 0

∫(√(a^2 −cos^2 x ))dx=? (a≥1)

$$\int\sqrt{{a}^{\mathrm{2}} −\mathrm{cos}^{\mathrm{2}} \:{x}\:}{dx}=?\:\:\:\left({a}\geqslant\mathrm{1}\right) \\ $$

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