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Question Number 6838 Answers: 0 Comments: 2

Find the value of x 4^x = ((243)/x)

$${Find}\:{the}\:{value}\:{of}\:{x} \\ $$$$\mathrm{4}^{{x}} \:=\:\frac{\mathrm{243}}{{x}} \\ $$

Question Number 6834 Answers: 0 Comments: 2

a∈Z, (2a^3 +3a^2 −3a+7 : a^2 +a−2)≠1 ⇔ a=3k+1, k∈Z

$${a}\in\mathbb{Z},\:\left(\mathrm{2}{a}^{\mathrm{3}} +\mathrm{3}{a}^{\mathrm{2}} −\mathrm{3}{a}+\mathrm{7}\::\:{a}^{\mathrm{2}} +{a}−\mathrm{2}\right)\neq\mathrm{1}\:\Leftrightarrow\:{a}=\mathrm{3}{k}+\mathrm{1},\:{k}\in\mathbb{Z} \\ $$

Question Number 6832 Answers: 0 Comments: 4

What remainder does 3^(6n) − 2^(6n) leave when divided by 35 for any value of n

$${What}\:{remainder}\:{does}\:\mathrm{3}^{\mathrm{6}{n}} \:−\:\mathrm{2}^{\mathrm{6}{n}} \:\:{leave}\:{when}\:{divided}\:{by}\:\mathrm{35}\:{for}\: \\ $$$${any}\:{value}\:{of}\:{n}\: \\ $$

Question Number 6831 Answers: 0 Comments: 5

Find the 100th digits of this sum 1 + 50 + 50^2 + ..... + 50^(999)

$${Find}\:{the}\:\mathrm{100}{th}\:{digits}\:{of}\:{this}\:{sum}\: \\ $$$$\mathrm{1}\:+\:\mathrm{50}\:+\:\mathrm{50}^{\mathrm{2}} \:+\:.....\:+\:\mathrm{50}^{\mathrm{999}} \\ $$

Question Number 6830 Answers: 0 Comments: 0

A, B, and C are three non-collinear points such that ∣AB∣=c , ∣BC∣=a and ∣CA∣=b. What will be the condition for a point D to be concircle with A, B and C when all the four points belong to same plane?

$${A},\:{B},\:{and}\:{C}\:{are}\:{three}\:{non}-{collinear}\:{points}\:{such} \\ $$$${that}\:\mid{AB}\mid={c}\:,\:\mid{BC}\mid={a}\:\:{and}\:\:\mid{CA}\mid={b}. \\ $$$${What}\:{will}\:{be}\:{the}\:{condition}\:{for}\:{a}\:{point}\:{D}\:{to}\:{be} \\ $$$${concircle}\:{with}\:{A},\:{B}\:\:{and}\:{C}\:{when}\:{all}\:{the}\:{four} \\ $$$${points}\:{belong}\:{to}\:{same}\:{plane}? \\ $$

Question Number 6829 Answers: 0 Comments: 0

Question Number 6825 Answers: 1 Comments: 0

is there a triple integral that equals to the golden ratio φ 1.618?

$${is}\:{there}\:{a}\:{triple}\:{integral}\:{that}\:{equals} \\ $$$${to}\:{the}\:{golden}\:{ratio}\:\phi\:\mathrm{1}.\mathrm{618}? \\ $$$$ \\ $$$$ \\ $$

Question Number 6824 Answers: 0 Comments: 2

Solve simultaneously (1/u) + (1/v) = (1/3) .......... equation (i) (u^2 /v) + (v^2 /u) = 12 ........ equation (ii)

$${Solve}\:{simultaneously} \\ $$$$\frac{\mathrm{1}}{{u}}\:+\:\frac{\mathrm{1}}{{v}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\:\:..........\:{equation}\:\left({i}\right) \\ $$$$\frac{{u}^{\mathrm{2}} }{{v}}\:+\:\frac{{v}^{\mathrm{2}} }{{u}}\:=\:\mathrm{12}\:\:\:\:\:\:........\:{equation}\:\left({ii}\right) \\ $$

Question Number 6820 Answers: 0 Comments: 2

The position vector of the point P at time t is given by (αtant)i + (αsect)k where α is positive constant and 0≤t≤(Π/2). Show that the velocity and acceleration of P when t = 0 are at right angle to each other. If A is the point with position vector αj, obtain the vector equation for the straight line AP at time t. If the point Q divides AP internally in the ratio (cost) : (1 − cost). Show that the acceleration of the point Q is constant in magnitude and is always directed towards a fixed point.

