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Question Number 20764    Answers: 1   Comments: 0

A small bead is slipped on a horizontal rod of length l. The rod starts moving with a horizontal acceleration a in a direction making an angle α with the length of the rod. Assuming that initially the bead is in the middle of the rod, find the time elapsed before the bead leaves the rod. Coefficient of friction between the bead and the rod is μ. (Neglect gravity).

$${A}\:{small}\:{bead}\:{is}\:{slipped}\:{on}\:{a}\:{horizontal} \\ $$$${rod}\:{of}\:{length}\:{l}.\:{The}\:{rod}\:{starts}\:{moving} \\ $$$${with}\:{a}\:{horizontal}\:{acceleration}\:{a}\:{in}\:{a} \\ $$$${direction}\:{making}\:{an}\:{angle}\:\alpha\:{with}\:{the} \\ $$$${length}\:{of}\:{the}\:{rod}.\:{Assuming}\:{that} \\ $$$${initially}\:{the}\:{bead}\:{is}\:{in}\:{the}\:{middle}\:{of} \\ $$$${the}\:{rod},\:{find}\:{the}\:{time}\:{elapsed}\:{before} \\ $$$${the}\:{bead}\:{leaves}\:{the}\:{rod}.\:{Coefficient}\:{of} \\ $$$${friction}\:{between}\:{the}\:{bead}\:{and}\:{the}\:{rod} \\ $$$${is}\:\mu.\:\left({Neglect}\:{gravity}\right). \\ $$

Question Number 20745    Answers: 1   Comments: 3

Consider a disc rotating in the horizontal plane with a constant angular speed ω about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the Figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R. The velocity of projection is in the y-z plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed (1/8) rotation, (ii) their range is less than half the disc radius, and (iii) ω remains constant throughout. Then (a) P lands in the shaded region and Q in the unshaded region (b) P lands in the unshaded region and Q in the shaded region (c) Both P and Q land in the unshaded region (d) Both P and Q land in the shaded region

$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{disc}\:\mathrm{rotating}\:\mathrm{in}\:\mathrm{the}\:\mathrm{horizontal} \\ $$$$\mathrm{plane}\:\mathrm{with}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{angular}\:\mathrm{speed}\:\omega \\ $$$$\mathrm{about}\:\mathrm{its}\:\mathrm{centre}\:{O}.\:\mathrm{The}\:\mathrm{disc}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{shaded}\:\mathrm{region}\:\mathrm{on}\:\mathrm{one}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diameter} \\ $$$$\mathrm{and}\:\mathrm{an}\:\mathrm{unshaded}\:\mathrm{region}\:\mathrm{on}\:\mathrm{the}\:\mathrm{other} \\ $$$$\mathrm{side}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Figure}.\:\mathrm{When}\:\mathrm{the} \\ $$$$\mathrm{disc}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:\mathrm{orientation}\:\mathrm{as}\:\mathrm{shown},\:\mathrm{two} \\ $$$$\mathrm{pebbles}\:{P}\:\mathrm{and}\:{Q}\:\mathrm{are}\:\mathrm{simultaneously} \\ $$$$\mathrm{projected}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{towards}\:{R}.\:\mathrm{The} \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{projection}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:{y}-{z} \\ $$$$\mathrm{plane}\:\mathrm{and}\:\mathrm{is}\:\mathrm{same}\:\mathrm{for}\:\mathrm{both}\:\mathrm{pebbles}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{the}\:\mathrm{disc}.\:\mathrm{Assume}\:\mathrm{that}\:\left({i}\right)\:\mathrm{they} \\ $$$$\mathrm{land}\:\mathrm{back}\:\mathrm{on}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{before}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{has} \\ $$$$\mathrm{completed}\:\frac{\mathrm{1}}{\mathrm{8}}\:\mathrm{rotation},\:\left({ii}\right)\:\mathrm{their}\:\mathrm{range} \\ $$$$\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{half}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{radius},\:\mathrm{and} \\ $$$$\left({iii}\right)\:\omega\:\mathrm{remains}\:\mathrm{constant}\:\mathrm{throughout}. \\ $$$$\mathrm{Then} \\ $$$$\left({a}\right)\:{P}\:\mathrm{lands}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shaded}\:\mathrm{region}\:\mathrm{and}\:{Q} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{unshaded}\:\mathrm{region} \\ $$$$\left({b}\right)\:{P}\:\mathrm{lands}\:\mathrm{in}\:\mathrm{the}\:\mathrm{unshaded}\:\mathrm{region}\:\mathrm{and} \\ $$$${Q}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shaded}\:\mathrm{region} \\ $$$$\left({c}\right)\:\mathrm{Both}\:{P}\:\mathrm{and}\:{Q}\:\mathrm{land}\:\mathrm{in}\:\mathrm{the}\:\mathrm{unshaded} \\ $$$$\mathrm{region} \\ $$$$\left({d}\right)\:\mathrm{Both}\:{P}\:\mathrm{and}\:{Q}\:\mathrm{land}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shaded} \\ $$$$\mathrm{region} \\ $$

