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AllQuestion and Answers: Page 1831

Question Number 15865    Answers: 2   Comments: 0

a,b,c∈R^(+ ) andIf a+b+c=18 then maximum value of a^2 b^3 c^4 is

$${a},{b},{c}\in\mathbb{R}^{+\:} \mathrm{andIf}\:{a}+{b}+{c}=\mathrm{18}\:\mathrm{then}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{of}\:{a}^{\mathrm{2}} {b}^{\mathrm{3}} {c}^{\mathrm{4}} \:\mathrm{is} \\ $$

Question Number 15856    Answers: 1   Comments: 0

A particle is moving along a straight line with uniform acceleration has velocities 7 m/s at P and 17 m/s at Q. If R is the midpoint of PQ, then the average velocity between P and R is?

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{along}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{with}\:\mathrm{uniform}\:\mathrm{acceleration}\:\mathrm{has} \\ $$$$\mathrm{velocities}\:\mathrm{7}\:\mathrm{m}/\mathrm{s}\:\mathrm{at}\:{P}\:\mathrm{and}\:\mathrm{17}\:\mathrm{m}/\mathrm{s}\:\mathrm{at}\:{Q}. \\ $$$$\mathrm{If}\:{R}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:{PQ},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{average}\:\mathrm{velocity}\:\mathrm{between}\:{P}\:\mathrm{and}\:{R}\:\mathrm{is}? \\ $$

Question Number 15850    Answers: 0   Comments: 0

Let (X, T) be any topological space . Verify that the intersection of any finite number of members of T is a member of T. Use mathematical induction to prove your result.

$$\mathrm{Let}\:\left(\mathrm{X},\:\mathrm{T}\right)\:\mathrm{be}\:\mathrm{any}\:\mathrm{topological}\:\mathrm{space}\:.\:\mathrm{Verify}\:\mathrm{that}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{of}\:\mathrm{any}\: \\ $$$$\mathrm{finite}\:\mathrm{number}\:\mathrm{of}\:\mathrm{members}\:\mathrm{of}\:\mathrm{T}\:\mathrm{is}\:\mathrm{a}\:\mathrm{member}\:\mathrm{of}\:\mathrm{T}.\:\mathrm{Use}\:\mathrm{mathematical} \\ $$$$\mathrm{induction}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{your}\:\mathrm{result}. \\ $$

Question Number 15840    Answers: 1   Comments: 2

Solve the ODE (dy/dx) + x^2 y = x^4

$$\mathrm{Solve}\:\mathrm{the}\:\:\mathrm{ODE} \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{y}\:=\:\mathrm{x}^{\mathrm{4}} \\ $$$$ \\ $$

Question Number 15839    Answers: 0   Comments: 1

Question Number 15835    Answers: 0   Comments: 1

Number of decimal digits in 50! is

$$\:\mathrm{Number}\:\mathrm{of}\: \\ $$$$\mathrm{decimal}\:\:\mathrm{digits}\: \\ $$$$\mathrm{in}\:\:\mathrm{50}!\:\mathrm{is} \\ $$

Question Number 15832    Answers: 1   Comments: 0

A passenger in a train moving at an acceleration a, drops a stone from the window. A person, standing on the ground, by the sides of the rails, observes the ball falling (1) Vertically with an acceleration (√(g^2 + a^2 )) (2) Horizontally with an acceleration (√(g^2 + a^2 )) (3) Along a parabola with an acceleration (√(g^2 + a^2 )) (4) Along a parabola with an acceleration g

$$\mathrm{A}\:\mathrm{passenger}\:\mathrm{in}\:\mathrm{a}\:\mathrm{train}\:\mathrm{moving}\:\mathrm{at}\:\mathrm{an} \\ $$$$\mathrm{acceleration}\:{a},\:\mathrm{drops}\:\mathrm{a}\:\mathrm{stone}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{window}.\:\mathrm{A}\:\mathrm{person},\:\mathrm{standing}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{ground},\:\mathrm{by}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rails}, \\ $$$$\mathrm{observes}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{falling} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Vertically}\:\mathrm{with}\:\mathrm{an}\:\mathrm{acceleration} \\ $$$$\sqrt{{g}^{\mathrm{2}} \:+\:{a}^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Horizontally}\:\mathrm{with}\:\mathrm{an}\:\mathrm{acceleration} \\ $$$$\sqrt{{g}^{\mathrm{2}} \:+\:{a}^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Along}\:\mathrm{a}\:\mathrm{parabola}\:\mathrm{with}\:\mathrm{an}\:\mathrm{acceleration} \\ $$$$\sqrt{{g}^{\mathrm{2}} \:+\:{a}^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Along}\:\mathrm{a}\:\mathrm{parabola}\:\mathrm{with}\:\mathrm{an}\:\mathrm{acceleration} \\ $$$${g} \\ $$

