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AllQuestion and Answers: Page 1518

Question Number 58480 Answers: 0 Comments: 2

Question Number 58479 Answers: 0 Comments: 1

Question Number 58478 Answers: 2 Comments: 1

{1} ∫((x^2 −2)/(x^4 +8x^2 +4)) dx = ? {2} Shortest distance between the parabolas y^2 =4x and y^2 =2x−6 is ?

$$\left\{\mathrm{1}\right\}\:\:\:\int\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{4}} +\mathrm{8}{x}^{\mathrm{2}} +\mathrm{4}}\:{dx}\:=\:? \\ $$$$\left\{\mathrm{2}\right\}\:\:{Shortest}\:{distance}\:{between}\:{the} \\ $$$${parabolas}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:{and}\:{y}^{\mathrm{2}} =\mathrm{2}{x}−\mathrm{6}\:{is}\:? \\ $$

Question Number 58467 Answers: 1 Comments: 3

Question Number 58462 Answers: 1 Comments: 0

a, b, c, d ∈ R^+ a + b + c + d = 1 Prove that : abc + bcd + cda + dab ≤ (1/(27)) + ((176)/(27)) abcd

$${a},\:{b},\:{c},\:{d}\:\:\in\:\:\mathbb{R}^{+} \\ $$$${a}\:+\:{b}\:+\:{c}\:+\:{d}\:\:=\:\:\mathrm{1} \\ $$$${Prove}\:\:{that}\:\:: \\ $$$${abc}\:+\:{bcd}\:+\:{cda}\:+\:{dab}\:\:\leqslant\:\:\frac{\mathrm{1}}{\mathrm{27}}\:\:+\:\:\frac{\mathrm{176}}{\mathrm{27}}\:{abcd} \\ $$

Question Number 58454 Answers: 0 Comments: 0

Question Number 58447 Answers: 2 Comments: 1

Question Number 58438 Answers: 1 Comments: 4

lim_(x→∞) (((x−1)/x))^x

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{x}−\mathrm{1}}{{x}}\right)^{{x}} \\ $$

Question Number 58422 Answers: 1 Comments: 0

Question Number 58406 Answers: 2 Comments: 0

1)Value of 20!+((21!)/(1!))+((22!)/(2!))+....+((60!)/(40!)) is ? 2) Sum of all solutions of eq^n : cos 3θ=sin 2θ in interval [−(π/2),(π/2)] is ?

$$\left.\mathrm{1}\right){Value}\:{of}\:\mathrm{20}!+\frac{\mathrm{21}!}{\mathrm{1}!}+\frac{\mathrm{22}!}{\mathrm{2}!}+....+\frac{\mathrm{60}!}{\mathrm{40}!}\:{is}\:\:? \\ $$$$\left.\mathrm{2}\right)\:{Sum}\:{of}\:{all}\:{solutions}\:{of}\:{eq}^{{n}} \:: \\ $$$$\mathrm{cos}\:\mathrm{3}\theta=\mathrm{sin}\:\mathrm{2}\theta\:{in}\:{interval}\:\left[−\frac{\pi}{\mathrm{2}},\frac{\pi}{\mathrm{2}}\right]\:{is}\:? \\ $$

Question Number 58405 Answers: 3 Comments: 2

1) lim_(x→0) ((e^(ax) −bx−1)/x^2 )=2 . find a,b ? 2) 6 balls are placed randomly into 6 cells. Then the probability that exactly one cell remains empty is ?

$$\left.\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{ax}} −{bx}−\mathrm{1}}{{x}^{\mathrm{2}} }=\mathrm{2}\:. \\ $$$${find}\:{a},{b}\:? \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{6}\:{balls}\:{are}\:{placed}\:{randomly}\:{into} \\ $$$$\mathrm{6}\:{cells}.\:{Then}\:{the}\:{probability}\:{that}\:{exactly} \\ $$$${one}\:{cell}\:{remains}\:{empty}\:{is}\:? \\ $$

Question Number 58409 Answers: 0 Comments: 3

Prove without mathematical induction that the expression (1 + (√2))^(2n) + (1 − (√2))^(2n) is even for every natural number n.

$$\mathrm{Prove}\:\mathrm{without}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{expression}\:\:\:\left(\mathrm{1}\:+\:\sqrt{\mathrm{2}}\right)^{\mathrm{2n}} \:+\:\left(\mathrm{1}\:−\:\sqrt{\mathrm{2}}\right)^{\mathrm{2n}} \:\:\mathrm{is}\:\mathrm{even}\:\mathrm{for}\:\mathrm{every} \\ $$$$\mathrm{natural}\:\mathrm{number}\:\:\mathrm{n}. \\ $$

