Question and Answers Forum

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 94080 Answers: 1 Comments: 0

3.(a) Find the complex number z which satisfy the equation z^3 = 8i , giving your answer in the form a + bi where a and b are real.

$$\mathrm{3}.\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{number}\:{z}\:\mathrm{which}\:\mathrm{satisfy}\: \\ $$$$\mathrm{the}\:\mathrm{equation}\:{z}^{\mathrm{3}} \:=\:\mathrm{8}{i}\:,\:\mathrm{giving}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{form}\:{a}\:+\:{bi}\:\mathrm{where}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{real}. \\ $$

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 94071 Answers: 1 Comments: 2

Question Number 94125 Answers: 1 Comments: 0

Given a function H(x) = ∣sin x+cos x∣ + (√2) cos x with x ∈ [ 0, 2π ] find H(x)_(max) and H(x)_(min)

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\: \\ $$$$\mathrm{H}\left(\mathrm{x}\right)\:=\:\mid\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\mid\:+\:\sqrt{\mathrm{2}}\:\mathrm{cos}\:\mathrm{x} \\ $$$$\mathrm{with}\:\mathrm{x}\:\in\:\left[\:\mathrm{0},\:\mathrm{2}\pi\:\right]\: \\ $$$$\mathrm{find}\:\mathrm{H}\left(\mathrm{x}\right)_{\mathrm{max}} \:\mathrm{and}\:\mathrm{H}\left(\mathrm{x}\right)_{\mathrm{min}} \\ $$

Question Number 94124 Answers: 0 Comments: 3

20+a=a cosh(((75)/a)) a=?

$$\mathrm{20}+{a}={a}\:{cosh}\left(\frac{\mathrm{75}}{{a}}\right) \\ $$$${a}=? \\ $$

Question Number 94123 Answers: 0 Comments: 0

prove that ∫_0 ^∞ ((sin^(2n) (x))/x^2 )d=∫_0 ^∞ ((sin^(2n−1) (x))/x)dx

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{sin}^{\mathrm{2}{n}} \left({x}\right)}{{x}^{\mathrm{2}} }{d}=\int_{\mathrm{0}} ^{\infty} \frac{{sin}^{\mathrm{2}{n}−\mathrm{1}} \left({x}\right)}{{x}}{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 94114 Answers: 0 Comments: 1

Find ∫_( 1) ^( ∞) ((sin^2 x)/x^2 ) dx

$${Find}\:\:\:\underset{\:\mathrm{1}} {\int}\overset{\:\infty} {\:}\:\:\frac{\mathrm{sin}^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} }\:\:{dx}\:\: \\ $$

Question Number 94110 Answers: 1 Comments: 2

Question Number 94025 Answers: 0 Comments: 5

LCM(a,(3/5)a)=3a ∧ HCF(a,(3/5)a)=(1/5)a a=?

$$\mathrm{LCM}\left({a},\frac{\mathrm{3}}{\mathrm{5}}{a}\right)=\mathrm{3}{a}\:\wedge\:\mathrm{HCF}\left({a},\frac{\mathrm{3}}{\mathrm{5}}{a}\right)=\frac{\mathrm{1}}{\mathrm{5}}{a} \\ $$$${a}=? \\ $$

Question Number 94020 Answers: 0 Comments: 4

Question Number 94383 Answers: 4 Comments: 0

Integrate: (i).∫ (1/(1+x^4 ))dx (ii).∫_β ^( α) (√((x−𝛂)(β−x))) dx (iii). ∫_0 ^( 1) ∫_0 ^( x^2 ) e^(y/x) dx dy

$$\boldsymbol{\mathrm{Integrate}}: \\ $$$$\:\:\left(\boldsymbol{\mathrm{i}}\right).\int\:\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{4}} }\boldsymbol{\mathrm{dx}} \\ $$$$\:\left(\boldsymbol{\mathrm{ii}}\right).\int_{\beta} ^{\:\alpha} \sqrt{\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\alpha}\right)\left(\beta−\boldsymbol{\mathrm{x}}\right)}\:\:\boldsymbol{\mathrm{dx}} \\ $$$$\:\left(\boldsymbol{\mathrm{iii}}\right).\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\int_{\mathrm{0}} ^{\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{y}}/\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{dx}}\:\boldsymbol{\mathrm{dy}} \\ $$

