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Question Number 19194    Answers: 2   Comments: 0

Question Number 19193    Answers: 1   Comments: 0

The sum of two positive integers is 52 and their LCM is 168. Find the numbers.

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{is}\:\mathrm{52} \\ $$$$\mathrm{and}\:\mathrm{their}\:\mathrm{LCM}\:\mathrm{is}\:\mathrm{168}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{numbers}. \\ $$

Question Number 19192    Answers: 0   Comments: 2

Find a natural number ′n′ such that 3^9 + 3^(12) + 3^(15) + 3^n is a perfect cube of an integer.

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:'\mathrm{n}'\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{3}^{\mathrm{9}} \:+\:\mathrm{3}^{\mathrm{12}} \:+\:\mathrm{3}^{\mathrm{15}} \:+\:\mathrm{3}^{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{cube}\:\mathrm{of} \\ $$$$\mathrm{an}\:\mathrm{integer}. \\ $$

Question Number 19262    Answers: 1   Comments: 0

Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p ×{x−q)×p}+q×{x−r)×q} + r×{x−p)×r}=0, then x is given by

$$\mathrm{Let}\:\boldsymbol{\mathrm{p}},\:\boldsymbol{\mathrm{q}},\:\boldsymbol{\mathrm{r}}\:\mathrm{be}\:\mathrm{three}\:\mathrm{mutually}\:\mathrm{perpendicular} \\ $$$$\mathrm{vectors}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{magnitude}.\:\mathrm{If}\:\mathrm{a}\:\mathrm{vector} \\ $$$$\boldsymbol{\mathrm{x}}\:\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\left.\boldsymbol{\mathrm{p}}\left.\:×\left\{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{q}}\right)×\boldsymbol{\mathrm{p}}\right\}+\boldsymbol{\mathrm{q}}×\left\{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{r}}\right)×\boldsymbol{\mathrm{q}}\right\} \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\:\boldsymbol{\mathrm{r}}×\left\{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{p}}\right)×\boldsymbol{\mathrm{r}}\right\}=\mathrm{0},\:\mathrm{then}\:\boldsymbol{\mathrm{x}}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{by} \\ $$

Question Number 19182    Answers: 2   Comments: 0

Find all three digit numbers abc (with a ≠ 0) such that a^2 + b^2 + c^2 , is divisible by 26.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}\:\mathrm{digit}\:\mathrm{numbers}\:{abc}\:\left(\mathrm{with}\right. \\ $$$$\left.\mathrm{a}\:\neq\:\mathrm{0}\right)\:\mathrm{such}\:\mathrm{that}\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} ,\:\mathrm{is}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{26}. \\ $$

Question Number 19171    Answers: 1   Comments: 1

log_(√2) (√(2(√(2(√(2(√(2 ))))))))

$${log}_{\sqrt{\mathrm{2}}} \:\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}\:\:\:\:}}}} \\ $$

Question Number 19154    Answers: 0   Comments: 0

If f(x) = (9^x /(9^x + 3)), find the value f((1/(2007))) + f((2/(2007))) + f((3/(2007))) + ... + f(((2006)/(2006)))

$$\mathrm{If}\:{f}\left({x}\right)\:=\:\frac{\mathrm{9}^{{x}} }{\mathrm{9}^{{x}} \:+\:\mathrm{3}},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2007}}\right)\:+\:{f}\left(\frac{\mathrm{2}}{\mathrm{2007}}\right)\:+\:{f}\left(\frac{\mathrm{3}}{\mathrm{2007}}\right)\:+\:...\:+\:{f}\left(\frac{\mathrm{2006}}{\mathrm{2006}}\right) \\ $$

Question Number 19150    Answers: 1   Comments: 4

A semicircle is tangent to both legs of a right triangle and has its centre on the hypotenuse. The hypotenuse is partitioned into 4 segments, with lengths 3, 12, 12, and x, as shown in the figure. Determine the value of ′x′.

$$\mathrm{A}\:\mathrm{semicircle}\:\mathrm{is}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{both}\:\mathrm{legs}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{right}\:\mathrm{triangle}\:\mathrm{and}\:\mathrm{has}\:\mathrm{its}\:\mathrm{centre}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{hypotenuse}.\:\mathrm{The}\:\mathrm{hypotenuse}\:\mathrm{is} \\ $$$$\mathrm{partitioned}\:\mathrm{into}\:\mathrm{4}\:\mathrm{segments},\:\mathrm{with}\:\mathrm{lengths} \\ $$$$\mathrm{3},\:\mathrm{12},\:\mathrm{12},\:\mathrm{and}\:{x},\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{figure}. \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:'{x}'. \\ $$

