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Question Number 31292 Answers: 1 Comments: 3

Question Number 31290 Answers: 1 Comments: 1

Question Number 31259 Answers: 0 Comments: 4

Question Number 31255 Answers: 1 Comments: 0

2n boys are randomly divided into two subgroups containing n boys each. The probability that the two tallest boys are in different groups is

$$\mathrm{2}{n}\:\mathrm{boys}\:\mathrm{are}\:\mathrm{randomly}\:\mathrm{divided}\:\mathrm{into}\:\mathrm{two} \\ $$$$\mathrm{subgroups}\:\mathrm{containing}\:{n}\:\mathrm{boys}\:\mathrm{each}.\:\mathrm{The} \\ $$$$\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{two}\:\mathrm{tallest}\:\mathrm{boys}\:\mathrm{are} \\ $$$$\mathrm{in}\:\mathrm{different}\:\mathrm{groups}\:\mathrm{is} \\ $$

Question Number 31249 Answers: 0 Comments: 10

2 lines through the point A(5, 1) are tangent to the circle x^2 + y^2 − 4x + 6y + 4 = 0 Find the equation of these 2 lines

$$\mathrm{2}\:\mathrm{lines}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:{A}\left(\mathrm{5},\:\mathrm{1}\right)\:\mathrm{are}\:\mathrm{tangent} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{circle}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:−\:\mathrm{4}{x}\:+\:\mathrm{6}{y}\:+\:\mathrm{4}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{these}\:\mathrm{2}\:\mathrm{lines} \\ $$

Question Number 31246 Answers: 1 Comments: 2

without using lohpital find lim_(x→π/6) ((1−2sinx)/(cos 3x))

$$\mathrm{without}\:\mathrm{using}\:\mathrm{lohpital} \\ $$$$\mathrm{find} \\ $$$$\underset{\mathrm{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{2sinx}}{\mathrm{cos}\:\mathrm{3x}} \\ $$

Question Number 31237 Answers: 0 Comments: 3

Show that A−B = B′ ∩ A.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{A}−\mathrm{B}\:=\:\mathrm{B}'\:\cap\:\mathrm{A}. \\ $$

Question Number 31229 Answers: 0 Comments: 1

simplify A_n = C_n ^1 +2 C_n ^2 +3 C_n ^3 +... + n C_n ^n .

$${simplify}\:{A}_{{n}} =\:{C}_{{n}} ^{\mathrm{1}} \:\:+\mathrm{2}\:{C}_{{n}} ^{\mathrm{2}} \:+\mathrm{3}\:{C}_{{n}} ^{\mathrm{3}} \:+...\:+\:{n}\:{C}_{{n}} ^{{n}} \:. \\ $$

Question Number 31214 Answers: 2 Comments: 2

Question Number 31209 Answers: 1 Comments: 2

Question Number 31194 Answers: 2 Comments: 1

Find the remainder when x^(203) −1 is divided by x^4 −1.

$${Find}\:{the}\:{remainder}\:{when}\:{x}^{\mathrm{203}} −\mathrm{1} \\ $$$${is}\:{divided}\:{by}\:{x}^{\mathrm{4}} −\mathrm{1}. \\ $$

Question Number 31193 Answers: 1 Comments: 0

Question Number 31188 Answers: 0 Comments: 2

Question Number 31187 Answers: 0 Comments: 4

Question Number 31149 Answers: 1 Comments: 1

here is a question really troubling me. A cylindrical tube rolling down a slope of inclination θ moves a distance L in the time T. The equation relating these quantities is L(3+(a^2 /P))=QT^2 sin θ where a is the internal radius of the tube and P and Q are constants.What are the units of P and Q?

$${here}\:{is}\:{a}\:{question}\:{really}\:{troubling} \\ $$$${me}. \\ $$$$ \\ $$$${A}\:{cylindrical}\:{tube}\:{rolling}\:{down}\:{a} \\ $$$${slope}\:{of}\:{inclination}\:\theta\:{moves}\:{a} \\ $$$${distance}\:{L}\:{in}\:{the}\:{time}\:{T}.\:{The} \\ $$$${equation}\:{relating}\:{these}\:{quantities}\:{is} \\ $$$$ \\ $$$$\:\:\:{L}\left(\mathrm{3}+\frac{{a}^{\mathrm{2}} }{{P}}\right)={QT}^{\mathrm{2}} \mathrm{sin}\:\theta\:{where}\:{a}\:{is} \\ $$$${the}\:{internal}\:{radius}\:{of}\:{the}\:{tube}\:{and} \\ $$$${P}\:\:{and}\:{Q}\:{are}\:{constants}.{What}\:{are} \\ $$$${the}\:{units}\:{of}\:{P}\:{and}\:{Q}? \\ $$