Question Number 94098 Answers: 1 Comments: 1

Integrate: ∫ (( dx)/(a sin x+ b cos x))

$$\:\:\mathrm{Integrate}: \\ $$$$\:\:\int\:\frac{\:\:\mathrm{dx}}{\mathrm{a}\:\mathrm{sin}\:\mathrm{x}+\:\mathrm{b}\:\mathrm{cos}\:\mathrm{x}} \\ $$

Question Number 94097 Answers: 2 Comments: 1

find all integers n for which 13 ∣4(n^2 +1).

$$\mathrm{find}\:\mathrm{all}\:\mathrm{integers}\:{n}\:\:\mathrm{for}\:\mathrm{which}\:\:\mathrm{13}\:\mid\mathrm{4}\left({n}^{\mathrm{2}} +\mathrm{1}\right). \\ $$

Question Number 94119 Answers: 0 Comments: 1

∫ cot^(−1) ((√x)) dx

$$\int\:\mathrm{cot}^{−\mathrm{1}} \left(\sqrt{\mathrm{x}}\right)\:\mathrm{dx}\: \\ $$

Question Number 94096 Answers: 1 Comments: 1

The result of adding the odd natural numbers is: 1 = 1 1 + 3=4 1 + 3+ 5 = 9 1 + 3 + 5 +7 = 16 1 + 3 + 5 + 7+9 = 25 show that from this result, Σ_(i=1) ^n (2i−1) = n^2 .

$$\mathrm{The}\:\mathrm{result}\:\mathrm{of}\:\mathrm{adding}\:\mathrm{the}\:\mathrm{odd}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{is}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:=\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{1}\:+\:\mathrm{3}=\mathrm{4} \\ $$$$\:\:\mathrm{1}\:+\:\mathrm{3}+\:\mathrm{5}\:=\:\mathrm{9} \\ $$$$\mathrm{1}\:+\:\mathrm{3}\:+\:\mathrm{5}\:+\mathrm{7}\:=\:\mathrm{16} \\ $$$$\mathrm{1}\:+\:\mathrm{3}\:+\:\mathrm{5}\:+\:\mathrm{7}+\mathrm{9}\:=\:\mathrm{25} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:\mathrm{from}\:\mathrm{this}\:\mathrm{result},\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{2}{i}−\mathrm{1}\right)\:=\:{n}^{\mathrm{2}} . \\ $$

Question Number 94095 Answers: 0 Comments: 0

How many subgroups do Z_3 ⊕Z_(16 ) has? Justify.

$$\mathrm{How}\:\mathrm{many}\:\mathrm{subgroups}\:\mathrm{do}\:\mathrm{Z}_{\mathrm{3}} \oplus\mathrm{Z}_{\mathrm{16}\:} \:\mathrm{has}?\:\mathrm{Justify}. \\ $$

Question Number 94093 Answers: 0 Comments: 0

evaluate the inequality for n≥2 ((π/2)−(1/n))(1/(n)^(1/n) )<∫_(1/n) ^(π/2) ((sin(t)))^(1/n) dt

$${evaluate}\:{the}\:{inequality}\:{for}\:{n}\geqslant\mathrm{2} \\ $$$$\left(\frac{\pi}{\mathrm{2}}−\frac{\mathrm{1}}{{n}}\right)\frac{\mathrm{1}}{\sqrt[{{n}}]{{n}}}<\int_{\frac{\mathrm{1}}{{n}}} ^{\frac{\pi}{\mathrm{2}}} \sqrt[{{n}}]{{sin}\left({t}\right)}{dt} \\ $$

Question Number 94117 Answers: 0 Comments: 8

Another update available to provide ability to bookmark.

$$\mathrm{Another}\:\mathrm{update}\:\mathrm{available}\:\mathrm{to} \\ $$$$\mathrm{provide}\:\mathrm{ability}\:\mathrm{to}\:\mathrm{bookmark}. \\ $$

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