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Question Number 15202 Answers: 0 Comments: 0

if z = x + jy , find the equations of the two loci defined by: (a) ∣z − 4∣ = 3 (b) arg(z + 2) = (π/6)

$$\mathrm{if}\:\mathrm{z}\:=\:\mathrm{x}\:+\:\mathrm{jy}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{loci}\:\mathrm{defined}\:\mathrm{by}: \\ $$$$\left(\mathrm{a}\right)\:\mid\mathrm{z}\:−\:\mathrm{4}\mid\:=\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{b}\right)\:\mathrm{arg}\left(\mathrm{z}\:+\:\mathrm{2}\right)\:=\:\frac{\pi}{\mathrm{6}} \\ $$

Question Number 15201 Answers: 0 Comments: 0

If z = x + jy , where x and y are real, show that the locus ∣((z − 2)/(z + 1))∣ = 2 is a circle and determine its centre and radius.

$$\mathrm{If}\:\:\mathrm{z}\:=\:\mathrm{x}\:+\:\mathrm{jy}\:,\:\mathrm{where}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{real},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{locus}\:\:\mid\frac{\mathrm{z}\:−\:\mathrm{2}}{\mathrm{z}\:+\:\mathrm{1}}\mid\:=\:\mathrm{2}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\mathrm{and}\:\mathrm{determine}\:\mathrm{its}\:\mathrm{centre}\:\mathrm{and}\:\mathrm{radius}. \\ $$

Question Number 15195 Answers: 0 Comments: 3

lim_(x→∞) (((2x − 5)/(2x + 1)))^(x + 3)

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{2}{x}\:−\:\mathrm{5}}{\mathrm{2}{x}\:+\:\mathrm{1}}\right)^{{x}\:+\:\mathrm{3}} \\ $$

Question Number 14949 Answers: 0 Comments: 2

Find the largest prime factor of 203203. Anyone please suggest the method without calculators or log tables.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{prime}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{203203}. \\ $$$$\mathrm{Anyone}\:\mathrm{please}\:\mathrm{suggest}\:\mathrm{the}\:\mathrm{method} \\ $$$$\mathrm{without}\:\mathrm{calculators}\:\mathrm{or}\:\mathrm{log}\:\mathrm{tables}. \\ $$

Question Number 14940 Answers: 2 Comments: 12

For those who are interested in Geometry: A triangle has an area of 1 unit. Each of its sides is divided into 4 equal parts through 3 points. The first and the last point of each side will be connected with each other to form 2 inscribed triangles and these 2 triangles form a hexagon. Find the area of the hexagon. What is the result, if each side is equally divided into 5 parts, or generally into n parts?

$${For}\:{those}\:{who}\:{are}\:{interested}\:{in}\: \\ $$$${Geometry}:\: \\ $$$${A}\:{triangle}\:{has}\:{an}\:{area}\:{of}\:\mathrm{1}\:{unit}.\:{Each} \\ $$$${of}\:{its}\:{sides}\:{is}\:{divided}\:{into}\:\mathrm{4}\:{equal}\:{parts} \\ $$$${through}\:\mathrm{3}\:{points}.\:{The}\:{first}\:{and}\:{the}\:{last} \\ $$$${point}\:{of}\:{each}\:{side}\:{will}\:{be}\:{connected} \\ $$$${with}\:{each}\:{other}\:{to}\:{form}\:\mathrm{2}\:{inscribed} \\ $$$${triangles}\:{and}\:{these}\:\mathrm{2}\:{triangles}\:{form} \\ $$$${a}\:{hexagon}.\:{Find}\:{the}\:{area}\:{of}\:{the}\:{hexagon}. \\ $$$$ \\ $$$${What}\:{is}\:{the}\:{result},\:{if}\:{each}\:{side}\:{is} \\ $$$${equally}\:{divided}\:{into}\:\mathrm{5}\:{parts},\:{or} \\ $$$${generally}\:{into}\:{n}\:{parts}? \\ $$

