Question and Answers Forum

AllQuestion and Answers: Page 1884

Question Number 13237 Answers: 1 Comments: 2

Question Number 13236 Answers: 0 Comments: 16

For positive a,b,c such that a b c=1 show that a^(b+c) b^(c+a) c^(a+b) ≤1 solution: a^(b+c) b^(c+a) c^(a+b) =(a^b a^c b^c b^a c^a c^b ) =(b×c)^a (a×c)^b (a×b)^c =(a^0 ×b×c)^a (a×b^0 ×c)^b (a×b×c^0 )^c ≤(a×b×c)^(a ) (a×b^ × c)^b (a×b×c)^c ≤(1)^a (1)^b (1)^c ;since a b c=1 ≤1

$${For}\:{positive}\:\:{a},{b},{c}\:\:{such}\:{that}\:\:{a}\:{b}\:{c}=\mathrm{1} \\ $$$${show}\:{that}\:\:{a}^{{b}+{c}} \:\:{b}^{{c}+{a}} \:\:{c}^{{a}+{b}} \:\leqslant\mathrm{1} \\ $$$${solution}: \\ $$$${a}^{{b}+{c}} \:\:{b}^{{c}+{a}} \:\:{c}^{{a}+{b}} \:=\left({a}^{{b}} {a}^{{c}} \:{b}^{{c}} {b}^{{a}} \:\:{c}^{{a}} {c}^{{b}} \right) \\ $$$$\:\:\:\:\:\:=\left({b}×{c}\right)^{{a}} \:\left({a}×{c}\right)^{{b}} \:\left({a}×{b}\right)^{{c}} \\ $$$$\:\:\:\:\:\:\:=\left({a}^{\mathrm{0}} ×{b}×{c}\right)^{{a}} \:\left({a}×{b}^{\mathrm{0}} ×{c}\right)^{{b}} \left({a}×{b}×{c}^{\mathrm{0}} \right)^{{c}} \\ $$$$\:\:\:\:\:\:\:\leqslant\left({a}×{b}×{c}\right)^{{a}\:\:\:\:} \left({a}×{b}^{} ×\:{c}\right)^{{b}} \:\left({a}×{b}×{c}\right)^{{c}} \\ $$$$\:\:\:\:\:\:\:\leqslant\left(\mathrm{1}\right)^{{a}} \:\left(\mathrm{1}\right)^{{b}} \:\left(\mathrm{1}\right)^{{c}} \:\:\:\:;{since}\:\:{a}\:{b}\:{c}=\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\leqslant\mathrm{1} \\ $$

Question Number 13234 Answers: 2 Comments: 0

The angle of elevation of the sun is 27°. A man is 180 cm tall . How long is his shadow. Give your answer to the nearest 10cm

$$\mathrm{The}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{elevation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sun}\:\mathrm{is}\:\mathrm{27}°.\:\:\mathrm{A}\:\mathrm{man}\:\mathrm{is}\:\mathrm{180}\:\mathrm{cm}\:\mathrm{tall}\:.\:\mathrm{How}\:\mathrm{long}\:\mathrm{is} \\ $$$$\mathrm{his}\:\mathrm{shadow}.\:\mathrm{Give}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\:\mathrm{10cm} \\ $$

Question Number 13228 Answers: 0 Comments: 5

Question Number 13226 Answers: 2 Comments: 0

if x^3 +y^3 =3axy,find dy/dx in terms of x and y and prove that dy/dx cannot be equal to -1 for finite values of x and y except x=y. please help

$$\mathrm{if}\:\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} =\mathrm{3axy},\mathrm{find}\:\mathrm{dy}/\mathrm{dx}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{dy}/\mathrm{dx}\: \\ $$$$\mathrm{cannot}\:\mathrm{be}\:\mathrm{equal}\:\mathrm{to}\:-\mathrm{1}\:\mathrm{for}\:\mathrm{finite} \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{except}\:\mathrm{x}=\mathrm{y}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{help}\: \\ $$

Question Number 13224 Answers: 1 Comments: 3

Question Number 13206 Answers: 0 Comments: 0

An open box whose shape is a cuboid has dimensions 9 cm by 7 cm by 6 cm. Find (i) The outer surface area of the box (ii) The volume of the box

$$\mathrm{An}\:\mathrm{open}\:\mathrm{box}\:\mathrm{whose}\:\mathrm{shape}\:\mathrm{is}\:\mathrm{a}\:\mathrm{cuboid}\:\mathrm{has}\:\mathrm{dimensions}\:\mathrm{9}\:\mathrm{cm}\:\mathrm{by}\:\mathrm{7}\:\mathrm{cm}\:\mathrm{by}\:\mathrm{6}\:\mathrm{cm}. \\ $$$$\mathrm{Find} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{The}\:\mathrm{outer}\:\mathrm{surface}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{box} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{The}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{box} \\ $$

Question Number 13201 Answers: 5 Comments: 1

In a ΔABC prove that: ((sin A + sin B)/2) ≤ sin (((A + B)/2))

$$\mathrm{In}\:\mathrm{a}\:\Delta{ABC}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{sin}\:{A}\:+\:\mathrm{sin}\:\mathrm{B}}{\mathrm{2}}\:\leqslant\:\mathrm{sin}\:\left(\frac{{A}\:+\:{B}}{\mathrm{2}}\right) \\ $$

Question Number 13200 Answers: 2 Comments: 0

(6)^(1/(5)^(1/(2)^(1/(√3)) ) ) = x How to write x in standard form?

$$\sqrt[{\sqrt[{\sqrt[{\sqrt{\mathrm{3}}}]{\mathrm{2}}}]{\mathrm{5}}}]{\mathrm{6}}\:=\:{x} \\ $$$$\mathrm{How}\:\mathrm{to}\:\mathrm{write}\:{x}\:\mathrm{in}\:\mathrm{standard}\:\mathrm{form}? \\ $$

Question Number 13194 Answers: 1 Comments: 0

Question Number 13191 Answers: 0 Comments: 4

Can we ask here Chemistry or Physics doubts because I am preparing for JEE?

$$\mathrm{Can}\:\mathrm{we}\:\mathrm{ask}\:\mathrm{here}\:\mathrm{Chemistry}\:\mathrm{or}\:\mathrm{Physics} \\ $$$$\mathrm{doubts}\:\mathrm{because}\:\mathrm{I}\:\mathrm{am}\:\mathrm{preparing}\:\mathrm{for}\:\mathrm{JEE}? \\ $$

Question Number 13166 Answers: 2 Comments: 0

Find the values of x and y x^2 − 2xy − y^2 = 14 ............ equation (i) 2x^2 + 3xy + y^2 = − 2 ............ equation (ii)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{x}\:\:\mathrm{and}\:\:\mathrm{y} \\ $$$$ \\ $$$$\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{2xy}\:−\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{14}\:\:\:\:\:\:\:\:\:............\:\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{3xy}\:+\:\mathrm{y}^{\mathrm{2}} \:=\:−\:\mathrm{2}\:\:\:\:\:............\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 13155 Answers: 2 Comments: 0

Question Number 13154 Answers: 2 Comments: 0

Find the smallest number such that when divided by 18 the remainder is 17, When divided by 20 the remainder is 19. and when divided by 24 the remainder is 23.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{18}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{17}, \\ $$$$\mathrm{When}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{20}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{19}.\:\mathrm{and}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{24}\:\mathrm{the}\:\mathrm{remainder}\: \\ $$$$\mathrm{is}\:\mathrm{23}.\: \\ $$

Question Number 13153 Answers: 1 Comments: 3