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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

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

Question Number 15707 Answers: 1 Comments: 0

x^3 + (y + 1)^2 . (dy/dx) = 0 Find y

$${x}^{\mathrm{3}} \:+\:\left({y}\:+\:\mathrm{1}\right)^{\mathrm{2}} \:.\:\frac{{dy}}{{dx}}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:{y} \\ $$

Question Number 15706 Answers: 0 Comments: 2

(d^2 y/dx^2 ) + (dy/dx) − 6y = 2 + sin x Find y

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\frac{{dy}}{{dx}}\:−\:\mathrm{6}{y}\:=\:\mathrm{2}\:+\:\mathrm{sin}\:{x} \\ $$$$\mathrm{Find}\:{y} \\ $$

Question Number 15700 Answers: 1 Comments: 1