Question Number 31133 Answers: 0 Comments: 3

Question Number 31125 Answers: 2 Comments: 1

Let n be a positive integer. Then x^2 + 1 is a factor of (x^4 + 3)^n − [(x^2 + 3)(x^2 − 1)]^n for ... (A) All n (B) Odd n (C) Even n (D) n ≥ 3 (E) None of these options

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}.\:\mathrm{Then}\:{x}^{\mathrm{2}} \:+\:\mathrm{1}\: \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\left({x}^{\mathrm{4}} \:+\:\mathrm{3}\right)^{{n}} \:−\:\left[\left({x}^{\mathrm{2}} \:+\:\mathrm{3}\right)\left({x}^{\mathrm{2}} \:−\:\mathrm{1}\right)\right]^{{n}} \\ $$$$\mathrm{for}\:... \\ $$$$\left(\mathrm{A}\right)\:\mathrm{All}\:{n} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{Odd}\:{n} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{Even}\:{n} \\ $$$$\left(\mathrm{D}\right)\:{n}\:\geqslant\:\mathrm{3} \\ $$$$\left(\mathrm{E}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these}\:\mathrm{options} \\ $$

Question Number 31118 Answers: 1 Comments: 1

Question Number 31114 Answers: 1 Comments: 2

Question Number 31113 Answers: 2 Comments: 0

Two lines through the point (1,−3) are tamgent to the curve y=x^2 . Find the equation of these two lines and make a sketch to verify your results.

$${Two}\:{lines}\:{through}\:{the}\:{point}\:\left(\mathrm{1},−\mathrm{3}\right) \\ $$$${are}\:{tamgent}\:{to}\:{the}\:{curve}\:{y}={x}^{\mathrm{2}} . \\ $$$${Find}\:{the}\:{equation}\:{of}\:{these}\:{two} \\ $$$${lines}\:{and}\:{make}\:{a}\:{sketch}\:{to}\:{verify} \\ $$$${your}\:{results}. \\ $$

Question Number 31111 Answers: 1 Comments: 1

Question Number 31109 Answers: 0 Comments: 5

a^4 + b^4 + 13 is a possible largest prime number . a and b are prime numbers . Find a and b .

$$\boldsymbol{{a}}^{\mathrm{4}} \:+\:\boldsymbol{{b}}^{\mathrm{4}} \:+\:\mathrm{13}\:\:\:{is}\:\:{a}\:\:{possible}\:\:{largest}\:\:{prime}\:\:{number}\:. \\ $$$$\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:\:{are}\:\:{prime}\:\:{numbers}\:. \\ $$$${Find}\:\:\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:. \\ $$

Question Number 31107 Answers: 0 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^n )) with n>1.

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{{n}} }\:\:{with}\:{n}>\mathrm{1}. \\ $$

Question Number 31106 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−x^2 ) =lim_(n→+∞) ∫_0 ^∞ (dx/((1+x^2 )^n )) . 2) prove that (1/(√π)) =lim_(n→∞) ((1.3.5....(2n−3))/(2.4.6....(2n−2))) (√n) (wallis formula).

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } ={lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\frac{\mathrm{1}}{\sqrt{\pi}}\:={lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}....\left(\mathrm{2}{n}−\mathrm{3}\right)}{\mathrm{2}.\mathrm{4}.\mathrm{6}....\left(\mathrm{2}{n}−\mathrm{2}\right)}\:\sqrt{{n}} \\ $$$$\left({wallis}\:{formula}\right). \\ $$

Question Number 31105 Answers: 0 Comments: 1

prove that ∫_0 ^x e^(−t^2 ) dt =((√π)/2) −(e^(−x^2 ) /(√π)) ∫_0 ^∞ (e^(−x^2 t^2 ) /(1+t^2 )) dt with x>0

$${prove}\:{that}\:\int_{\mathrm{0}} ^{{x}} \:\:\:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:−\frac{{e}^{−{x}^{\mathrm{2}} } }{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} {t}^{\mathrm{2}} } }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 31104 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) e^(−(x^2 +2x−1)) dx .

$${find}\:\:\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{1}\right)} {dx}\:. \\ $$

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