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Question Number 25201 Answers: 1 Comments: 0

Question Number 25200 Answers: 1 Comments: 0

The point A has coordinate (− 1, − 5) and the point B has coordinates (7, 1). The perpendicular bisector of AB meets the x − axis at C and the y − axis at D. Calculate the length of CD

$$\mathrm{The}\:\mathrm{point}\:\mathrm{A}\:\mathrm{has}\:\mathrm{coordinate}\:\left(−\:\mathrm{1},\:\:−\:\mathrm{5}\right)\:\mathrm{and}\:\mathrm{the}\:\mathrm{point}\:\mathrm{B}\:\mathrm{has}\:\mathrm{coordinates}\:\left(\mathrm{7},\:\mathrm{1}\right). \\ $$$$\mathrm{The}\:\mathrm{perpendicular}\:\mathrm{bisector}\:\mathrm{of}\:\mathrm{AB}\:\mathrm{meets}\:\mathrm{the}\:\mathrm{x}\:−\:\mathrm{axis}\:\mathrm{at}\:\mathrm{C}\:\mathrm{and}\:\mathrm{the}\:\mathrm{y}\:−\:\mathrm{axis}\:\mathrm{at} \\ $$$$\mathrm{D}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{CD} \\ $$

Question Number 25199 Answers: 1 Comments: 1

A line has equation y = 2x − 7 and a curve has equation y = x^2 − 4x + c, where c is a constant. Find the set of possible values of c for which the line does not intersect the curve.

$$\mathrm{A}\:\mathrm{line}\:\mathrm{has}\:\mathrm{equation}\:\mathrm{y}\:=\:\mathrm{2x}\:−\:\mathrm{7}\:\mathrm{and}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{has}\:\mathrm{equation}\:\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\mathrm{c}, \\ $$$$\mathrm{where}\:\mathrm{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{c}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{intersect}\:\mathrm{the}\:\mathrm{curve}. \\ $$

Question Number 25198 Answers: 0 Comments: 0

Given sin (K + L) cos M = 2 sin K cos (L − M) Prove that tan M = ((sin (L − K))/(2 sin K sin L))

$$\mathrm{Given} \\ $$$$\mathrm{sin}\:\left({K}\:+\:{L}\right)\:\mathrm{cos}\:{M}\:=\:\mathrm{2}\:\mathrm{sin}\:{K}\:\mathrm{cos}\:\left({L}\:−\:{M}\right) \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{tan}\:{M}\:=\:\frac{\mathrm{sin}\:\left({L}\:−\:{K}\right)}{\mathrm{2}\:\mathrm{sin}\:{K}\:\mathrm{sin}\:{L}} \\ $$

Question Number 25191 Answers: 0 Comments: 3

A man is climbing a ladder which is inclined to the wall at an angle of 30° . If he ascends at a rate of 2 m/s then he approaches the wall at the rate of−

$$\mathrm{A}\:\mathrm{man}\:\mathrm{is}\:\mathrm{climbing}\:\mathrm{a}\:\mathrm{ladder}\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{inclined}\:\mathrm{to}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{30}°\:. \\ $$$$\mathrm{If}\:\mathrm{he}\:\mathrm{ascends}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{2}\:\mathrm{m}/\mathrm{s}\:\mathrm{then}\: \\ $$$$\mathrm{he}\:\mathrm{approaches}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{at}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}− \\ $$$$ \\ $$

Question Number 25188 Answers: 0 Comments: 1

the straight line x+y=0 3x+y−4i=0 and x+3y−4=0 forms a triangle which triangle it is

$${the}\:{straight}\:{line}\:{x}+{y}=\mathrm{0}\:\: \\ $$$$\mathrm{3}{x}+{y}−\mathrm{4}{i}=\mathrm{0} \\ $$$${and}\:{x}+\mathrm{3}{y}−\mathrm{4}=\mathrm{0}\:{forms}\:{a}\:{triangle}\:{which}\:{triangle}\: \\ $$$${it}\:{is} \\ $$

Question Number 25187 Answers: 0 Comments: 1

if in a quadratic eqution x^2 +ax+b=0 and x^2 +bx+a=0 have a common root then the numerical value of (a+b) is

$${if}\:{in}\:{a}\:{quadratic}\:{eqution}\:{x}^{\mathrm{2}} +{ax}+{b}=\mathrm{0} \\ $$$${and}\:{x}^{\mathrm{2}} +{bx}+{a}=\mathrm{0}\:{have}\:{a}\:{common}\:{root} \\ $$$${then}\:{the}\:{numerical}\:{value}\:{of}\:\left({a}+{b}\right)\:{is} \\ $$

Question Number 25184 Answers: 1 Comments: 1

if a and b are the root of the quadratic equation x^2 +2x+3 then find the value of α^2 /β+β^2 /α

$${if}\:{a}\:{and}\:{b}\:{are}\:{the}\:{root}\:{of}\:{the}\:{quadratic}\:{equation}\:{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}\:{then}\:{find}\:{the}\:{value}\:{of}\:\alpha^{\mathrm{2}} /\beta+\beta^{\mathrm{2}} /\alpha \\ $$

Question Number 25183 Answers: 0 Comments: 1

Prove that ∫_0 ^( ∞) ((x^6 dx)/((a^4 +x^4 )^2 ))= ((3π(√2))/(16a)) , (a>0) .

$${Prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\:\:\infty} \frac{{x}^{\mathrm{6}} {dx}}{\left({a}^{\mathrm{4}} +{x}^{\mathrm{4}} \right)^{\mathrm{2}} }=\:\frac{\mathrm{3}\pi\sqrt{\mathrm{2}}}{\mathrm{16}{a}}\:\:\:,\:\:\left({a}>\mathrm{0}\right)\:. \\ $$

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

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

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