$${The}\:{position}\:{vector}\:{of}\:{the}\:{point}\:{P}\:\:{at}\:{time}\:{t}\:{is}\:{given}\:{by}\: \\ $$$$\left(\alpha{tant}\right){i}\:+\:\left(\alpha{sect}\right){k}\:{where}\:\alpha\:{is}\:{positive}\:{constant}\:{and} \\ $$$$\mathrm{0}\leqslant{t}\leqslant\frac{\Pi}{\mathrm{2}}.\:\:{Show}\:{that}\:{the}\:{velocity}\:{and}\:{acceleration}\:{of}\:{P}\:\:{when} \\ $$$${t}\:=\:\mathrm{0}\:{are}\:{at}\:{right}\:{angle}\:{to}\:{each}\:{other}.\:{If}\:{A}\:{is}\:{the}\:{point}\:{with}\: \\ $$$${position}\:{vector}\:\alpha{j},\:{obtain}\:{the}\:{vector}\:{equation}\:{for}\:{the}\:{straight}\: \\ $$$${line}\:{AP}\:{at}\:{time}\:{t}.\:\:{If}\:{the}\:{point}\:{Q}\:{divides}\:{AP}\:\:{internally}\:{in}\:{the} \\ $$$${ratio}\:\left({cost}\right)\::\:\left(\mathrm{1}\:−\:{cost}\right).\:{Show}\:{that}\:{the}\:{acceleration}\:{of}\:{the}\:{point} \\ $$$${Q}\:{is}\:{constant}\:{in}\:{magnitude}\:{and}\:{is}\:{always}\:{directed}\:{towards}\:{a}\: \\ $$$${fixed}\:{point}. \\ $$

Question Number 6811 Answers: 0 Comments: 4

Let Σ_(k = 1) ^n log_2 k = 1994 find the positive integer n ?

$${Let}\:\:\:\underset{{k}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\:{log}_{\mathrm{2}} {k}\:\:=\:\:\mathrm{1994} \\ $$$${find}\:{the}\:{positive}\:{integer}\:{n}\:? \\ $$

Question Number 6810 Answers: 0 Comments: 2

Up to which prime number should 9 divide 9631 to ensure that this number is a prime number or a composite number ?

$${Up}\:{to}\:{which}\:{prime}\:{number}\:{should}\:\:\:\mathrm{9}\:\:\:{divide}\:\:\:\mathrm{9631}\:\:\:{to}\:{ensure}\:{that} \\ $$$${this}\:{number}\:{is}\:{a}\:{prime}\:{number}\:{or}\:{a}\:{composite}\:{number}\:? \\ $$

Question Number 6809 Answers: 0 Comments: 2

If x^(1/3) + y^(1/3) + z^(1/3) = 0, Then pove that (x + y + z)^3 = 3xyz

$${If}\:\:{x}^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\:{y}^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\:{z}^{\frac{\mathrm{1}}{\mathrm{3}}} \:\:=\:\:\mathrm{0},\:\:{Then}\:\:{pove}\:{that}\: \\ $$$$\left({x}\:+\:{y}\:+\:{z}\right)^{\mathrm{3}} \:\:=\:\mathrm{3}{xyz} \\ $$

Question Number 6806 Answers: 1 Comments: 0

Question Number 6802 Answers: 0 Comments: 1

Solve for x x(x^2 +1)^2 =1

$$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$${x}\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} =\mathrm{1} \\ $$

Question Number 6799 Answers: 0 Comments: 2

What is the probability that 3 customers waiting in bank will be served in sequence of their arrival.

$${What}\:{is}\:{the}\:{probability}\:{that}\:\mathrm{3}\:{customers}\:{waiting}\:{in}\:{bank}\:{will}\: \\ $$$${be}\:{served}\:{in}\:{sequence}\:{of}\:{their}\:{arrival}. \\ $$

Question Number 6792 Answers: 0 Comments: 2

When the price of egg was raised by #2 an egg, the number of eggs which can be bought for #120 is reduced by #5. What is the present price of the egg.

$${When}\:{the}\:{price}\:{of}\:{egg}\:\:{was}\:{raised}\:{by}\:#\mathrm{2}\:{an}\:{egg},\:\:{the}\:{number}\:{of}\:{eggs} \\ $$$${which}\:{can}\:{be}\:{bought}\:{for}\:#\mathrm{120}\:{is}\:{reduced}\:{by}\:#\mathrm{5}.\:{What}\:{is}\:{the} \\ $$$${present}\:{price}\:{of}\:{the}\:{egg}. \\ $$

Question Number 6791 Answers: 1 Comments: 2

Question Number 6790 Answers: 0 Comments: 0

∫_0 ^1 ∫_y ^1 x^(−3/2) cos((πy)/(2x)) dx dy=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\int_{{y}} ^{\mathrm{1}} \:{x}^{−\mathrm{3}/\mathrm{2}} {cos}\frac{\pi{y}}{\mathrm{2}{x}}\:{dx}\:{dy}=? \\ $$

