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Question Number 17530 Answers: 0 Comments: 7

Two masses 5 kg and M are hanging with the help of light rope and pulley as shown below. If the system is in equilibrium then M =

$$\mathrm{Two}\:\mathrm{masses}\:\mathrm{5}\:\mathrm{kg}\:\mathrm{and}\:{M}\:\mathrm{are}\:\mathrm{hanging} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{help}\:\mathrm{of}\:\mathrm{light}\:\mathrm{rope}\:\mathrm{and}\:\mathrm{pulley} \\ $$$$\mathrm{as}\:\mathrm{shown}\:\mathrm{below}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}\:\mathrm{then}\:{M}\:= \\ $$

Question Number 17525 Answers: 0 Comments: 1

Evaluate ∫_(−1) ^1 (1−x^2 )^(n/2) dx for n ∈ Z∩[0;∞) (i.e. 0, 1, 2, ...) and: a) n ≡ 0(mod 2) b) n ≡ 1(mod 2)

$$\mathrm{Evaluate}\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} {dx}\:\:\mathrm{for}\: \\ $$$${n}\:\in\:\mathbb{Z}\cap\left[\mathrm{0};\infty\right)\:\left(\mathrm{i}.\mathrm{e}.\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:...\right)\:\mathrm{and}: \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\:\:{n}\:\equiv\:\mathrm{0}\left({mod}\:\mathrm{2}\right) \\ $$$$\left.\boldsymbol{{b}}\right)\:{n}\:\equiv\:\mathrm{1}\left({mod}\:\mathrm{2}\right) \\ $$

Question Number 17524 Answers: 1 Comments: 0

The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY be a diameter of Ω which is also tangent to ω. Assume PY > PX. Let PY intersect ω at Z. If YZ = 2PZ, what is the magnitude of ∠PYX in degrees?

$$\mathrm{The}\:\mathrm{circle}\:\omega\:\mathrm{touches}\:\mathrm{the}\:\mathrm{circle}\:\Omega \\ $$$$\mathrm{internally}\:\mathrm{at}\:{P}.\:\mathrm{The}\:\mathrm{centre}\:{O}\:\mathrm{of}\:\Omega\:\mathrm{is} \\ $$$$\mathrm{outside}\:\omega.\:\mathrm{Let}\:{XY}\:\mathrm{be}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{of}\:\Omega \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{also}\:\mathrm{tangent}\:\mathrm{to}\:\omega.\:\mathrm{Assume} \\ $$$${PY}\:>\:{PX}.\:\mathrm{Let}\:{PY}\:\mathrm{intersect}\:\omega\:\mathrm{at}\:{Z}.\:\mathrm{If} \\ $$$${YZ}\:=\:\mathrm{2}{PZ},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{magnitude}\:\mathrm{of} \\ $$$$\angle{PYX}\:\mathrm{in}\:\mathrm{degrees}? \\ $$

Question Number 17522 Answers: 0 Comments: 1

∫(dx/(sin^5 x+cos^5 x))

$$\int\frac{{dx}}{\mathrm{sin}\:^{\mathrm{5}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}} \\ $$

Question Number 17520 Answers: 1 Comments: 1

Find the coordinate of the point in RΛ3 which is the reflection the point (1,2,3) with respect to plane X+Y+Z=1 .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coordinate}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{in} \\ $$$$\mathrm{R}\Lambda\mathrm{3}\:\mathrm{which}\:\mathrm{is}\:\mathrm{the}\:\mathrm{reflection}\:\mathrm{the}\:\mathrm{point} \\ $$$$\left(\mathrm{1},\mathrm{2},\mathrm{3}\right)\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{plane}\: \\ $$$$\mathrm{X}+\mathrm{Y}+\mathrm{Z}=\mathrm{1}\:. \\ $$

Question Number 17514 Answers: 1 Comments: 2

S(n)=∫_0 ^1 x^(2n) sin(2nπx)dx

$${S}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}{n}} {sin}\left(\mathrm{2}{n}\pi{x}\right){dx} \\ $$

Question Number 17512 Answers: 1 Comments: 0

∫ ((e^(−t) ln(1 + e^(−t) ))/(1 + e^(−t) )) dt

$$\int\:\frac{\mathrm{e}^{−\mathrm{t}} \:\mathrm{ln}\left(\mathrm{1}\:+\:\mathrm{e}^{−\mathrm{t}} \right)}{\mathrm{1}\:+\:\mathrm{e}^{−\mathrm{t}} }\:\mathrm{dt} \\ $$