Question Number 20744    Answers: 1   Comments: 0

Question Number 20739    Answers: 1   Comments: 0

If ∣z_1 ∣ = 2, ∣z_2 ∣ = 3, ∣z_3 ∣ = 4 and ∣2z_1 + 3z_2 + 4z_3 ∣ = 4, then the expression ∣8z_2 z_3 + 27z_3 z_1 + 64z_1 z_2 ∣ equals

$$\mathrm{If}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mathrm{2},\:\mid{z}_{\mathrm{2}} \mid\:=\:\mathrm{3},\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{4}\:\mathrm{and} \\ $$$$\mid\mathrm{2}{z}_{\mathrm{1}} \:+\:\mathrm{3}{z}_{\mathrm{2}} \:+\:\mathrm{4}{z}_{\mathrm{3}} \mid\:=\:\mathrm{4},\:\mathrm{then}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\mid\mathrm{8}{z}_{\mathrm{2}} {z}_{\mathrm{3}} \:+\:\mathrm{27}{z}_{\mathrm{3}} {z}_{\mathrm{1}} \:+\:\mathrm{64}{z}_{\mathrm{1}} {z}_{\mathrm{2}} \mid\:\mathrm{equals} \\ $$

Question Number 20733    Answers: 1   Comments: 0

Find the volume of the solid of revolution obtained by revolving area bounded by x = 4 + 6y − 2y^2 , x = −4, x = 0 about the y−axis

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid}\:\mathrm{of}\:\mathrm{revolution} \\ $$$$\mathrm{obtained}\:\mathrm{by}\:\mathrm{revolving}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{by} \\ $$$${x}\:=\:\mathrm{4}\:+\:\mathrm{6}{y}\:−\:\mathrm{2}{y}^{\mathrm{2}} ,\:{x}\:=\:−\mathrm{4},\:{x}\:=\:\mathrm{0}\:\mathrm{about} \\ $$$$\mathrm{the}\:{y}−\mathrm{axis} \\ $$

Question Number 20724    Answers: 0   Comments: 3

A sphere is rolling without slipping on a fixed horizontal plane surface. In the Figure, A is a point of contact, B is the centre of the sphere and C is its topmost point. Then (a) v_C ^→ − v_A ^→ = 2(v_B ^→ − v_C ^→ ) (b) v_C ^→ − v_B ^→ = v_B ^→ − v_A ^→ (c) ∣v_C ^→ − v_A ^→ ∣ = 2∣v_B ^→ − v_C ^→ ∣ (d) ∣v_C ^→ − v_A ^→ ∣ = 4∣v_B ^→ ∣