Question Number 15828    Answers: 1   Comments: 0

In a ΔABC, let M_a , M_b and M_c denote the length of medians, s = ((M_a + M_b + M_c )/2) and Δ = ar(ΔABC). Prove that Δ = (4/3)(√(s(s − M_a )(s − M_b )(s − M_c )))

$$\mathrm{In}\:\mathrm{a}\:\Delta{ABC},\:\mathrm{let}\:{M}_{{a}} ,\:{M}_{{b}} \:\mathrm{and}\:{M}_{{c}} \:\mathrm{denote} \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{medians}, \\ $$$${s}\:=\:\frac{{M}_{{a}} \:+\:{M}_{{b}} \:+\:{M}_{{c}} }{\mathrm{2}}\:\:\mathrm{and}\:\Delta\:=\:\mathrm{ar}\left(\Delta{ABC}\right). \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\Delta\:=\:\frac{\mathrm{4}}{\mathrm{3}}\sqrt{{s}\left({s}\:−\:{M}_{{a}} \right)\left({s}\:−\:{M}_{{b}} \right)\left({s}\:−\:{M}_{{c}} \right)} \\ $$

Question Number 15824    Answers: 2   Comments: 0

If in a ΔABC, cos A + cos B + cos C = (3/2) . Prove that ΔABC is an equilateral triangle.

$$\mathrm{If}\:\mathrm{in}\:\mathrm{a}\:\Delta{ABC},\:\mathrm{cos}\:{A}\:+\:\mathrm{cos}\:{B}\:+\:\mathrm{cos}\:{C}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\Delta{ABC}\:\mathrm{is}\:\mathrm{an}\:\mathrm{equilateral} \\ $$$$\mathrm{triangle}. \\ $$

Question Number 15821    Answers: 2   Comments: 2

Five engineers A,B,C,D and E can complete a process in 8 hours, assuming that every engineer works with the same efficiency. They started working at 10:00am. If after 4:00pm..,one engineer is removed from the group every hour,what is the time when they will finish the work? (a)6:00pm (b)7:00pm (c)4:00pm (d)8:00pm

$$\mathrm{Five}\:\mathrm{engineers}\:\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\:\mathrm{and}\:\mathrm{E}\:\mathrm{can} \\ $$$$\mathrm{complete}\:\mathrm{a}\:\mathrm{process}\:\mathrm{in}\:\mathrm{8}\:\mathrm{hours}, \\ $$$$\mathrm{assuming}\:\mathrm{that}\:\mathrm{every}\:\mathrm{engineer}\: \\ $$$$\mathrm{works}\:\mathrm{with}\:\mathrm{the}\:\mathrm{same}\:\mathrm{efficiency}. \\ $$$$\mathrm{They}\:\mathrm{started}\:\mathrm{working}\:\mathrm{at}\:\mathrm{10}:\mathrm{00am}. \\ $$$$\mathrm{If}\:\mathrm{after}\:\mathrm{4}:\mathrm{00pm}..,\mathrm{one}\:\mathrm{engineer}\:\mathrm{is} \\ $$$$\mathrm{removed}\:\mathrm{from}\:\mathrm{the}\:\mathrm{group}\:\mathrm{every}\: \\ $$$$\mathrm{hour},\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{time}\:\mathrm{when}\:\mathrm{they} \\ $$$$\mathrm{will}\:\mathrm{finish}\:\mathrm{the}\:\mathrm{work}? \\ $$$$\left(\mathrm{a}\right)\mathrm{6}:\mathrm{00pm} \\ $$$$\left(\mathrm{b}\right)\mathrm{7}:\mathrm{00pm} \\ $$$$\left(\mathrm{c}\right)\mathrm{4}:\mathrm{00pm} \\ $$$$\left(\mathrm{d}\right)\mathrm{8}:\mathrm{00pm} \\ $$