Question Number 17506 Answers: 1 Comments: 0

∫ tan^(−1) ((√((x + 1)/(x − 1)))) dx

$$\int\:\mathrm{tan}^{−\mathrm{1}} \left(\sqrt{\frac{\mathrm{x}\:+\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{1}}}\right)\:\mathrm{dx} \\ $$

Question Number 17497 Answers: 1 Comments: 0

Evaluate: sin(9)°

$$\mathrm{Evaluate}:\:\:\:\mathrm{sin}\left(\mathrm{9}\right)° \\ $$

Question Number 17481 Answers: 0 Comments: 0

Question Number 17480 Answers: 1 Comments: 0

why (d/dx)(∫_0 ^( y) e^t dt)=e^y (dy/dx)

$$\mathrm{why}\:\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{0}} ^{\:\:\mathrm{y}} \mathrm{e}^{\mathrm{t}} \mathrm{dt}\right)=\mathrm{e}^{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}} \\ $$

Question Number 17475 Answers: 1 Comments: 0

Prove that ∫_e ^e^2 ((lnx)/((1+lnx)^2 ))dx=(e/6)(2e−3)

$$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{e}} ^{\mathrm{e}^{\mathrm{2}} } \frac{\mathrm{ln}{x}}{\left(\mathrm{1}+\mathrm{ln}{x}\right)^{\mathrm{2}} }\mathrm{d}{x}=\frac{\mathrm{e}}{\mathrm{6}}\left(\mathrm{2e}−\mathrm{3}\right) \\ $$

Question Number 17473 Answers: 1 Comments: 0

Prove that ∫_0 ^( 2a) (√(2ax−x^2 ))dx=((Πa^2 )/2)

$$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\mathrm{2a}} \sqrt{\mathrm{2ax}−\mathrm{x}^{\mathrm{2}} }\mathrm{dx}=\frac{\Pi\mathrm{a}^{\mathrm{2}} }{\mathrm{2}} \\ $$

Question Number 17472 Answers: 1 Comments: 0

Prove that ∫_6 ^(11) (dx/(√((x−2)(x−3))))=2ln((3+2(√2))/(2+(√3)))

$$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{6}} ^{\mathrm{11}} \frac{\mathrm{dx}}{\sqrt{\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}}=\mathrm{2ln}\frac{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$

Question Number 17479 Answers: 0 Comments: 0

If f(x) is a periodic function with period time t ; prove that∫_a ^( a+t) f(x)dx is a indipendent.

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{function}\:\mathrm{with}\:\mathrm{period} \\ $$$$\mathrm{time}\:\mathrm{t}\:;\:\mathrm{prove}\:\mathrm{that}\int_{\mathrm{a}} ^{\:\mathrm{a}+\mathrm{t}} {f}\left({x}\right)\mathrm{d}{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{indipendent}. \\ $$

Question Number 17449 Answers: 1 Comments: 4

Between 2:00 and 2:15, what time is it exactly when the hour, minute, and second′s hand of a clock occupy the same angular position.

$$\mathrm{Between}\:\mathrm{2}:\mathrm{00}\:\mathrm{and}\:\mathrm{2}:\mathrm{15},\:\mathrm{what}\:\mathrm{time} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{exactly}\:\mathrm{when}\:\mathrm{the}\:\mathrm{hour}, \\ $$$$\mathrm{minute},\:\mathrm{and}\:\mathrm{second}'\mathrm{s}\:\mathrm{hand}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{clock}\:\mathrm{occupy}\:\mathrm{the}\:\mathrm{same}\:\mathrm{angular} \\ $$$$\mathrm{position}. \\ $$$$ \\ $$

Question Number 17492 Answers: 1 Comments: 0

The number of values of x lying in [−π, π] and satisfying 2 sin^2 θ = cos 2θ and sin 2θ + 2 cos 2θ − cos θ − 1 = 0 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{lying}\:\mathrm{in} \\ $$$$\left[−\pi,\:\pi\right]\:\mathrm{and}\:\mathrm{satisfying}\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:\theta\:=\:\mathrm{cos}\:\mathrm{2}\theta \\ $$$$\mathrm{and}\:\mathrm{sin}\:\mathrm{2}\theta\:+\:\mathrm{2}\:\mathrm{cos}\:\mathrm{2}\theta\:−\:\mathrm{cos}\:\theta\:−\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{is} \\ $$

Question Number 17446 Answers: 0 Comments: 2

Find the integer closest to 100(12 − (√(143))).

