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Question Number 111159 Answers: 2 Comments: 0

z is a complex number with Re(z) , Im(z)∈N. Determine z if z.z^− =1000

$${z}\:{is}\:{a}\:{complex}\:{number}\:{with}\: \\ $$$${Re}\left({z}\right)\:,\:{Im}\left({z}\right)\in\mathbb{N}. \\ $$$${Determine}\:{z}\:\:{if} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}.\overset{−} {{z}}=\mathrm{1000} \\ $$

Question Number 111157 Answers: 2 Comments: 0

question proposed MN july 1970 ∫_0 ^(π/4) tanx(ln(1+tan^2 x))dx my solution followed

$${question}\:{proposed}\:{MN}\:{july}\:\mathrm{1970} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{tan}{x}\left(\mathrm{ln}\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} {x}\right)\right){dx} \\ $$$${my}\:{solution}\:{followed} \\ $$

Question Number 111155 Answers: 1 Comments: 0

Find the number of rational numbers r, 0<r<1, such that when r is written as a fraction in lowest term. The numerator and the denominator have a sum of 1000.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers} \\ $$$$\mathrm{r},\:\mathrm{0}<\mathrm{r}<\mathrm{1},\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{r}\:\mathrm{is}\:\mathrm{written} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{lowest}\:\mathrm{term}.\:\mathrm{The} \\ $$$$\mathrm{numerator}\:\mathrm{and}\:\mathrm{the}\:\mathrm{denominator} \\ $$$$\mathrm{have}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{1000}. \\ $$

Question Number 111154 Answers: 0 Comments: 2

Version 2.202 adds calculator. place cursor on a line which you want to evalute. Select calculate from menu in 3 vertical dots at top. Calculator accepts complex numbers and will return result in complex when needed. principal values are return for asin, acos etc. Angle are in radian unless ° symbol is used e.g. (30+60i)° Result will be added in immediate next line can be edited to form another expression for further computation. sin^(−1) 2 (1.570796 − 1.316958i)

$$\mathrm{Version}\:\mathrm{2}.\mathrm{202}\:\mathrm{adds}\:\mathrm{calculator}. \\ $$$$\mathrm{place}\:\mathrm{cursor}\:\mathrm{on}\:\mathrm{a}\:\mathrm{line}\:\mathrm{which}\:\mathrm{you} \\ $$$$\mathrm{want}\:\mathrm{to}\:\mathrm{evalute}.\: \\ $$$$\mathrm{Select}\:\mathrm{calculate}\:\mathrm{from}\:\mathrm{menu}\:\mathrm{in}\:\mathrm{3}\:\mathrm{vertical} \\ $$$$\mathrm{dots}\:\mathrm{at}\:\mathrm{top}. \\ $$$$\mathrm{Calculator}\:\mathrm{accepts}\:\mathrm{complex}\:\mathrm{numbers} \\ $$$$\mathrm{and}\:\mathrm{will}\:\mathrm{return}\:\mathrm{result}\:\mathrm{in}\:\mathrm{complex} \\ $$$$\mathrm{when}\:\mathrm{needed}. \\ $$$$\mathrm{principal}\:\mathrm{values}\:\mathrm{are}\:\mathrm{return}\:\mathrm{for} \\ $$$$\mathrm{asin},\:\mathrm{acos}\:\mathrm{etc}. \\ $$$$\mathrm{Angle}\:\mathrm{are}\:\mathrm{in}\:\mathrm{radian}\:\mathrm{unless}\:°\:\mathrm{symbol} \\ $$$$\mathrm{is}\:\mathrm{used}\:\mathrm{e}.\mathrm{g}.\:\left(\mathrm{30}+\mathrm{60i}\right)° \\ $$$$\mathrm{Result}\:\mathrm{will}\:\mathrm{be}\:\mathrm{added}\:\mathrm{in}\:\mathrm{immediate} \\ $$$$\mathrm{next}\:\mathrm{line}\:\mathrm{can}\:\mathrm{be}\:\mathrm{edited}\:\mathrm{to}\:\mathrm{form} \\ $$$$\mathrm{another}\:\mathrm{expression}\:\mathrm{for}\:\mathrm{further} \\ $$$$\mathrm{computation}. \\ $$$$\mathrm{sin}^{−\mathrm{1}} \mathrm{2} \\ $$$$\left(\mathrm{1}.\mathrm{570796}\:−\:\mathrm{1}.\mathrm{316958i}\right) \\ $$

Question Number 111276 Answers: 1 Comments: 0

A coin that comes up head with probability p and tail with probability 1−p independently of each flip is flipped five times. The probability of two heads and three tails is equal to (1/7) of the probability of three heads and two tails. Let p=(x/y), where gcd(x,y) =1 . Find x+y.

$$\mathrm{A}\:\mathrm{coin}\:\mathrm{that}\:\mathrm{comes}\:\mathrm{up}\:\mathrm{head}\:\mathrm{with} \\ $$$$\mathrm{probability}\:\mathrm{p}\:\mathrm{and}\:\mathrm{tail}\:\mathrm{with}\:\mathrm{probability} \\ $$$$\mathrm{1}−\mathrm{p}\:\mathrm{independently}\:\mathrm{of}\:\mathrm{each}\:\mathrm{flip}\:\mathrm{is}\:\mathrm{flipped}\:\mathrm{five}\:\mathrm{times}. \\ $$$$\mathrm{The}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{two}\:\mathrm{heads}\:\mathrm{and}\:\mathrm{three}\:\mathrm{tails}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\frac{\mathrm{1}}{\mathrm{7}}\:\mathrm{of}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{three} \\ $$$$\mathrm{heads}\:\mathrm{and}\:\mathrm{two}\:\mathrm{tails}.\:\mathrm{Let}\:\mathrm{p}=\frac{\mathrm{x}}{\mathrm{y}},\:\mathrm{where} \\ $$$$\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)\:=\mathrm{1}\:.\:\mathrm{Find}\:\mathrm{x}+\mathrm{y}. \\ $$

