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AllQuestion and Answers: Page 358

Question Number 184061 Answers: 1 Comments: 0

∫((sin^(−1) (√x)−cos^(−1) (√x))/(sin^(−1) (√x)+cos^(−1) (√x)))dx=?

$$\int\frac{{sin}^{−\mathrm{1}} \sqrt{{x}}−{cos}^{−\mathrm{1}} \sqrt{{x}}}{{sin}^{−\mathrm{1}} \sqrt{{x}}+{cos}^{−\mathrm{1}} \sqrt{{x}}}{dx}=? \\ $$

Question Number 184048 Answers: 5 Comments: 0

{ ((u_0 = 2)),((u_(n+1) = ((2u_n −1)/u_n ))) :} Find u_n .

$$\:\:\begin{cases}{{u}_{\mathrm{0}} \:=\:\mathrm{2}}\\{{u}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}{u}_{{n}} \:−\mathrm{1}}{{u}_{{n}} }}\end{cases} \\ $$$$\:\:\:{Find}\:{u}_{{n}} . \\ $$

Question Number 184041 Answers: 2 Comments: 0

Question Number 184040 Answers: 4 Comments: 0

Use implicit differentiation to find (d^2 y/dx^2 ) for siny = x

$$\mathrm{Use}\:\mathrm{implicit}\:\mathrm{differentiation}\:\mathrm{to}\:\mathrm{find}\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} } \\ $$$$\mathrm{for}\:\mathrm{sin}{y}\:=\:{x} \\ $$

Question Number 184037 Answers: 2 Comments: 0

i^2 =−1 Σ_(j=1) ^(2023) ji^j =?

$${i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{\mathrm{2023}} {\sum}}{ji}^{{j}} =? \\ $$

Question Number 184030 Answers: 1 Comments: 0

How many words can be made from 5 letters if (a) all letters are different (b) 2 letters are identical (c) all letters are different but 2 partucular letters cannot be adjacent. M.m

$$\mathrm{How}\:\mathrm{many}\:\mathrm{words}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\: \\ $$$$\mathrm{from}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{if} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{different} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{2}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{identical} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{all}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{2} \\ $$$$\mathrm{partucular}\:\mathrm{letters}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{adjacent}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Question Number 184029 Answers: 1 Comments: 0

∫_0 ^( 1) (( x−1)/(x+1))∙(1/(ln(x))) dx =?

$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\boldsymbol{\mathrm{x}}−\mathrm{1}}{\boldsymbol{\mathrm{x}}+\mathrm{1}}\centerdot\frac{\mathrm{1}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)}\:\boldsymbol{\mathrm{dx}}\:=? \\ $$

Question Number 184028 Answers: 0 Comments: 0

H^(A^P P) Y Y_(E_A R) ! ⌊e⌋⌊i-i⌋⌊e⌋⌊𝛑⌋^(−)

Question Number 184027 Answers: 1 Comments: 0

∫_0 ^( (𝛑/2)) e^(−tg^2 (x)) dx = ???

$$\:\:\: \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\boldsymbol{\mathrm{e}}^{−\boldsymbol{{tg}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)} \:\boldsymbol{{d}\mathrm{x}}\:=\:??? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 184019 Answers: 1 Comments: 2

{ ((y^x = 64)),((y^((x + 1)/(x − 1)) = 16)) :} find “x”

$$\begin{cases}{{y}^{{x}} =\:\mathrm{64}}\\{{y}^{\frac{{x}\:+\:\mathrm{1}}{{x}\:−\:\mathrm{1}}} \:=\:\mathrm{16}}\end{cases} \\ $$$$\:{find}\:``{x}'' \\ $$

Question Number 184018 Answers: 1 Comments: 0

evaluate 𝚺_(n=1) ^∞ (1/(n^3 (((6n)),((3n)) )))

$$\boldsymbol{\mathrm{evaluate}}\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\boldsymbol{\sum}}}\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{3}} \begin{pmatrix}{\mathrm{6}\boldsymbol{\mathrm{n}}}\\{\mathrm{3}\boldsymbol{\mathrm{n}}}\end{pmatrix}} \\ $$

Question Number 184012 Answers: 1 Comments: 4

3^x −2^x =x^2 +10 find (3^x −10)^2 (2^x −1)

$$\mathrm{3}^{{x}} −\mathrm{2}^{{x}} ={x}^{\mathrm{2}} +\mathrm{10} \\ $$$${find} \\ $$$$\left(\mathrm{3}^{{x}} −\mathrm{10}\right)^{\mathrm{2}} \left(\mathrm{2}^{{x}} −\mathrm{1}\right) \\ $$

