Question and Answers Forum

AllQuestion and Answers: Page 1927

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!

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

Question Number 15741 Answers: 1 Comments: 0

Find the minimum value of x^2 +y^2 +z^2 , with the condition ax+by+cz=p .

$${Find}\:{the}\:{minimum}\:{value}\:\:{of} \\ $$$$\:\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} ,\:{with}\:{the}\:{condition} \\ $$$$\:{ax}+{by}+{cz}={p}\:. \\ $$

Question Number 15740 Answers: 1 Comments: 0

Question Number 15736 Answers: 1 Comments: 0

From a point A on the circum- ference of a circle of radius r, a perpendicular AF is dropped on a tangent to the circle at P. Find the maximum possible area of ΔAPF .

$${From}\:{a}\:{point}\:{A}\:{on}\:{the}\:{circum}- \\ $$$${ference}\:{of}\:{a}\:{circle}\:{of}\:{radius}\:\boldsymbol{{r}},\:{a} \\ $$$${perpendicular}\:{AF}\:\:{is}\:{dropped}\:{on} \\ $$$${a}\:{tangent}\:{to}\:{the}\:{circle}\:{at}\:{P}. \\ $$$${Find}\:{the}\:\:{maximum}\:{possible}\: \\ $$$${area}\:{of}\:\Delta{APF}\:. \\ $$

Question Number 15721 Answers: 1 Comments: 0

If x+y+z=1 with 0<x, y, z <(1/2) then find tbe range of values of (1/(x+y))+(1/(y+z))+(1/(z+x)) .

$${If}\:\:\:\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}=\mathrm{1} \\ $$$$\:\:\:\:\:{with}\:\mathrm{0}<{x},\:{y},\:{z}\:<\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:{then}\:{find}\:{tbe}\:\:{range}\:{of}\:{values}\:{of} \\ $$$$\:\:\:\frac{\mathrm{1}}{\boldsymbol{{x}}+\boldsymbol{{y}}}+\frac{\mathrm{1}}{\boldsymbol{{y}}+\boldsymbol{{z}}}+\frac{\mathrm{1}}{\boldsymbol{{z}}+\boldsymbol{{x}}}\:. \\ $$

Question Number 15734 Answers: 0 Comments: 3

An elastic cord can be stretched to its elastic limit by a load of 2N.If a 35cm lemgth of the cord is extended 0.6cm by a force of 0.5N, what will be the length of the cord when the stretching force is 2.5N? (a)350.8cm (b)352.8cm (c)353.0cm (d)356cm (e)cannot be determined from the data given

$$\mathrm{An}\:\mathrm{elastic}\:\mathrm{cord}\:\mathrm{can}\:\mathrm{be}\:\mathrm{stretched}\:\mathrm{to} \\ $$$$\mathrm{its}\:\mathrm{elastic}\:\mathrm{limit}\:\mathrm{by}\:\mathrm{a}\:\mathrm{load}\:\mathrm{of}\:\mathrm{2N}.\mathrm{If}\: \\ $$$$\mathrm{a}\:\mathrm{35cm}\:\mathrm{lemgth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cord}\:\mathrm{is} \\ $$$$\mathrm{extended}\:\mathrm{0}.\mathrm{6cm}\:\mathrm{by}\:\mathrm{a}\:\mathrm{force}\:\mathrm{of}\:\mathrm{0}.\mathrm{5N}, \\ $$$$\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cord} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{stretching}\:\mathrm{force}\:\mathrm{is}\:\mathrm{2}.\mathrm{5N}? \\ $$$$\left(\mathrm{a}\right)\mathrm{350}.\mathrm{8cm} \\ $$$$\left(\mathrm{b}\right)\mathrm{352}.\mathrm{8cm} \\ $$$$\left(\mathrm{c}\right)\mathrm{353}.\mathrm{0cm} \\ $$$$\left(\mathrm{d}\right)\mathrm{356cm} \\ $$$$\left(\mathrm{e}\right)\mathrm{cannot}\:\mathrm{be}\:\mathrm{determined}\:\mathrm{from}\:\mathrm{the}\: \\ $$$$\mathrm{data}\:\mathrm{given} \\ $$

Question Number 15737 Answers: 0 Comments: 2

Sir mrW1 and Ajfour, please which advance or physics textbook pdf can i get

$$\mathrm{Sir}\:\:\:\mathrm{mrW1}\:\mathrm{and}\:\mathrm{Ajfour},\:\mathrm{please}\:\mathrm{which}\:\mathrm{advance}\:\mathrm{or}\:\mathrm{physics}\:\mathrm{textbook}\:\mathrm{pdf}\:\mathrm{can}\:\mathrm{i}\:\mathrm{get} \\ $$

Pg 1922 Pg 1923 Pg 1924 Pg 1925 Pg 1926 Pg 1927 Pg 1928 Pg 1929 Pg 1930 Pg 1931

Contact: info@tinkutara.com