Question Number 14939 Answers: 1 Comments: 0

A point moves in x-y plane according to the law x = 4 sin 6t and y = 4(1 − cos 6t). Find distance traversed by the particle in 5 seconds, when x and y are in metres.

$$\mathrm{A}\:\mathrm{point}\:\mathrm{moves}\:\mathrm{in}\:{x}-{y}\:\mathrm{plane}\:\mathrm{according} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{law}\:{x}\:=\:\mathrm{4}\:\mathrm{sin}\:\mathrm{6}{t}\:\mathrm{and} \\ $$$${y}\:=\:\mathrm{4}\left(\mathrm{1}\:−\:\mathrm{cos}\:\mathrm{6}{t}\right).\:\mathrm{Find}\:\mathrm{distance}\:\mathrm{traversed} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{in}\:\mathrm{5}\:\mathrm{seconds},\:\mathrm{when}\:{x}\:\mathrm{and} \\ $$$${y}\:\mathrm{are}\:\mathrm{in}\:\mathrm{metres}. \\ $$

Question Number 14938 Answers: 0 Comments: 0

Question Number 14923 Answers: 1 Comments: 0

A plane is inclined at an angle of 30° with horizontal. Find the component of a force F^→ = −10k^∧ N perpendicular to the plane. Given that z-direction is vertically upwards.

$$\mathrm{A}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{inclined}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{30}° \\ $$$$\mathrm{with}\:\mathrm{horizontal}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{component} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{force}\:\overset{\rightarrow} {{F}}\:=\:−\mathrm{10}\overset{\wedge} {{k}}\mathrm{N}\:\mathrm{perpendicular}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{plane}.\:\mathrm{Given}\:\mathrm{that}\:{z}-\mathrm{direction}\:\mathrm{is} \\ $$$$\mathrm{vertically}\:\mathrm{upwards}. \\ $$

Question Number 14922 Answers: 1 Comments: 2

The velocity of particle P due East is 4 m/s, and that of Q is 3 m/s due South. What is the velocity of P w.r.t. Q?

$$\mathrm{The}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{particle}\:{P}\:\mathrm{due}\:\mathrm{East}\:\mathrm{is} \\ $$$$\mathrm{4}\:\mathrm{m}/\mathrm{s},\:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:{Q}\:\mathrm{is}\:\mathrm{3}\:\mathrm{m}/\mathrm{s}\:\mathrm{due} \\ $$$$\mathrm{South}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:{P}\:\mathrm{w}.\mathrm{r}.\mathrm{t}. \\ $$$${Q}? \\ $$

Question Number 14921 Answers: 1 Comments: 2

A car travelling at 36 km/h due North turns West in 5 seconds and maintains the same speed. What is the acceleration of the car?

$$\mathrm{A}\:\mathrm{car}\:\mathrm{travelling}\:\mathrm{at}\:\mathrm{36}\:\mathrm{km}/\mathrm{h}\:\mathrm{due}\:\mathrm{North} \\ $$$$\mathrm{turns}\:\mathrm{West}\:\mathrm{in}\:\mathrm{5}\:\mathrm{seconds}\:\mathrm{and}\:\mathrm{maintains} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{speed}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{acceleration} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{car}? \\ $$

Question Number 14920 Answers: 1 Comments: 0

A swimmer crosses a flowing river of width d to and fro in time t_1 . The time taken to cover the same distance up and down the stream is t_2 . If t_3 is the time the swimmer would take to swim a distance 2d in still water, then prove that t_1 ^2 = t_2 t_3 .