Question Number 19148    Answers: 1   Comments: 0

If f(x)= determinant (((sin x+sin2x+sin 3x),(sin 2x),(sin 3x)),(( 3+4 sin x),( 3),(4 sin x)),(( 1+sin x),( sin x),( 1))) then the value of ∫_( 0) ^(π/2) f(x) dx is

$$\mathrm{If} \\ $$$${f}\left({x}\right)=\begin{vmatrix}{\mathrm{sin}\:{x}+\mathrm{sin2}{x}+\mathrm{sin}\:\mathrm{3}{x}}&{\mathrm{sin}\:\mathrm{2}{x}}&{\mathrm{sin}\:\mathrm{3}{x}}\\{\:\:\:\:\:\:\:\:\mathrm{3}+\mathrm{4}\:\mathrm{sin}\:{x}}&{\:\:\:\:\mathrm{3}}&{\mathrm{4}\:\mathrm{sin}\:{x}}\\{\:\:\:\:\:\:\:\:\:\:\mathrm{1}+\mathrm{sin}\:{x}}&{\:\mathrm{sin}\:{x}}&{\:\:\:\:\mathrm{1}}\end{vmatrix} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:\:\:\mathrm{is} \\ $$

Question Number 19140    Answers: 0   Comments: 9

A racing car travels on a track (without banking) ABCDEFA. ABC is a circular arc of radius 2R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is μ = 0.1. The maximum speed of the car is 50 ms^(−1) . Find the minimum time for completing one round.

$$\mathrm{A}\:\mathrm{racing}\:\mathrm{car}\:\mathrm{travels}\:\mathrm{on}\:\mathrm{a}\:\mathrm{track}\:\left(\mathrm{without}\right. \\ $$$$\left.\mathrm{banking}\right)\:{ABCDEFA}.\:{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{arc}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{2}{R}.\:{CD}\:\mathrm{and}\:{FA}\:\mathrm{are} \\ $$$$\mathrm{straight}\:\mathrm{paths}\:\mathrm{of}\:\mathrm{length}\:{R}\:\mathrm{and}\:{DEF}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{circular}\:\mathrm{arc}\:\mathrm{of}\:\mathrm{radius}\:{R}\:=\:\mathrm{100}\:\mathrm{m}.\:\mathrm{The} \\ $$$$\mathrm{co}-\mathrm{efficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{on}\:\mathrm{the}\:\mathrm{road}\:\mathrm{is}\:\mu\:= \\ $$$$\mathrm{0}.\mathrm{1}.\:\mathrm{The}\:\mathrm{maximum}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{is} \\ $$$$\mathrm{50}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{time}\:\mathrm{for} \\ $$$$\mathrm{completing}\:\mathrm{one}\:\mathrm{round}. \\ $$

Question Number 19137    Answers: 1   Comments: 1

Figure shows (x, t), (y, t) diagram of a particle moving in 2-dimensions. If the particle has a mass of 500 g, find the force (direction and magnitude) acting on the particle.

$$\mathrm{Figure}\:\mathrm{shows}\:\left({x},\:{t}\right),\:\left({y},\:{t}\right)\:\mathrm{diagram}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{particle}\:\mathrm{moving}\:\mathrm{in}\:\mathrm{2}-\mathrm{dimensions}.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{has}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{500}\:\mathrm{g},\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{force}\:\left(\mathrm{direction}\:\mathrm{and}\:\mathrm{magnitude}\right)\:\mathrm{acting} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{particle}. \\ $$

Question Number 19135    Answers: 1   Comments: 0

solve for x: 2^(∣x+2∣) −∣2^(x+1) −1∣=2^(x+1) +1

$${solve}\:{for}\:{x}: \\ $$$$\mathrm{2}^{\mid{x}+\mathrm{2}\mid} −\mid\mathrm{2}^{{x}+\mathrm{1}} −\mathrm{1}\mid=\mathrm{2}^{{x}+\mathrm{1}} +\mathrm{1} \\ $$

Question Number 19134    Answers: 1   Comments: 0

If (1/((243)^x )) = (729)^y = 3^3 , then find the value of 5x + 6y.

$$\mathrm{If}\:\frac{\mathrm{1}}{\left(\mathrm{243}\right)^{{x}} }\:=\:\left(\mathrm{729}\right)^{{y}} =\:\mathrm{3}^{\mathrm{3}} ,\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{5}{x}\:+\:\mathrm{6}{y}. \\ $$

Question Number 19127    Answers: 0   Comments: 0

Question Number 19123    Answers: 1   Comments: 0

{ ((xf(x)−g(x)+h(x)=2x+1)),((f(x)−(2x−2)g(x)−3h(x)=x)),((ln (x)f(x)−(x−3)h(x)=1)) :} Find f(x),g(x),h(x)

$$\begin{cases}{\mathrm{xf}\left(\mathrm{x}\right)−\mathrm{g}\left(\mathrm{x}\right)+\mathrm{h}\left(\mathrm{x}\right)=\mathrm{2x}+\mathrm{1}}\\{\mathrm{f}\left(\mathrm{x}\right)−\left(\mathrm{2x}−\mathrm{2}\right)\mathrm{g}\left(\mathrm{x}\right)−\mathrm{3h}\left(\mathrm{x}\right)=\mathrm{x}}\\{\mathrm{ln}\:\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{x}\right)−\left(\mathrm{x}−\mathrm{3}\right)\mathrm{h}\left(\mathrm{x}\right)=\mathrm{1}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{f}\left(\mathrm{x}\right),\mathrm{g}\left(\mathrm{x}\right),\mathrm{h}\left(\mathrm{x}\right) \\ $$