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{closest}\:\mathrm{to} \\ $$$$\mathrm{100}\left(\mathrm{12}\:−\:\sqrt{\mathrm{143}}\right). \\ $$

Question Number 17444 Answers: 1 Comments: 0

Evaluate: ∫_0 ^(π/4) (dx/(cos^3 x (√(2 sin 2x))))

$$\mathrm{Evaluate}:\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\frac{{dx}}{\mathrm{cos}^{\mathrm{3}} \:{x}\:\sqrt{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}{x}}} \\ $$

Question Number 17440 Answers: 1 Comments: 0

3x−4y=0,4y−5z=0,5z−3x=0 then x,y,z is AP,GP,HP,AGP??????

$$\mathrm{3}{x}−\mathrm{4}{y}=\mathrm{0},\mathrm{4}{y}−\mathrm{5}{z}=\mathrm{0},\mathrm{5}{z}−\mathrm{3}{x}=\mathrm{0} \\ $$$${then}\:{x},{y},{z}\:{is}\:{AP},{GP},{HP},{AGP}?????? \\ $$

Question Number 17645 Answers: 2 Comments: 1

Suppose that the point M lying in the interior of the parallelogram ABCD, two parallels to AB and AD are drawn, intersecting the sides of ABCD at the points P, Q, R, S (See Figure). Prove that M lies on the diagonal AC if and only if [MRDS] = [MPBQ].

$$\mathrm{Suppose}\:\mathrm{that}\:\mathrm{the}\:\mathrm{point}\:{M}\:\mathrm{lying}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{interior}\:\mathrm{of}\:\mathrm{the}\:\mathrm{parallelogram}\:{ABCD}, \\ $$$$\mathrm{two}\:\mathrm{parallels}\:\mathrm{to}\:{AB}\:\mathrm{and}\:{AD}\:\mathrm{are}\:\mathrm{drawn}, \\ $$$$\mathrm{intersecting}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:{ABCD}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{points}\:{P},\:{Q},\:{R},\:{S}\:\left(\mathrm{See}\:\mathrm{Figure}\right).\:\mathrm{Prove} \\ $$$$\mathrm{that}\:{M}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{diagonal}\:{AC}\:\mathrm{if}\:\mathrm{and} \\ $$$$\mathrm{only}\:\mathrm{if}\:\left[{MRDS}\right]\:=\:\left[{MPBQ}\right]. \\ $$

Question Number 17438 Answers: 0 Comments: 0

find the mean value and root mean square of i=25sin100Πt ranging from 0 to 10

$$\mathrm{find}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value}\:\mathrm{and}\:\mathrm{root}\:\mathrm{mean}\:\mathrm{square}\:\mathrm{of}\: \\ $$$$\mathrm{i}=\mathrm{25sin100}\Pi\mathrm{t}\:\:\:\:\:\mathrm{ranging}\:\mathrm{from}\:\mathrm{0}\:\mathrm{to}\:\mathrm{10} \\ $$

Question Number 17435 Answers: 1 Comments: 0

Find the value of 4 sin (π/(24)) cos (π/(12)) cos(π/6).

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{4}\:\mathrm{sin}\:\frac{\pi}{\mathrm{24}}\:\:\mathrm{cos}\:\frac{\pi}{\mathrm{12}}\:\:\mathrm{cos}\frac{\pi}{\mathrm{6}}. \\ $$

Question Number 17421 Answers: 1 Comments: 0

The number of solutions of the equation 2^(∣x∣) = 1 + 2∣cos x∣ is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{2}^{\mid{x}\mid} \:=\:\mathrm{1}\:+\:\mathrm{2}\mid\mathrm{cos}\:{x}\mid\:\mathrm{is} \\ $$

Question Number 17420 Answers: 1 Comments: 0

tan^6 (π/9)−33tan^4 (π/9)+27tan^2 (π/9)=?

$$\mathrm{tan}^{\mathrm{6}} \frac{\pi}{\mathrm{9}}−\mathrm{33tan}^{\mathrm{4}} \frac{\pi}{\mathrm{9}}+\mathrm{27tan}^{\mathrm{2}} \frac{\pi}{\mathrm{9}}=? \\ $$

Question Number 17454 Answers: 1 Comments: 0

Find all integers n such that (n^2 − n − 1)^(n + 2) = 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left({n}^{\mathrm{2}} \:−\:{n}\:−\:\mathrm{1}\right)^{{n}\:+\:\mathrm{2}} \:=\:\mathrm{1} \\ $$

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