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AllQuestion and Answers: Page 1895

Question Number 9589 Answers: 3 Comments: 0

Question Number 9586 Answers: 1 Comments: 0

Question Number 9585 Answers: 0 Comments: 1

((3x−5)/(x+2))=(3/4)

$$\frac{\mathrm{3}{x}−\mathrm{5}}{{x}+\mathrm{2}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 9596 Answers: 0 Comments: 0

Question Number 9581 Answers: 1 Comments: 0

∫sin^4 x.cos^4 xdx

$$\int\mathrm{sin}^{\mathrm{4}} {x}.\mathrm{cos}^{\mathrm{4}} {xdx} \\ $$

Question Number 9577 Answers: 1 Comments: 0

The mean of n number is 20.if the same numbers together with 30 give a new mean of 22, find n.

$$\mathrm{The}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{n}\:\mathrm{number}\:\mathrm{is}\:\mathrm{20}.\mathrm{if}\: \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{numbers}\:\mathrm{together}\:\mathrm{with}\:\mathrm{30} \\ $$$$\mathrm{give}\:\mathrm{a}\:\mathrm{new}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{22}, \\ $$$$\mathrm{find}\:\mathrm{n}. \\ $$

Question Number 9576 Answers: 0 Comments: 1

show that x is small enough for its cube and higher power to be neghected (√((1−x)/(1+x)))=1−x+ (1/2)x^2 . by putting =(1/8),show that (√7)≈2((83)/(128)).

$$\mathrm{show}\:\mathrm{that}\:\mathrm{x}\:\mathrm{is}\:\mathrm{small}\:\mathrm{enough}\:\mathrm{for}\:\mathrm{its}\:\mathrm{cube}\:\mathrm{and}\:\mathrm{higher}\:\mathrm{power}\:\mathrm{to}\:\mathrm{be}\:\mathrm{neghected} \\ $$$$\sqrt{\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}}=\mathrm{1}−\mathrm{x}+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}^{\mathrm{2}} . \\ $$$$\mathrm{by}\:\mathrm{putting}\:=\frac{\mathrm{1}}{\mathrm{8}},\mathrm{show}\:\mathrm{that}\:\sqrt{\mathrm{7}}\approx\mathrm{2}\frac{\mathrm{83}}{\mathrm{128}}. \\ $$

Question Number 9575 Answers: 0 Comments: 1

the expression ax^2 + bx + c is divisible by x−1,has reminder 2 when divided by x+1,and has reminder 8 when divided by x−2.find the value of a,b and c.

$$\mathrm{the}\:\mathrm{expression}\:\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{bx}\:+\:\mathrm{c}\:\mathrm{is}\:\mathrm{divisible}\:\:\mathrm{by}\:\mathrm{x}−\mathrm{1},\mathrm{has}\:\mathrm{reminder}\:\mathrm{2}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{x}+\mathrm{1},\mathrm{and}\:\mathrm{has}\:\mathrm{reminder}\:\mathrm{8}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{x}−\mathrm{2}.\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}. \\ $$

Question Number 9573 Answers: 0 Comments: 1

If n is positive integer prove that the cofficient of x^(2 ) and x^3 in the expansion of (x^2 +2x+2)^n are 2^(n−1) .n^2 and 2^(n−1) n(n−1)(1/3).

$${If}\:{n}\:{is}\:{positive}\:{integer}\:{prove}\:{that}\: \\ $$$${the}\:{cofficient}\:{of}\:{x}^{\mathrm{2}\:} {and}\:{x}^{\mathrm{3}} \:{in}\:{the}\: \\ $$$${expansion}\:{of}\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{{n}} \:{are}\:\mathrm{2}^{{n}−\mathrm{1}} .{n}^{\mathrm{2}} \\ $$$${and}\:\mathrm{2}^{{n}−\mathrm{1}} {n}\left({n}−\mathrm{1}\right)\frac{\mathrm{1}}{\mathrm{3}}. \\ $$

Question Number 9571 Answers: 1 Comments: 0

Question Number 9569 Answers: 0 Comments: 0

Question Number 9560 Answers: 0 Comments: 2

Question Number 9553 Answers: 0 Comments: 3

Find : u∙v (i) f(z) = ∣z∣^2 (ii) f(z) = z + (1/z)

$$\mathrm{Find}\::\:\:\:\mathrm{u}\centerdot\mathrm{v} \\ $$$$\left(\mathrm{i}\right)\:\:\:\mathrm{f}\left(\mathrm{z}\right)\:=\:\mid\mathrm{z}\mid^{\mathrm{2}} \\ $$$$\left(\mathrm{ii}\right)\:\:\mathrm{f}\left(\mathrm{z}\right)\:=\:\mathrm{z}\:+\:\frac{\mathrm{1}}{\mathrm{z}} \\ $$

Question Number 9548 Answers: 1 Comments: 0

a − 7b + 8c = 4 8a + 4b − c = 7 a, b, c ∈ R 60(a^2 − b^2 + c^2 ) = ?

$${a}\:−\:\mathrm{7}{b}\:+\:\mathrm{8}{c}\:=\:\mathrm{4} \\ $$$$\mathrm{8}{a}\:+\:\mathrm{4}{b}\:−\:{c}\:=\:\mathrm{7} \\ $$$${a},\:{b},\:{c}\:\in\:\mathbb{R} \\ $$$$ \\ $$$$\mathrm{60}\left({a}^{\mathrm{2}} \:−\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \right)\:=\:? \\ $$