Question Number 58402 Answers: 2 Comments: 2

The imaginary part of ((1/2)+(1/2)i)^(10) is ?

$${The}\:{imaginary}\:{part}\:{of}\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{i}\right)^{\mathrm{10}} {is}\:? \\ $$

Question Number 58400 Answers: 1 Comments: 1

If parabola y=−x^2 −2x+k touches the parabola y=−(1/2)x^2 −4x+3 , then the value of k is ? a) 1 b)2 c)3 d)4

$${If}\:{parabola}\:{y}=−{x}^{\mathrm{2}} −\mathrm{2}{x}+{k}\:{touches} \\ $$$${the}\:{parabola}\:{y}=−\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}\:,\:{then} \\ $$$${the}\:{value}\:{of}\:{k}\:{is}\:? \\ $$$$\left.{a}\left.\right)\left.\:\left.\mathrm{1}\:\:\:\:{b}\right)\mathrm{2}\:\:\:\:\:{c}\right)\mathrm{3}\:\:\:\:\:{d}\right)\mathrm{4} \\ $$

Question Number 58410 Answers: 1 Comments: 0

Show that the sum of the cube of three consecutive number gives a multiple of 9.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{of}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{number}\:\mathrm{gives}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\:\mathrm{9}. \\ $$

Question Number 58526 Answers: 0 Comments: 1

Question Number 58390 Answers: 2 Comments: 2

write without roots in denominator if possible (1) (1/(√a)) (2) (1/((√a)+(√b))) (3) (1/((√a)+(√b)+(√c))) (4) (1/((√a)+(√b)+(√c)+(√d))) (5) (1/((√a)+(√b)+(√c)+(√d)+(√e)))

$$\mathrm{write}\:\mathrm{without}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{denominator}\:\mathrm{if}\:\mathrm{possible} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}+\sqrt{{d}}} \\ $$$$\left(\mathrm{5}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}+\sqrt{{d}}+\sqrt{{e}}} \\ $$

Question Number 58387 Answers: 2 Comments: 0

Question Number 58529 Answers: 3 Comments: 4

Solve for x and y (1/x) + (1/y) = 4 ....... (i) (x^2 /y) + (y^2 /x) = 9 ....... (ii)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:\mathrm{4}\:\:\:\:\:\:\:.......\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{x}}\:\:=\:\:\mathrm{9}\:\:\:\:\:\:.......\:\left(\mathrm{ii}\right) \\ $$

Question Number 58380 Answers: 0 Comments: 3

Question Number 58378 Answers: 0 Comments: 1

Question Number 58377 Answers: 1 Comments: 0

Question Number 58374 Answers: 1 Comments: 0

Question Number 58373 Answers: 1 Comments: 1

Find the coefficient of x^6 in (2x + 1)^6 (x^2 + x + (1/4))^4

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\:\mathrm{x}^{\mathrm{6}} \:\:\mathrm{in}\:\:\:\left(\mathrm{2x}\:+\:\mathrm{1}\right)^{\mathrm{6}} \:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} \\ $$

Question Number 58364 Answers: 2 Comments: 0

Let a_n = 10 × ((1/(√2)))^n a_n is a geometrical sequence S_n = a_0 + a_1 + ... + a_(n−1) S_n = 10 × ((1 − ((1/(√2)))^n )/(1 − ((1/(√2))))) Proove that : S_n = ((10(√2))/((√2) − 1)) × (1−((1/(√2)))^n )

$$\mathrm{Let}\:{a}_{{n}} =\:\mathrm{10}\:×\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} \\ $$$$\:\:\:\:{a}_{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{geometrical}\:\mathrm{sequence} \\ $$$$\:\:{S}_{{n}} \:=\:{a}_{\mathrm{0}} \:+\:{a}_{\mathrm{1}} \:+\:...\:+\:{a}_{{n}−\mathrm{1}} \\ $$$$\:\:\:\:\:{S}_{{n}} =\:\mathrm{10}\:×\:\frac{\mathrm{1}\:−\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} }{\mathrm{1}\:−\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)} \\ $$$$\boldsymbol{\mathrm{Proove}}\:\boldsymbol{\mathrm{that}}\:: \\ $$$$ \\ $$$${S}_{{n}} \:=\:\frac{\mathrm{10}\sqrt{\mathrm{2}}}{\sqrt{\mathrm{2}}\:−\:\mathrm{1}}\:×\:\left(\mathrm{1}−\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} \right) \\ $$$$ \\ $$

Question Number 58362 Answers: 1 Comments: 0

∫((sin x)/(sin 3x))dx

$$\int\frac{\mathrm{sin}\:{x}}{\mathrm{sin}\:\mathrm{3}{x}}{dx} \\ $$

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