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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

Question Number 32591 Answers: 0 Comments: 0

Question Number 32590 Answers: 1 Comments: 0

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Question Number 32585 Answers: 3 Comments: 2

A body undergoing SHM about the origin has its equation x=0.2cos 5πt. Find its average speed from t=0 to t=0.7 sec.

$$\boldsymbol{{A}}\:{body}\:{undergoing}\:{SHM}\:{about}\:{the} \\ $$$${origin}\:{has}\:{its}\:{equation}\:{x}=\mathrm{0}.\mathrm{2cos}\:\mathrm{5}\pi{t}. \\ $$$${Find}\:{its}\:{average}\:{speed}\:{from}\:{t}=\mathrm{0}\:{to} \\ $$$${t}=\mathrm{0}.\mathrm{7}\:{sec}. \\ $$

Question Number 32578 Answers: 0 Comments: 1

Question Number 32569 Answers: 0 Comments: 1

Compute the number of ordered quadruple (a, b, c, d) of distinct positive integers so that (( ((a),(b) )),( ((c),(d) )) ) = 21 .

$${Compute}\:\:{the}\:\:{number}\:\:{of}\:\:\:{ordered}\:\:{quadruple}\:\:\left({a},\:{b},\:{c},\:{d}\right)\:\:{of}\:\:{distinct}\:\:{positive}\:\:{integers}\:\:\:{so}\:\:{that}\:\:\begin{pmatrix}{\begin{pmatrix}{{a}}\\{{b}}\end{pmatrix}}\\{\begin{pmatrix}{{c}}\\{{d}}\end{pmatrix}}\end{pmatrix}\:\:\:=\:\:\mathrm{21}\:. \\ $$

Question Number 32563 Answers: 2 Comments: 2

The set of values of ′a′ for which all the solutions of the equation 4sin^4 x+asin^2 x+3=0 are real and distinct is ?

$$\boldsymbol{{T}}{he}\:{set}\:{of}\:{values}\:{of}\:'{a}'\:{for}\:{which}\: \\ $$$${all}\:{the}\:{solutions}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{4sin}\:^{\mathrm{4}} {x}+{a}\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{3}=\mathrm{0}\:{are}\:{real}\:{and}\: \\ $$$${distinct}\:{is}\:? \\ $$

Question Number 32584 Answers: 1 Comments: 0

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Question Number 32543 Answers: 1 Comments: 0

The coefficient of x^4 in the expansion of (1+5x+9x^2 +.....∞)(1+x^2 )^(11) is a) 171 b) 172 c) 173 d) 176

$$\boldsymbol{{T}}{he}\:{coefficient}\:{of}\:{x}^{\mathrm{4}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\mathrm{1}+\mathrm{5}{x}+\mathrm{9}{x}^{\mathrm{2}} +.....\infty\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{11}} {is} \\ $$$$\left.{a}\right)\:\mathrm{171} \\ $$$$\left.{b}\right)\:\mathrm{172} \\ $$$$\left.{c}\right)\:\mathrm{173} \\ $$$$\left.{d}\right)\:\mathrm{176} \\ $$

Question Number 32541 Answers: 2 Comments: 0

Coefficient of x^5 in the expansion of (x^2 −x−2)^5 is

$${Coefficient}\:{of}\:{x}^{\mathrm{5}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left({x}^{\mathrm{2}} −{x}−\mathrm{2}\right)^{\mathrm{5}} \:{is} \\ $$

Question Number 32538 Answers: 1 Comments: 1

Question Number 32535 Answers: 0 Comments: 0

k ≤ 2018 f (f (n) ) = 2n f (k) = 2018 how many the possible of k integers ?

$${k}\:\:\leqslant\:\:\mathrm{2018} \\ $$$${f}\:\left({f}\:\left({n}\right)\:\right)\:\:=\:\:\mathrm{2}{n} \\ $$$${f}\:\left({k}\right)\:\:=\:\:\mathrm{2018} \\ $$$${how}\:\:{many}\:\:\:{the}\:{possible}\:{of}\:\:\:\boldsymbol{{k}}\:\:{integers}\:? \\ $$

Question Number 32534 Answers: 1 Comments: 0

Question Number 32532 Answers: 0 Comments: 3

If a,b,c are 3 positive numbers in an A.P and T= ((a+8b)/(2b−a))+((8b+c)/(2b−c)). Then the value of T^( 2 ) is ? Ans. given is 361.

$$\boldsymbol{{I}}{f}\:{a},{b},{c}\:{are}\:\mathrm{3}\:{positive}\:{numbers}\:{in}\:{an} \\ $$$$\boldsymbol{{A}}.\boldsymbol{{P}}\:{and}\: \\ $$$${T}=\:\frac{{a}+\mathrm{8}{b}}{\mathrm{2}{b}−{a}}+\frac{\mathrm{8}{b}+{c}}{\mathrm{2}{b}−{c}}. \\ $$$${Then}\:{the}\:{value}\:{of}\:{T}^{\:\:\mathrm{2}\:} \:{is}\:? \\ $$$${Ans}.\:{given}\:{is}\:\mathrm{361}. \\ $$

Question Number 32529 Answers: 1 Comments: 2

Question Number 32517 Answers: 0 Comments: 1

calculatelim_(x→0^+ ) ((x^(sinx) −(sinx)^x )/x) .

$${calculatelim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\frac{{x}^{{sinx}} \:\:−\left({sinx}\right)^{{x}} }{{x}}\:. \\ $$

Question Number 32510 Answers: 2 Comments: 1

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Question Number 32503 Answers: 0 Comments: 0

algebra1ic

$${algebra}\mathrm{1}{ic} \\ $$

Question Number 32501 Answers: 1 Comments: 0

(x−2y+3)^2 +(3x+4y−1)^2 =100 what is the area of the ellipse?

$$\left(\mathrm{x}−\mathrm{2y}+\mathrm{3}\right)^{\mathrm{2}} +\left(\mathrm{3x}+\mathrm{4y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{100} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ellipse}? \\ $$

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