$$\mathrm{A}\:\mathrm{sphere}\:\mathrm{is}\:\mathrm{rolling}\:\mathrm{without}\:\mathrm{slipping}\:\mathrm{on} \\ $$$$\mathrm{a}\:\mathrm{fixed}\:\mathrm{horizontal}\:\mathrm{plane}\:\mathrm{surface}.\:\mathrm{In}\:\mathrm{the} \\ $$$$\mathrm{Figure},\:\mathrm{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{of}\:\mathrm{contact},\:{B}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere}\:\mathrm{and}\:{C}\:\mathrm{is}\:\mathrm{its}\:\mathrm{topmost} \\ $$$$\mathrm{point}.\:\mathrm{Then} \\ $$$$\left({a}\right)\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \:=\:\mathrm{2}\left(\overset{\rightarrow} {{v}}_{\mathrm{B}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \right) \\ $$$$\left({b}\right)\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{B}} \:=\:\overset{\rightarrow} {{v}}_{\mathrm{B}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \\ $$$$\left({c}\right)\:\mid\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \mid\:=\:\mathrm{2}\mid\overset{\rightarrow} {{v}}_{\mathrm{B}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \mid \\ $$$$\left({d}\right)\:\mid\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \mid\:=\:\mathrm{4}\mid\overset{\rightarrow} {{v}}_{\mathrm{B}} \mid \\ $$

Question Number 20723    Answers: 0   Comments: 0

∫(√x) .sinx.dx

$$\int\sqrt{{x}}\:.{sinx}.{dx} \\ $$

Question Number 20721    Answers: 1   Comments: 0

tan xtan z=3 tan ytan z=6 x+y+z = π Solve for x, y, and z.

$$\mathrm{tan}\:{x}\mathrm{tan}\:{z}=\mathrm{3} \\ $$$$\mathrm{tan}\:{y}\mathrm{tan}\:{z}=\mathrm{6} \\ $$$${x}+{y}+{z}\:=\:\pi \\ $$$${Solve}\:{for}\:{x},\:{y},\:{and}\:{z}. \\ $$

Question Number 20705    Answers: 1   Comments: 0

Question Number 20704    Answers: 1   Comments: 2

4cos θcos (((2π)/3)+θ)cos (((4π)/3)+θ)=cos 3θ

$$\mathrm{4cos}\:\theta\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{3}}+\theta\right)\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{3}}+\theta\right)=\mathrm{cos}\:\mathrm{3}\theta \\ $$

Question Number 20703    Answers: 1   Comments: 1

((sinα+sin 3α+sin5α)/(cos α+cos 3α+cos 5α))=tan 3α

$$\frac{\mathrm{sin}\alpha+\mathrm{sin}\:\mathrm{3}\alpha+\mathrm{sin5}\alpha}{\mathrm{cos}\:\alpha+\mathrm{cos}\:\mathrm{3}\alpha+\mathrm{cos}\:\mathrm{5}\alpha}=\mathrm{tan}\:\mathrm{3}\alpha \\ $$

Question Number 20702    Answers: 1   Comments: 0

tan 20°tan 40°tan 80°=(√3)

$$\mathrm{tan}\:\mathrm{20}°\mathrm{tan}\:\mathrm{40}°\mathrm{tan}\:\mathrm{80}°=\sqrt{\mathrm{3}} \\ $$

Question Number 20693    Answers: 1   Comments: 0

Integers 1, 2, 3, ...., n, where n > 2, are written on a board. Two numbers m, k such that 1 < m < n, 1 < k < n are removed and the average of the remaining numbers is found to be 17. What is the maximum sum of the two removed numbers?

$$\mathrm{Integers}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:....,\:{n},\:\mathrm{where}\:{n}\:>\:\mathrm{2},\:\mathrm{are} \\ $$$$\mathrm{written}\:\mathrm{on}\:\mathrm{a}\:\mathrm{board}.\:\mathrm{Two}\:\mathrm{numbers}\:{m},\:{k} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{1}\:<\:{m}\:<\:{n},\:\mathrm{1}\:<\:{k}\:<\:{n}\:\mathrm{are} \\ $$$$\mathrm{removed}\:\mathrm{and}\:\mathrm{the}\:\mathrm{average}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{remaining}\:\mathrm{numbers}\:\mathrm{is}\:\mathrm{found}\:\mathrm{to}\:\mathrm{be}\:\mathrm{17}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{removed}\:\mathrm{numbers}? \\ $$