Question Number 19122    Answers: 1   Comments: 0

Prove that r_1 + r_2 + r_3 = 4R + r

$$\mathrm{Prove}\:\mathrm{that}\:{r}_{\mathrm{1}} \:+\:{r}_{\mathrm{2}} \:+\:{r}_{\mathrm{3}} \:=\:\mathrm{4}{R}\:+\:{r} \\ $$

Question Number 19121    Answers: 0   Comments: 0

Question Number 19118    Answers: 0   Comments: 0

Question Number 19104    Answers: 1   Comments: 1

Let ABC be an acute-angled triangle with AC ≠ BC and let O be the circumcenter and F be the foot of altitude through C. Further, let X and Y be the feet of perpendiculars dropped from A and B respectively to (the extension of) CO. The line FO intersects the circumcircle of ΔFXY, second time at P. Prove that OP < OF.

$$\mathrm{Let}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{AC}\:\neq\:\mathrm{BC}\:\mathrm{and}\:\mathrm{let}\:\mathrm{O}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{circumcenter}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:\mathrm{C}.\:\mathrm{Further},\:\mathrm{let}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{perpendiculars}\:\mathrm{dropped} \\ $$$$\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{respectively}\:\mathrm{to}\:\left(\mathrm{the}\right. \\ $$$$\left.\mathrm{extension}\:\mathrm{of}\right)\:\mathrm{CO}.\:\mathrm{The}\:\mathrm{line}\:\mathrm{FO}\:\mathrm{intersects} \\ $$$$\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\Delta\mathrm{FXY},\:\mathrm{second}\:\mathrm{time} \\ $$$$\mathrm{at}\:\mathrm{P}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{OP}\:<\:\mathrm{OF}. \\ $$

Question Number 19101    Answers: 0   Comments: 3

A polynomial f(x) with rational coefficients leaves remainder 15, when divided by x − 3 and remainder 2x + 1, when divided by (x − 1)^2 . Find the remainder when f(x) is divided by (x − 3)(x − 1)^2 .

$$\mathrm{A}\:\mathrm{polynomial}\:{f}\left({x}\right)\:\mathrm{with}\:\mathrm{rational} \\ $$$$\mathrm{coefficients}\:\mathrm{leaves}\:\mathrm{remainder}\:\mathrm{15},\:\mathrm{when} \\ $$$$\mathrm{divided}\:\mathrm{by}\:{x}\:−\:\mathrm{3}\:\mathrm{and}\:\mathrm{remainder}\:\mathrm{2}{x}\:+\:\mathrm{1}, \\ $$$$\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} .\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remainder}\:\mathrm{when}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left({x}\:−\:\mathrm{3}\right)\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} . \\ $$

Question Number 19097    Answers: 1   Comments: 0

If tan ((π/4) + x) = tan^3 ((π/4) + α) then prove that cosec 2x = ((1 + 3 sin^2 2α)/(3 sin 2α + sin^3 2α))

$$\mathrm{If}\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}\:+\:{x}\right)\:=\:\mathrm{tan}^{\mathrm{3}} \:\left(\frac{\pi}{\mathrm{4}}\:+\:\alpha\right)\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{cosec}\:\mathrm{2}{x}\:=\:\frac{\mathrm{1}\:+\:\mathrm{3}\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{2}\alpha}{\mathrm{3}\:\mathrm{sin}\:\mathrm{2}\alpha\:+\:\mathrm{sin}^{\mathrm{3}} \:\mathrm{2}\alpha} \\ $$

Question Number 19095    Answers: 0   Comments: 3

Question Number 19085    Answers: 0   Comments: 0

f_n (x)=(√(f_(n−1) (x)×(f_(n−1) (x))′)) f_1 (x)=x^(2017) +x^8 +x^4 lim_(n→∞) f_n (x)=?

$$\mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)=\sqrt{\mathrm{f}_{\mathrm{n}−\mathrm{1}} \left(\mathrm{x}\right)×\left(\mathrm{f}_{\mathrm{n}−\mathrm{1}} \left(\mathrm{x}\right)\right)'} \\ $$$$\mathrm{f}_{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2017}} +\mathrm{x}^{\mathrm{8}} +\mathrm{x}^{\mathrm{4}} \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}f}_{\mathrm{n}} \left(\mathrm{x}\right)=? \\ $$

Question Number 19083    Answers: 1   Comments: 4

Question Number 19080    Answers: 0   Comments: 1

Question Number 19064    Answers: 1   Comments: 2

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