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Question Number 32629 Answers: 1 Comments: 0

If x^2 +y^2 =9 , 4a^2 +9b^2 =16, then maximum value of 4a^2 x^2 +9b^2 y^2 −12abxy is ?

$$\boldsymbol{{I}}{f}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{9}\:,\:\mathrm{4}{a}^{\mathrm{2}} +\mathrm{9}{b}^{\mathrm{2}} =\mathrm{16}, \\ $$$${then}\:\boldsymbol{{maximum}}\:{value}\:{of}\: \\ $$$$\mathrm{4}{a}^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{9}{b}^{\mathrm{2}} {y}^{\mathrm{2}} −\mathrm{12}{abxy}\:{is}\:? \\ $$

Question Number 32627 Answers: 0 Comments: 1

plzz help ne differentiate between ∫sin(2x)= −(1/2)cox(2x)+c is not change to ∫2sin(x)cos(x) but ∫_b ^a sin(2x)= is change to ∫_b ^a 2sin(x)cos(x)

$${plzz}\:{help}\:{ne}\:{differentiate}\: \\ $$$${between} \\ $$$$\int{sin}\left(\mathrm{2}{x}\right)=\:−\frac{\mathrm{1}}{\mathrm{2}}{cox}\left(\mathrm{2}{x}\right)+{c}\: \\ $$$${is}\:{not}\:{change}\:{to}\:\int\mathrm{2}{sin}\left({x}\right){cos}\left({x}\right) \\ $$$${but}\:\underset{{b}} {\overset{{a}} {\int}}{sin}\left(\mathrm{2}{x}\right)=\:{is}\:{change}\:{to} \\ $$$$\underset{{b}} {\overset{{a}} {\int}}\mathrm{2}{sin}\left({x}\right){cos}\left({x}\right) \\ $$

Question Number 32610 Answers: 0 Comments: 1

2. Find the number of ordered triples (a, b, c) of integers satisfying 0 ≤ a, b, c ≤ 1000 for which a^3 + b^3 + c^3 ≡ 3abc + 1 (mod 1001)

$$\mathrm{2}.\:\:{Find}\:\:{the}\:\:{number}\:\:{of}\:\:{ordered}\:\:{triples}\:\:\left({a},\:{b},\:{c}\right)\:\:{of}\:\:{integers}\:\:{satisfying}\:\:\:\:\mathrm{0}\:\leqslant\:\:{a},\:{b},\:{c}\:\:\leqslant\:\:\mathrm{1000}\:\:\:{for}\:\:{which} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:+\:{c}^{\mathrm{3}} \:\:\equiv\:\:\mathrm{3}{abc}\:+\:\mathrm{1}\:\:\:\left({mod}\:\:\mathrm{1001}\right)\: \\ $$

Question Number 32609 Answers: 0 Comments: 0

1. Suppose that a, b, and c are real numbers such that a < b < c and a^3 − 3a + 1 = b^3 − 3b + 1 = c^3 − 3c + 1 = 0 . Then (1/(a^2 + b)) + (1/(b^2 + c)) + (1/(c^2 + a)) can be written as (p/q) for relatively prime of positive integers p and q. Find 100p + q

$$\mathrm{1}.\:{Suppose}\:\:{that}\:\:{a},\:{b},\:{and}\:\:{c}\:\:{are}\:\:{real}\:\:{numbers}\:\:{such}\:\:{that}\:\:{a}\:<\:{b}\:<\:{c}\:\:{and}\:\:\:{a}^{\mathrm{3}} \:−\:\mathrm{3}{a}\:+\:\mathrm{1}\:\:=\:\:{b}^{\mathrm{3}} \:−\:\mathrm{3}{b}\:+\:\mathrm{1}\:\:=\:\:{c}^{\mathrm{3}} \:−\:\mathrm{3}{c}\:+\:\mathrm{1}\:=\:\:\mathrm{0}\:. \\ $$$$\:\:\:\:{Then}\:\:\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} \:+\:{b}}\:+\:\frac{\mathrm{1}}{{b}^{\mathrm{2}} \:+\:{c}}\:+\:\frac{\mathrm{1}}{{c}^{\mathrm{2}} \:+\:{a}}\:\:\:\:{can}\:{be}\:\:{written}\:\:{as}\:\:\:\frac{{p}}{{q}}\:\:\:{for}\:\:{relatively}\:\:{prime}\:\:{of}\:\:{positive}\:\:{integers}\:\:\boldsymbol{{p}}\:\:{and}\:\:\:\boldsymbol{{q}}.\:\:\:{Find}\:\:\:\mathrm{100}{p}\:+\:{q} \\ $$

Question Number 32600 Answers: 0 Comments: 6