Question Number 6788 Answers: 0 Comments: 2

Question Number 6786 Answers: 0 Comments: 2

Three Arabs A,B and C were traveling together. On the lunch time A and B produce m and n loaves of bread respectively for eating but C have no loaf of bread. All the three ate together. After completing lunch C gave A and B m+n darhams. How shoud they divide this sum?

$${Three}\:{Arabs}\:{A},{B}\:{and}\:{C}\:{were}\:{traveling} \\ $$$${together}.\:{On}\:{the}\:{lunch}\:{time}\:{A}\:{and}\:\:{B} \\ $$$${produce}\:{m}\:{and}\:\:{n}\:{loaves}\:{of}\:{bread}\:{respectively} \\ $$$${for}\:{eating}\:{but}\:{C}\:{have}\:{no}\:{loaf}\:{of}\:{bread}.\:{All} \\ $$$${the}\:{three}\:{ate}\:{together}.\:\:{After}\:{completing} \\ $$$${lunch}\:{C}\:\:{gave}\:{A}\:{and}\:{B}\:{m}+{n}\:{darhams}. \\ $$$${How}\:{shoud}\:{they}\:{divide}\:{this}\:{sum}? \\ $$$$ \\ $$

Question Number 6775 Answers: 1 Comments: 0

∫_0 ^∞ e^t^2 e^(−st) dt please help

$$\int_{\mathrm{0}} ^{\infty} {e}^{{t}^{\mathrm{2}} } {e}^{−{st}} \:\:{dt} \\ $$$$ \\ $$$${please}\:{help} \\ $$

Question Number 6772 Answers: 0 Comments: 3

What is the smallest even number greater than 3 that cannot be expressed as the sum of two prime numbers ?

$${What}\:{is}\:{the}\:{smallest}\:{even}\:{number}\:{greater}\:{than}\:\mathrm{3}\:{that}\:{cannot} \\ $$$${be}\:{expressed}\:{as}\:{the}\:{sum}\:{of}\:{two}\:{prime}\:{numbers}\:? \\ $$

Question Number 6767 Answers: 0 Comments: 0

A molten plastic flows out of a tube that is 8.0cm long at a rate of 13cm^3 /min, when the pressure differential between the two ends of the tube is 18cm mercury. find the viscousity of the plastic. The internal diameter of the tube is 1.30mm. the density of mercury is 13.6g/cm^3

$${A}\:{molten}\:\:{plastic}\:{flows}\:{out}\:{of}\:{a}\:{tube}\:{that}\:{is}\:\mathrm{8}.\mathrm{0}{cm}\:{long} \\ $$$${at}\:{a}\:{rate}\:{of}\:\mathrm{13}{cm}^{\mathrm{3}} /{min},\:{when}\:{the}\:{pressure}\:{differential} \\ $$$${between}\:{the}\:{two}\:{ends}\:{of}\:{the}\:{tube}\:{is}\:\mathrm{18}{cm}\:{mercury}. \\ $$$${find}\:{the}\:{viscousity}\:{of}\:{the}\:{plastic}.\: \\ $$$${The}\:{internal}\:{diameter}\:{of}\:{the}\:{tube}\:{is}\:\:\mathrm{1}.\mathrm{30}{mm}.\: \\ $$$${the}\:{density}\:{of}\:{mercury}\:{is}\:\mathrm{13}.\mathrm{6}{g}/{cm}^{\mathrm{3}} \\ $$

Question Number 6778 Answers: 0 Comments: 1

If two forces P and Q acting at 0 are represented by line OA and OB with φ being the angle between the two forces . find their resultant in R in terms of P

$${If}\:{two}\:{forces}\:{P}\:{and}\:{Q}\:{acting}\:{at}\:\mathrm{0}\:{are}\:{represented}\:{by}\:{line}\:{OA} \\ $$$${and}\:{OB}\:{with}\:\phi\:{being}\:{the}\:{angle}\:{between}\:{the}\:{two}\:{forces}\:. \\ $$$${find}\:{their}\:{resultant}\:{in}\:{R}\:{in}\:{terms}\:{of}\:{P} \\ $$

Question Number 6762 Answers: 1 Comments: 2

Question Number 6758 Answers: 0 Comments: 4

A circle of radius r has a point O as its centre. Points A and B are points on the circumference. For △OAB, OA^(−) =OB^(−) =r, AB^(−) =d, ∠AOB=θ. What is (r/d)?

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:{r}\:\mathrm{has}\:\mathrm{a}\:\mathrm{point}\:{O}\:\mathrm{as}\:\mathrm{its} \\ $$$$\mathrm{centre}.\:\mathrm{Points}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{circumference}. \\ $$$$ \\ $$$$\mathrm{For}\:\bigtriangleup{OAB},\:\overline {{OA}}=\overline {{OB}}={r},\:\overline {{AB}}={d},\:\angle{AOB}=\theta. \\ $$$$\mathrm{What}\:\mathrm{is}\:\frac{{r}}{{d}}? \\ $$

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