$$\mathrm{A}\:\mathrm{swimmer}\:\mathrm{crosses}\:\mathrm{a}\:\mathrm{flowing}\:\mathrm{river}\:\mathrm{of} \\ $$$$\mathrm{width}\:{d}\:\mathrm{to}\:\mathrm{and}\:\mathrm{fro}\:\mathrm{in}\:\mathrm{time}\:{t}_{\mathrm{1}} .\:\mathrm{The}\:\mathrm{time} \\ $$$$\mathrm{taken}\:\mathrm{to}\:\mathrm{cover}\:\mathrm{the}\:\mathrm{same}\:\mathrm{distance}\:\mathrm{up} \\ $$$$\mathrm{and}\:\mathrm{down}\:\mathrm{the}\:\mathrm{stream}\:\mathrm{is}\:{t}_{\mathrm{2}} .\:\mathrm{If}\:{t}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{time}\:\mathrm{the}\:\mathrm{swimmer}\:\mathrm{would}\:\mathrm{take}\:\mathrm{to}\:\mathrm{swim} \\ $$$$\mathrm{a}\:\mathrm{distance}\:\mathrm{2}{d}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water},\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{t}_{\mathrm{1}} ^{\mathrm{2}} \:=\:{t}_{\mathrm{2}} {t}_{\mathrm{3}} . \\ $$

Question Number 14905 Answers: 0 Comments: 5

Question Number 14863 Answers: 1 Comments: 8

Question Number 14853 Answers: 1 Comments: 0

Solve for x 3^(2x) = 18x

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x} \\ $$$$\mathrm{3}^{\mathrm{2x}} \:=\:\mathrm{18x} \\ $$

Question Number 14821 Answers: 1 Comments: 0

A room has dimensions 3 m × 4 m × 5 m. A fly starting at one corner ends up at the diametrically opposite corner. If the fly were to walks, what is the length of the shortest path it can take?

$$\mathrm{A}\:\mathrm{room}\:\mathrm{has}\:\mathrm{dimensions}\:\mathrm{3}\:\mathrm{m}\:×\:\mathrm{4}\:\mathrm{m}\:×\:\mathrm{5}\:\mathrm{m}. \\ $$$$\mathrm{A}\:\mathrm{fly}\:\mathrm{starting}\:\mathrm{at}\:\mathrm{one}\:\mathrm{corner}\:\mathrm{ends}\:\mathrm{up}\:\mathrm{at} \\ $$$$\mathrm{the}\:\mathrm{diametrically}\:\mathrm{opposite}\:\mathrm{corner}.\:\mathrm{If} \\ $$$$\mathrm{the}\:\mathrm{fly}\:\mathrm{were}\:\mathrm{to}\:\mathrm{walks},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{length} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{path}\:\mathrm{it}\:\mathrm{can}\:\mathrm{take}? \\ $$

Question Number 14816 Answers: 1 Comments: 0

Question Number 14811 Answers: 1 Comments: 0

why (√x^2 )=∣x∣ ?

$${why}\:\:\:\:\:\sqrt{{x}^{\mathrm{2}} }=\mid{x}\mid\:\:\:\:?\: \\ $$

Question Number 14810 Answers: 0 Comments: 0

7 real numbers are given in the interval (1, 13). Prove that atleast 3 of them are the lengths of a triangle′s sides.

$$\mathrm{7}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{given}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left(\mathrm{1},\:\mathrm{13}\right).\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{atleast}\:\mathrm{3}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}'\mathrm{s}\:\mathrm{sides}. \\ $$

Question Number 14809 Answers: 1 Comments: 2

Let ABC be an acute triangle. Find the positions of the points M, N, P on the sides BC, CA, AB, respectively, such that the perimeter of the triangle MNP is minimal.

$$\mathrm{Let}\:{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}\:\mathrm{triangle}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{positions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{points}\:{M},\:{N},\:{P}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{BC},\:{CA},\:{AB},\:\mathrm{respectively}, \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle} \\ $$$${MNP}\:\mathrm{is}\:\mathrm{minimal}. \\ $$

Question Number 14807 Answers: 0 Comments: 0

Prove that the medians of a given triangle can form a triangle.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{medians}\:\mathrm{of}\:\mathrm{a}\:\mathrm{given} \\ $$$$\mathrm{triangle}\:\mathrm{can}\:\mathrm{form}\:\mathrm{a}\:\mathrm{triangle}. \\ $$

Question Number 14797 Answers: 1 Comments: 12