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Question Number 25173    Answers: 1   Comments: 0

Show that for all nεN−{0} 7^(2n+1) +1 is an integer multiple of 8.

$${Show}\:{that}\:{for}\:{all}\:{n}\epsilon{N}−\left\{\mathrm{0}\right\}\: \\ $$$$\mathrm{7}^{\mathrm{2}{n}+\mathrm{1}} +\mathrm{1}\:{is}\:{an}\:{integer}\:\:{multiple}\:{of} \\ $$$$\mathrm{8}. \\ $$

Question Number 25172    Answers: 1   Comments: 0

If the roots of the quadratic equation x^2 −3x−304=0 are α and β, then the quadratic equation with roots 3α and 3β is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic}\:\:\mathrm{equation}\: \\ $$$${x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{304}=\mathrm{0}\:\mathrm{are}\:\alpha\:\mathrm{and}\:\beta,\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{quadratic}\:\mathrm{equation}\:\mathrm{with}\:\mathrm{roots}\:\mathrm{3}\alpha\:\mathrm{and} \\ $$$$\mathrm{3}\beta\:\mathrm{is} \\ $$

Question Number 25171    Answers: 1   Comments: 0

100n>n^2 for integral n>100

$$\mathrm{100}{n}>{n}^{\mathrm{2}} \:{for}\:{integral}\:{n}>\mathrm{100} \\ $$$$ \\ $$

Question Number 25170    Answers: 2   Comments: 2

prove that n^2 >n−5 for integral n≥3

$${prove}\:{that}\:{n}^{\mathrm{2}} >{n}−\mathrm{5}\:{for}\:{integral}\: \\ $$$${n}\geqslant\mathrm{3}\: \\ $$

Question Number 25156    Answers: 2   Comments: 0

Question Number 25152    Answers: 2   Comments: 1

What is the real part and imaginary part of the complex number: z = (1 + i)^i

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{real}\:\mathrm{part}\:\mathrm{and}\:\mathrm{imaginary}\:\mathrm{part}\:\mathrm{of}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{number}:\:\:\:\:\mathrm{z}\:=\:\left(\mathrm{1}\:+\:\mathrm{i}\right)^{\mathrm{i}} \\ $$

Question Number 25148    Answers: 0   Comments: 1

difference between degree and radian.

$$\mathrm{difference}\:\mathrm{between}\:\mathrm{degree}\:\mathrm{and}\:\mathrm{radian}. \\ $$

Question Number 25139    Answers: 1   Comments: 0

What is the real and the imaginary part of the complex number z = (− 1)^(1000003)

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{real}\:\mathrm{and}\:\mathrm{the}\:\mathrm{imaginary}\:\mathrm{part}\:\mathrm{of}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{number}\:\:\:\mathrm{z}\:=\:\left(−\:\mathrm{1}\right)^{\mathrm{1000003}} \\ $$

Question Number 25129    Answers: 1   Comments: 1

Question Number 25122    Answers: 0   Comments: 1

3_C_1 + 4_C_2 + 5_C_3 +...........+ 49_C_(47) = ? where n_C_r = ((n!)/(r!×(n−r)!)) .

$$\:\mathrm{3}_{{C}_{\mathrm{1}} } \:+\:\mathrm{4}_{{C}_{\mathrm{2}} } \:+\:\mathrm{5}_{{C}_{\mathrm{3}} } \:+...........+\:\mathrm{49}_{{C}_{\mathrm{47}} } \:=\:? \\ $$$${where}\:{n}_{{C}_{{r}} } \:=\:\frac{{n}!}{{r}!×\left({n}−{r}\right)!}\:. \\ $$

Question Number 25125    Answers: 1   Comments: 1

A vertical stick 12 cm long casts a shadow of 8 cm long on the ground . At the same time, a tower casts a shadow of 40 m long on the ground. find the hight of the tower ?

$$\boldsymbol{\mathrm{A}}\:\mathrm{vertical}\:\mathrm{stick}\:\mathrm{12}\:\boldsymbol{\mathrm{cm}}\:\mathrm{long}\:\mathrm{casts}\:\mathrm{a}\: \\ $$$$\mathrm{shadow}\:\mathrm{of}\:\mathrm{8}\:\boldsymbol{\mathrm{cm}}\:\mathrm{long}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ground}\:. \\ $$$$\mathrm{At}\:\mathrm{the}\:\mathrm{same}\:\mathrm{time},\:\mathrm{a}\:\mathrm{tower}\:\mathrm{casts}\:\mathrm{a}\: \\ $$$$\mathrm{shadow}\:\mathrm{of}\:\mathrm{40}\:\boldsymbol{\mathrm{m}}\:\mathrm{long}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ground}.\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{hight}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tower}\:? \\ $$