Question Number 20761    Answers: 0   Comments: 5

A 5 kg block B is suspended from a cord attached to a 40 kg cart A. Find the accelerations of both the block and cart. (All surfaces are frictionless) (g = 10 m/s^2 )

$${A}\:\mathrm{5}\:{kg}\:{block}\:{B}\:{is}\:{suspended}\:{from}\:{a} \\ $$$${cord}\:{attached}\:{to}\:{a}\:\mathrm{40}\:{kg}\:{cart}\:{A}.\:{Find} \\ $$$${the}\:{accelerations}\:{of}\:{both}\:{the}\:{block}\:{and} \\ $$$${cart}.\:\left({All}\:{surfaces}\:{are}\:{frictionless}\right) \\ $$$$\left({g}\:=\:\mathrm{10}\:{m}/{s}^{\mathrm{2}} \right) \\ $$

Question Number 20686    Answers: 2   Comments: 1

∫(tanx)^(1/3) dx

$$\int\left({tanx}\right)^{\mathrm{1}/\mathrm{3}} {dx} \\ $$

Question Number 20684    Answers: 1   Comments: 0

If the equation x^2 + β^2 = 1 − 2βx and x^2 + α^2 = 1 − 2αx have one and only one root in common, then ∣α − β∣ is equal to

$${If}\:{the}\:{equation}\:{x}^{\mathrm{2}} \:+\:\beta^{\mathrm{2}} \:=\:\mathrm{1}\:−\:\mathrm{2}\beta{x}\:{and} \\ $$$${x}^{\mathrm{2}} \:+\:\alpha^{\mathrm{2}} \:=\:\mathrm{1}\:−\:\mathrm{2}\alpha{x}\:{have}\:{one}\:{and}\:{only} \\ $$$${one}\:{root}\:{in}\:{common},\:{then}\:\mid\alpha\:−\:\beta\mid\:{is} \\ $$$${equal}\:{to} \\ $$

Question Number 20671    Answers: 1   Comments: 0

The total number of positive integral solution(s) of the inequation ((x^2 (3x − 4)^3 (x − 2)^4 )/((x − 5)^5 (2x − 7)^6 )) ≤ 0 is/are

$${The}\:{total}\:{number}\:{of}\:{positive}\:{integral} \\ $$$${solution}\left({s}\right)\:{of}\:{the}\:{inequation} \\ $$$$\frac{{x}^{\mathrm{2}} \left(\mathrm{3}{x}\:−\:\mathrm{4}\right)^{\mathrm{3}} \left({x}\:−\:\mathrm{2}\right)^{\mathrm{4}} }{\left({x}\:−\:\mathrm{5}\right)^{\mathrm{5}} \left(\mathrm{2}{x}\:−\:\mathrm{7}\right)^{\mathrm{6}} }\:\leqslant\:\mathrm{0}\:{is}/{are} \\ $$

Question Number 20666    Answers: 1   Comments: 0

∫cot^4 xdx

$$\int\mathrm{cot}\:^{\mathrm{4}} {xdx} \\ $$

Question Number 20670    Answers: 1   Comments: 0

For the equation 3x^2 + px + 3 = 0, find the value(s) of p if one root is (i) square of the other (ii) fourth power of the other.

$${For}\:{the}\:{equation}\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{px}\:+\:\mathrm{3}\:=\:\mathrm{0}, \\ $$$${find}\:{the}\:{value}\left({s}\right)\:{of}\:{p}\:{if}\:{one}\:{root}\:{is} \\ $$$$\left({i}\right)\:{square}\:{of}\:{the}\:{other} \\ $$$$\left({ii}\right)\:{fourth}\:{power}\:{of}\:{the}\:{other}. \\ $$

Question Number 20653    Answers: 2   Comments: 0

Length of interval of range of function f(θ) = cos^2 θ − 6 sin θ cos θ + 3 sin^2 θ + 2 is (1) 8 (2) −8 (3) (√(10)) (4) 2(√(10))

$$\mathrm{Length}\:\mathrm{of}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{range}\:\mathrm{of}\:\mathrm{function} \\ $$$${f}\left(\theta\right)\:=\:\mathrm{cos}^{\mathrm{2}} \:\theta\:−\:\mathrm{6}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:+\:\mathrm{3}\:\mathrm{sin}^{\mathrm{2}} \:\theta\:+\:\mathrm{2} \\ $$$$\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{8} \\ $$$$\left(\mathrm{2}\right)\:−\mathrm{8} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\mathrm{10}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\sqrt{\mathrm{10}} \\ $$