Question Number 111149 Answers: 0 Comments: 4

Let f_0 (x) = (1/(1−x)) and f_n (x) =f_0 (f_(n−1) (x)), n=1,2,3,... Evaluate f_(2018) (2018)

$$\mathrm{Let}\:\mathrm{f}_{\mathrm{0}} \left(\mathrm{x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{x}}\:\mathrm{and}\:\mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right) \\ $$$$=\mathrm{f}_{\mathrm{0}} \left(\mathrm{f}_{\mathrm{n}−\mathrm{1}} \left(\mathrm{x}\right)\right),\:\mathrm{n}=\mathrm{1},\mathrm{2},\mathrm{3},...\:\mathrm{Evaluate} \\ $$$$\mathrm{f}_{\mathrm{2018}} \left(\mathrm{2018}\right) \\ $$

Question Number 111279 Answers: 1 Comments: 0

Triangle ABC has AB=2∙AC. Let D and E be on AB and BC respectively such that ∠BAE =∠ACD. Let F be the intersections of segments AE and CD, and suppose that △CFE is equilateral. What is ∠ACB?

$$\mathrm{Triangle}\:\mathrm{ABC}\:\mathrm{has}\:\mathrm{AB}=\mathrm{2}\centerdot\mathrm{AC}.\:\mathrm{Let} \\ $$$$\mathrm{D}\:\mathrm{and}\:\mathrm{E}\:\mathrm{be}\:\mathrm{on}\:\mathrm{AB}\:\mathrm{and}\:\mathrm{BC} \\ $$$$\mathrm{respectively}\:\mathrm{such}\:\mathrm{that}\:\angle\mathrm{BAE} \\ $$$$=\angle\mathrm{ACD}.\:\mathrm{Let}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{intersections}\:\mathrm{of} \\ $$$$\mathrm{segments}\:\mathrm{AE}\:\mathrm{and}\:\mathrm{CD},\:\mathrm{and}\:\mathrm{suppose} \\ $$$$\mathrm{that}\:\bigtriangleup\mathrm{CFE}\:\mathrm{is}\:\mathrm{equilateral}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\angle\mathrm{ACB}? \\ $$

Question Number 111147 Answers: 0 Comments: 0

Question Number 111477 Answers: 1 Comments: 0

Question Number 111140 Answers: 0 Comments: 0

Definite integral MATHEMATICS Full Marks : 40

$$\: \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{D}}\mathrm{efinite}\:\boldsymbol{\mathrm{integral}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{MATHEMATICS}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Full}}\:\:\boldsymbol{\mathrm{Marks}}\::\:\mathrm{40} \\ $$

Question Number 111135 Answers: 2 Comments: 0

Question Number 111134 Answers: 0 Comments: 0

Question Number 111132 Answers: 1 Comments: 0

prove by mathematical induction ⇒ 7^n −(3n+4)×4^(n−1) divided by 9

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\Rightarrow\:\mathrm{7}^{\mathrm{n}} −\left(\mathrm{3n}+\mathrm{4}\right)×\mathrm{4}^{\mathrm{n}−\mathrm{1}} \:\mathrm{divided}\:\mathrm{by}\:\mathrm{9} \\ $$

Question Number 111125 Answers: 1 Comments: 0

Question Number 111152 Answers: 0 Comments: 0

Question Number 111282 Answers: 1 Comments: 0

A particle of mass 500g is placed on a plane inclined at an angle 30° to the horizontal. What force is (i) acting parallel to the plane (ii) acting horizontally is required to hold the particle at rest (g=10m/s^2 )

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{500g}\:\mathrm{is}\:\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\: \\ $$$$\mathrm{plane}\:\mathrm{inclined}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{30}° \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{What}\:\mathrm{force}\:\mathrm{is} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{acting}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{acting}\:\mathrm{horizontally}\:\mathrm{is}\:\mathrm{required}\:\mathrm{to} \\ $$$$\mathrm{hold}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{at}\:\mathrm{rest}\:\left(\mathrm{g}=\mathrm{10m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$

Question Number 111278 Answers: 1 Comments: 0

Towns A,B,C and D are located on the vertices of a square whose area is 1000km^2 . There is a straight line highway passing through the centre of the square but not through any of the towns. Find the sum of the squares of the distances of the towns to the highway.

$$\mathrm{Towns}\:\mathrm{A},\mathrm{B},\mathrm{C}\:\mathrm{and}\:\mathrm{D}\:\mathrm{are}\:\mathrm{located}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{whose}\:\mathrm{area}\:\mathrm{is} \\ $$$$\mathrm{1000km}^{\mathrm{2}} .\:\mathrm{There}\:\mathrm{is}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\mathrm{highway}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{square}\:\mathrm{but}\:\mathrm{not}\:\mathrm{through}\:\mathrm{any}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{towns}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{squares} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{distances}\:\mathrm{of}\:\mathrm{the}\:\mathrm{towns}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{highway}. \\ $$

Question Number 111114 Answers: 2 Comments: 0

4 sin 36° cos 72° sin 108° ?

$$\mathrm{4}\:\mathrm{sin}\:\mathrm{36}°\:\mathrm{cos}\:\mathrm{72}°\:\mathrm{sin}\:\mathrm{108}°\:?\: \\ $$

Question Number 111109 Answers: 0 Comments: 0