Question Number 25114    Answers: 0   Comments: 1

Question Number 25113    Answers: 0   Comments: 1

Question Number 25112    Answers: 0   Comments: 1

Question Number 25109    Answers: 1   Comments: 0

Question Number 25103    Answers: 1   Comments: 0

Question Number 25091    Answers: 1   Comments: 0

A particle of mass m moving with speed u collides perfectly inelastically with a sphere of radius R and same mass, at rest, at an impact parameter d. Find (a) Angle between their final velocities (b) Magnitude of their final velocities

$${A}\:{particle}\:{of}\:{mass}\:{m}\:{moving}\:{with} \\ $$$${speed}\:{u}\:{collides}\:{perfectly}\:{inelastically} \\ $$$${with}\:{a}\:{sphere}\:{of}\:{radius}\:{R}\:{and}\:{same} \\ $$$${mass},\:{at}\:{rest},\:{at}\:{an}\:{impact}\:{parameter} \\ $$$${d}.\:{Find} \\ $$$$\left({a}\right)\:{Angle}\:{between}\:{their}\:{final}\:{velocities} \\ $$$$\left({b}\right)\:{Magnitude}\:{of}\:{their}\:{final} \\ $$$${velocities} \\ $$

Question Number 25094    Answers: 1   Comments: 0

Question Number 25088    Answers: 1   Comments: 1

Q...((x+7)/(x+4))>1, x∈R

$$ \\ $$$$ \\ $$$$ \\ $$$${Q}...\frac{{x}+\mathrm{7}}{{x}+\mathrm{4}}>\mathrm{1},\:\:\:\:\:{x}\in{R} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 25086    Answers: 0   Comments: 4

Question Number 25085    Answers: 1   Comments: 0

If a_n −a_(n−1) =1 for every positive integer greater than 1, then a_1 +a_2 +a_3 +...a_(100) equals (1) 5000 . a_1 (2) 5050 . a_1 (3) 5051 . a_1 (3) 5052 . a_2

$${If}\:{a}_{{n}} −{a}_{{n}−\mathrm{1}} =\mathrm{1}\:{for}\:{every}\:{positive} \\ $$$${integer}\:{greater}\:{than}\:\mathrm{1},\:{then}\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +{a}_{\mathrm{3}} \\ $$$$+...{a}_{\mathrm{100}} \:{equals} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{5000}\:.\:{a}_{\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{5050}\:.\:{a}_{\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{5051}\:.\:{a}_{\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{5052}\:.\:{a}_{\mathrm{2}} \\ $$

Question Number 25068    Answers: 1   Comments: 0

Evaluate lim_(x→(π/2)) (((tan2x)/(x−π/2)))

$${Evaluate}\: \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\left(\frac{{tan}\mathrm{2}{x}}{{x}−\pi/\mathrm{2}}\right) \\ $$

Question Number 25074    Answers: 0   Comments: 1

Question Number 25075    Answers: 1   Comments: 0

Let z_1 and z_2 be two roots of the equation z^2 +az+b=0, z being complex. Further assume that the origin, z_1 and z_2 form an equilateral triangle. Then,

$$\mathrm{Let}\:{z}_{\mathrm{1}} \mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${z}^{\mathrm{2}} +{az}+{b}=\mathrm{0},\:{z}\:\mathrm{being}\:\mathrm{complex}.\:\mathrm{Further} \\ $$$$\mathrm{assume}\:\mathrm{that}\:\mathrm{the}\:\mathrm{origin},\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{form} \\ $$$$\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}.\:\mathrm{Then}, \\ $$

Question Number 25066    Answers: 1   Comments: 0

let a,b,c,x,y and z be complex number such that a=((b+c)/(x−2)) ,b=((c+a)/(y−2)) c=((a+b)/(z−2)). xy +yz +zx=1000 and x+y+z=2016 find the value of xyz.

$${let}\:{a},{b},{c},{x},{y}\:{and}\:{z}\:{be}\:{complex}\:{number} \\ $$$${such}\:{that}\:{a}=\frac{{b}+{c}}{{x}−\mathrm{2}}\:,{b}=\frac{{c}+{a}}{{y}−\mathrm{2}}\:\:\:\:{c}=\frac{{a}+{b}}{{z}−\mathrm{2}}. \\ $$$${xy}\:+{yz}\:+{zx}=\mathrm{1000}\:{and}\:{x}+{y}+{z}=\mathrm{2016} \\ $$$${find}\:{the}\:{value}\:{of}\:{xyz}. \\ $$

Question Number 25076    Answers: 1   Comments: 0

A man can row 6 km/h in still water. When the river is running at 1.2 km/h, it takes]him 1 hour to row to a place and back. How far is the place?

$$\mathrm{A}\:\mathrm{man}\:\mathrm{can}\:\mathrm{row}\:\mathrm{6}\:\mathrm{km}/\mathrm{h}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water}. \\ $$$$\mathrm{When}\:\mathrm{the}\:\mathrm{river}\:\mathrm{is}\:\mathrm{running}\:\mathrm{at}\:\mathrm{1}.\mathrm{2}\:\mathrm{km}/\mathrm{h}, \\ $$$$\left.\mathrm{it}\:\mathrm{takes}\right]\mathrm{him}\:\mathrm{1}\:\mathrm{hour}\:\mathrm{to}\:\mathrm{row}\:\mathrm{to}\:\mathrm{a}\:\mathrm{place}\: \\ $$$$\mathrm{and}\:\mathrm{back}.\:\mathrm{How}\:\mathrm{far}\:\mathrm{is}\:\mathrm{the}\:\mathrm{place}? \\ $$

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