Question Number 20652    Answers: 0   Comments: 4

Suppose x is a positive real number such that {x}, [x] and x are in a geometric progression. Find the least positive integer n such that x^n > 100. (Here [x] denotes the integer part of x and {x} = x − [x].)

$$\mathrm{Suppose}\:{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left\{{x}\right\},\:\left[{x}\right]\:\mathrm{and}\:{x}\:\mathrm{are}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{geometric}\:\mathrm{progression}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{such}\:\mathrm{that}\:{x}^{{n}} \:>\:\mathrm{100}. \\ $$$$\left(\mathrm{Here}\:\left[{x}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{of}\:{x}\right. \\ $$$$\left.\mathrm{and}\:\left\{{x}\right\}\:=\:{x}\:−\:\left[{x}\right].\right) \\ $$

Question Number 20648    Answers: 1   Comments: 0

If (F/(sin (A − a))) = (W/(sin A)), prove that tan A = ((W sin a)/(W cos a − F))

$$\mathrm{If}\:\:\frac{{F}}{\mathrm{sin}\:\left({A}\:−\:{a}\right)}\:=\:\frac{{W}}{\mathrm{sin}\:{A}},\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{tan}\:{A}\:=\:\frac{{W}\:\mathrm{sin}\:{a}}{{W}\:\mathrm{cos}\:{a}\:−\:{F}} \\ $$

Question Number 20647    Answers: 1   Comments: 0

prove it, tan (α+(π/3))+tan (α−(π/3))=((4sin 2α)/(1−4sin^2 α))

$${prove}\:{it}, \\ $$$$\mathrm{tan}\:\left(\alpha+\frac{\pi}{\mathrm{3}}\right)+\mathrm{tan}\:\left(\alpha−\frac{\pi}{\mathrm{3}}\right)=\frac{\mathrm{4sin}\:\mathrm{2}\alpha}{\mathrm{1}−\mathrm{4sin}\:^{\mathrm{2}} \alpha} \\ $$

Question Number 20646    Answers: 1   Comments: 0

if cos (β−γ)+cos( γ−α)+cos (α−β)=−(3/2) so proof it, Σcos α=0,Σsin α=0

$${if}\:\mathrm{cos}\:\left(\beta−\gamma\right)+\mathrm{cos}\left(\:\gamma−\alpha\right)+\mathrm{cos}\:\left(\alpha−\beta\right)=−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${so}\:{proof}\:{it}, \\ $$$$\Sigma\mathrm{cos}\:\alpha=\mathrm{0},\Sigma\mathrm{sin}\:\alpha=\mathrm{0} \\ $$

Question Number 20664    Answers: 0   Comments: 0

In what time shall a carpet of length l and radius R unroll on a flat rough ground if given a little push to start the motion.

$${In}\:{what}\:{time}\:{shall}\:{a}\:{carpet}\:{of} \\ $$$${length}\:\boldsymbol{{l}}\:{and}\:{radius}\:\boldsymbol{{R}}\:{unroll}\:{on} \\ $$$${a}\:{flat}\:{rough}\:{ground}\:{if}\:{given}\:{a}\: \\ $$$${little}\:{push}\:{to}\:{start}\:{the}\:{motion}. \\ $$

Question Number 20639    Answers: 2   Comments: 0

if cos (A+B)sin (C+D)=cos (A−B) sin (C−D) so proof cot Acot Bcot C=cot D

$${if}\:\mathrm{cos}\:\left({A}+{B}\right)\mathrm{sin}\:\left({C}+{D}\right)=\mathrm{cos}\:\left({A}−{B}\right) \\ $$$$\mathrm{sin}\:\left({C}−{D}\right) \\ $$$${so}\:{proof}\:\mathrm{cot}\:{A}\mathrm{cot}\:{B}\mathrm{cot}\:{C}=\mathrm{cot}\:{D} \\ $$

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