Question Number 15803    Answers: 0   Comments: 0

∫(dx/((1+x)(√((1+x−x^2 ))) ))

$$\int\frac{\mathrm{dx}}{\left(\mathrm{1}+\mathrm{x}\right)\sqrt{\left(\mathrm{1}+\mathrm{x}−\mathrm{x}^{\mathrm{2}} \right)}\:} \\ $$

Question Number 15800    Answers: 1   Comments: 0

ODE The rate at which the ice melt is proportional to the amount of ice present at the instant. Find the amount of ice left after 2 hours if half the quantity melt in 30 minute.

$$\mathrm{ODE} \\ $$$$\mathrm{The}\:\mathrm{rate}\:\mathrm{at}\:\mathrm{which}\:\mathrm{the}\:\mathrm{ice}\:\mathrm{melt}\:\mathrm{is}\:\mathrm{proportional}\:\mathrm{to}\:\mathrm{the}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{ice}\:\mathrm{present} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{instant}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{ice}\:\mathrm{left}\:\mathrm{after}\:\mathrm{2}\:\mathrm{hours}\:\mathrm{if}\:\mathrm{half}\:\mathrm{the}\:\mathrm{quantity} \\ $$$$\mathrm{melt}\:\mathrm{in}\:\mathrm{30}\:\mathrm{minute}. \\ $$

Question Number 15797    Answers: 1   Comments: 1

If (a + bx)e^(y/x) = x, where a and b are constant, prove that,: x^3 y′′ = (xy′ − y)^2

$$\mathrm{If}\:\:\left(\mathrm{a}\:+\:\mathrm{bx}\right)\mathrm{e}^{\mathrm{y}/\mathrm{x}} \:=\:\mathrm{x},\:\:\:\mathrm{where}\:\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{constant},\: \\ $$$$\mathrm{prove}\:\mathrm{that},:\:\:\:\mathrm{x}^{\mathrm{3}} \mathrm{y}''\:=\:\left(\mathrm{xy}'\:−\:\mathrm{y}\right)^{\mathrm{2}} \\ $$

Question Number 15789    Answers: 0   Comments: 0

Prove that: 2^(2^(2n + 1) ) + 2^2^(2n) + 1 , is never a prime for any positive n.

$$\mathrm{Prove}\:\mathrm{that}:\:\:\mathrm{2}^{\mathrm{2}^{\mathrm{2n}\:+\:\mathrm{1}} \:} +\:\mathrm{2}^{\mathrm{2}^{\mathrm{2n}} } \:+\:\mathrm{1}\:,\:\:\mathrm{is}\:\mathrm{never}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{for}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{n}. \\ $$

Question Number 15788    Answers: 0   Comments: 2

Find all rational solution of the equation a + b = ab

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{rational}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\:\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{ab} \\ $$

Question Number 15787    Answers: 0   Comments: 2

Find the four digits number such that 4 ∙ abcd = dcba

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{four}\:\mathrm{digits}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{4}\:\centerdot\:\mathrm{abcd}\:=\:\mathrm{dcba} \\ $$

Question Number 15786    Answers: 1   Comments: 1

Solve for x cos((x/7)) − cos(((2x)/7)) + cos(((3x)/7)) = (1/2)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x} \\ $$$$\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{7}}\right)\:−\:\mathrm{cos}\left(\frac{\mathrm{2x}}{\mathrm{7}}\right)\:+\:\mathrm{cos}\left(\frac{\mathrm{3x}}{\mathrm{7}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 15784    Answers: 1   Comments: 1

Question Number 15782    Answers: 1   Comments: 1

Question Number 15773    Answers: 1   Comments: 0

lim_(h→0) ((sin^3 (2x − 2h) − sin^3 (2x) )/(3h))

$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\:−\:\:\mathrm{2}{h}\right)\:−\:\mathrm{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\right)\:}{\mathrm{3}{h}} \\ $$

Question Number 15770    Answers: 1   Comments: 0

How to calculate the last two digits of 2^(576)

$$\mathrm{How}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\:\:\mathrm{2}^{\mathrm{576}} \\ $$

Question Number 15765    Answers: 0   Comments: 2

Question Number 15761    Answers: 2   Comments: 0

If in a ΔABC, ((2 cos A)/a) + ((cos B)/b) + ((2 cos C)/c) = (a/(bc)) + (b/(ac)) , prove that ∠A = 90°.