Question Number 9544 Answers: 1 Comments: 0

Question Number 9543 Answers: 2 Comments: 0

(2−(√3))^x + (7−4(√3))(2+(√3))^x = 4(2−(√3)), x ≠ 0 What is the value of x ?

$$\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)^{{x}} \:+\:\left(\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\right)\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)^{{x}} \:=\:\mathrm{4}\left(\mathrm{2}−\sqrt{\mathrm{3}}\right),\:{x}\:\neq\:\mathrm{0} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:?\: \\ $$

Question Number 9541 Answers: 1 Comments: 0

∫(−1)^x xdx=??

$$\int\left(−\mathrm{1}\right)^{{x}} {xdx}=?? \\ $$

Question Number 9533 Answers: 3 Comments: 0

Determine and prove if true: ∫_0 ^( n) x^2 dx < Σ_(k=1) ^n k^2

$$\mathrm{Determine}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{if}\:\mathrm{true}: \\ $$$$\int_{\mathrm{0}} ^{\:{n}} {x}^{\mathrm{2}} {dx}\:<\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\mathrm{2}} \\ $$

Question Number 9532 Answers: 0 Comments: 3

S=Σ_(n=t) ^k (2n−1) S=?

$${S}=\underset{{n}={t}} {\overset{{k}} {\sum}}\left(\mathrm{2}{n}−\mathrm{1}\right) \\ $$$${S}=? \\ $$

Question Number 9530 Answers: 2 Comments: 1

Why i ≯ 0 and i ≮ 0 ????

$$\mathrm{Why}\:\:{i}\:\ngtr\:\mathrm{0}\:\mathrm{and}\:{i}\:\nless\:\mathrm{0}\:???? \\ $$

Question Number 9527 Answers: 0 Comments: 0

Prove that: x^(2n) ≥ (x−1)^(2n) + (2x−1)^n x ≥ (1/2), n is positive integers

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$${x}^{\mathrm{2}{n}} \:\geqslant\:\left({x}−\mathrm{1}\right)^{\mathrm{2}{n}} \:+\:\left(\mathrm{2}{x}−\mathrm{1}\right)^{{n}} \\ $$$${x}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{2}},\:\:{n}\:\mathrm{is}\:\mathrm{positive}\:\mathrm{integers} \\ $$

Question Number 9525 Answers: 1 Comments: 0

If x_1 , x_2 , x_3 , ..., x_(2009 ) ∈ R Find the minimum value from (cos x_1 )(sin x_2 ) + (cos x_2 )(sin x_3 ) + ... + (cos x_(2008) )(sin x_(2009) ) + (cos x_(2009) )(sin x_1 )

$$\mathrm{If}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} ,\:...,\:{x}_{\mathrm{2009}\:} \in\:\mathbb{R} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{from} \\ $$$$\left(\mathrm{cos}\:{x}_{\mathrm{1}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{2}} \right)\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{3}} \right)\:+\:...\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2008}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{2009}} \right)\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2009}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{1}} \right) \\ $$

Question Number 9523 Answers: 1 Comments: 0

3a = (b + c + d)^(2014) 3b = (a + c + d)^(2014) 3c = (a + b + d)^(2014) 3d = (a + b + c)^(2014) Find all the solution of (a, b, c, d) if a, b, c, d ∈ R

$$\mathrm{3}{a}\:=\:\left({b}\:+\:{c}\:+\:{d}\right)^{\mathrm{2014}} \\ $$$$\mathrm{3}{b}\:=\:\left({a}\:+\:{c}\:+\:{d}\right)^{\mathrm{2014}} \\ $$$$\mathrm{3}{c}\:=\:\left({a}\:+\:{b}\:+\:{d}\right)^{\mathrm{2014}} \\ $$$$\mathrm{3}{d}\:=\:\left({a}\:+\:{b}\:+\:{c}\right)^{\mathrm{2014}} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{of}\:\left({a},\:{b},\:{c},\:{d}\right)\:\mathrm{if}\:{a},\:{b},\:{c},\:{d}\:\in\:\mathbb{R} \\ $$

Question Number 9520 Answers: 1 Comments: 0

∫((sinxcosx)/(4+sin^4 x))dx=...?

$$\int\frac{\mathrm{sinxcosx}}{\mathrm{4}+\mathrm{sin}^{\mathrm{4}} \mathrm{x}}\mathrm{dx}=...? \\ $$

Question Number 9517 Answers: 0 Comments: 3

f(1) + f(2) + f(3) + ... + f(n) = n^2 . f(n) n > 1; f(1) = 2016 So, f(2016) = ?

$${f}\left(\mathrm{1}\right)\:+\:{f}\left(\mathrm{2}\right)\:+\:{f}\left(\mathrm{3}\right)\:+\:...\:+\:{f}\left({n}\right)\:=\:{n}^{\mathrm{2}} .\:{f}\left({n}\right) \\ $$$${n}\:>\:\mathrm{1};\:{f}\left(\mathrm{1}\right)\:=\:\mathrm{2016} \\ $$$$\mathrm{So},\:{f}\left(\mathrm{2016}\right)\:=\:? \\ $$

Question Number 9516 Answers: 0 Comments: 2

Σ_(k=1) ^n ⌊log_2 k⌋ = 2018 What is the value of n ?

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\lfloor\mathrm{log}_{\mathrm{2}} \:{k}\rfloor\:=\:\mathrm{2018} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}\:? \\ $$

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