$$\mathrm{If}\:\mathrm{in}\:\mathrm{a}\:\Delta{ABC},\:\frac{\mathrm{2}\:\mathrm{cos}\:{A}}{{a}}\:+\:\frac{\mathrm{cos}\:{B}}{{b}}\:+\:\frac{\mathrm{2}\:\mathrm{cos}\:{C}}{{c}} \\ $$$$=\:\frac{{a}}{{bc}}\:+\:\frac{{b}}{{ac}}\:,\:\mathrm{prove}\:\mathrm{that}\:\angle{A}\:=\:\mathrm{90}°. \\ $$

Question Number 15759    Answers: 1   Comments: 0

Let us call complex triangle which has either sides or angles are complex numbers. Let a,b,c ∈R which are sides of a complex triangle which need not satisfy triangle inequality. say a=1,b=2 and c=4. Prove (or counter example) (a/(sin A))=(b/(sin B))=(c/(sin C)) Is A+B+C=π? Assume only principle solution for A,B and C. For such a triangle A,B and C will take complex values.

$$\mathrm{Let}\:\mathrm{us}\:\mathrm{call}\:\mathrm{complex}\:\mathrm{triangle}\:\mathrm{which} \\ $$$$\mathrm{has}\:\mathrm{either}\:\mathrm{sides}\:\mathrm{or}\:\mathrm{angles}\:\mathrm{are} \\ $$$$\mathrm{complex}\:\mathrm{numbers}. \\ $$$$\mathrm{Let}\:{a},{b},{c}\:\in\mathbb{R}\:\mathrm{which}\:\mathrm{are}\:\mathrm{sides}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{complex}\:\mathrm{triangle}\:\mathrm{which}\:\mathrm{need} \\ $$$$\mathrm{not}\:\mathrm{satisfy}\:\mathrm{triangle}\:\mathrm{inequality}. \\ $$$$\mathrm{say}\:{a}=\mathrm{1},{b}=\mathrm{2}\:\mathrm{and}\:{c}=\mathrm{4}. \\ $$$$\mathrm{Prove}\:\left(\mathrm{or}\:\mathrm{counter}\:\mathrm{example}\right) \\ $$$$\frac{{a}}{\mathrm{sin}\:{A}}=\frac{{b}}{\mathrm{sin}\:{B}}=\frac{{c}}{\mathrm{sin}\:{C}} \\ $$$$\mathrm{Is}\:{A}+{B}+{C}=\pi? \\ $$$$\mathrm{Assume}\:\mathrm{only}\:\mathrm{principle}\:\mathrm{solution} \\ $$$$\mathrm{for}\:{A},{B}\:\mathrm{and}\:{C}. \\ $$$$\mathrm{For}\:\mathrm{such}\:\mathrm{a}\:\mathrm{triangle}\:{A},{B}\:\mathrm{and}\:{C} \\ $$$$\mathrm{will}\:\mathrm{take}\:\mathrm{complex}\:\mathrm{values}. \\ $$

Question Number 15760    Answers: 1   Comments: 0

If sides of triangle are x^2 + x + 1, 2x + 1 and x^2 − 1, prove that greatest angle is 120°. Also find the range of x such that triangle exist.

$$\mathrm{If}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{are}\:{x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}, \\ $$$$\mathrm{2}{x}\:+\:\mathrm{1}\:\mathrm{and}\:{x}^{\mathrm{2}} \:−\:\mathrm{1},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{greatest} \\ $$$$\mathrm{angle}\:\mathrm{is}\:\mathrm{120}°.\:\mathrm{Also}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{x} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{triangle}\:\mathrm{exist}. \\ $$

Question Number 15742    Answers: 1   Comments: 1

This question is posted on the request of mrW1 (See comments of my answer to Q#15543). Find the last last non-zero digit of the expansion of 2000!

$$\mathrm{This}\:\mathrm{question}\:\mathrm{is}\:\mathrm{posted}\:\mathrm{on}\:\mathrm{the}\:\mathrm{request}\:\mathrm{of}\:\mathrm{mrW1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{See}\:\mathrm{comments}\:\mathrm{of}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{Q}#\mathrm{15543}\right). \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{last}\:\:\boldsymbol{\mathrm{non}}-\boldsymbol{\mathrm{zero}}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\:\mathrm{2000}! \\ $$

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