Question Number 184010 Answers: 0 Comments: 3

determinant ((( determinant (((2023))) )))_( is^ _(a number_(which is divisible_(by_(•_• ) ) ) ) ) (i)its sum of digits & (ii)its sum of squares of digits

Question Number 184031 Answers: 1 Comments: 0

∫_1 ^∞ ((x^7 −x)/(x^(10) −1))dx = ?

$$\:\int_{\mathrm{1}} ^{\infty} \:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{7}} −\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}^{\mathrm{10}} −\mathrm{1}}\boldsymbol{\mathrm{dx}}\:=\:? \\ $$

Question Number 183998 Answers: 1 Comments: 1

Find the natural number n such n=7^a ∙17^b a∈N, b∈N and the sum of all its divisors (1 and n included) is 2456.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{natural}\:\mathrm{number}\:\boldsymbol{\mathrm{n}}\:{such} \\ $$$$\:\:\:\:\:\:\:{n}=\mathrm{7}^{{a}} \centerdot\mathrm{17}^{{b}} \:\:\:\:\:\:{a}\in\mathbb{N},\:\:\:{b}\in\mathbb{N} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{its}\:\mathrm{divisors}\:\left(\mathrm{1}\:\mathrm{and}\right. \\ $$$$\left.\boldsymbol{\mathrm{n}}\:\mathrm{included}\right)\:\mathrm{is}\:\mathrm{2456}. \\ $$

Question Number 183991 Answers: 2 Comments: 1

Question Number 183989 Answers: 4 Comments: 1

(((√x) + (√(x − 4a)))/( (√x) − (√(x − 4a)))) = a ≠ 0 find “x” in terms of “a”.

$$\frac{\sqrt{{x}}\:+\:\sqrt{{x}\:−\:\mathrm{4}{a}}}{\:\sqrt{{x}}\:−\:\sqrt{{x}\:−\:\mathrm{4}{a}}}\:=\:{a}\:\neq\:\mathrm{0} \\ $$$$\:{find}\:``{x}''\:{in}\:{terms}\:{of}\:``{a}''.\: \\ $$

Question Number 183977 Answers: 3 Comments: 0

∫_0 ^( ∞) (dx/(x^8 +x^4 +1)) =?

$$\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{dx}}{{x}^{\mathrm{8}} +{x}^{\mathrm{4}} +\mathrm{1}}\:=?\: \\ $$

Question Number 183976 Answers: 1 Comments: 0

∫ ((sin x−(√(1+sin x)))/(cos x−(√(1+cos x)))) dx =?

$$\:\:\int\:\frac{\mathrm{sin}\:{x}−\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}{\mathrm{cos}\:{x}−\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}\:{dx}\:=? \\ $$

Question Number 183974 Answers: 1 Comments: 0

Question Number 183967 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (((−1)^(n+1) )/(3n−1)) = ?

$$\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}+\mathrm{1}} }{\mathrm{3}\boldsymbol{\mathrm{n}}−\mathrm{1}}\:\:=\:\:\:? \\ $$

Question Number 183965 Answers: 1 Comments: 0

A linear transformation E, of the x−y plane is defined as E:(x, y) → (2x+y, 2x+3y) Find the equation of the line that remains invariant under the transformation.

$$\mathrm{A}\:\mathrm{linear}\:\mathrm{transformation}\:\boldsymbol{{E}},\:\mathrm{of}\:\mathrm{the} \\ $$$${x}−{y}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{E}}:\left({x},\:{y}\right)\:\rightarrow\:\left(\mathrm{2}{x}+{y},\:\mathrm{2}{x}+\mathrm{3}{y}\right) \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{that} \\ $$$$\mathrm{remains}\:\mathrm{invariant}\:\mathrm{under}\:\mathrm{the} \\ $$$$\mathrm{transformation}. \\ $$

Question Number 183962 Answers: 2 Comments: 1

Question Number 183951 Answers: 2 Comments: 1

Question Number 183945 Answers: 2 Comments: 0

Question Number 183943 Answers: 0 Comments: 1

∫_(1−(√(πx))(d^(1/2) /dx^(1/2) )(1)) ^(Σ_(n=1) ^∞ (4/(4n^2 −1)) ) ((arctan(((2−x)/(1+2x))))/(x^2 −4x−1))dx

$$\int_{\mathrm{1}−\sqrt{\pi{x}}\frac{{d}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{dx}^{\frac{\mathrm{1}}{\mathrm{2}}} }\left(\mathrm{1}\right)} ^{\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{4}}{\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}}\:} \frac{\mathrm{arctan}\left(\frac{\mathrm{2}−{x}}{\mathrm{1}+\mathrm{2}{x}}\right)}{{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{1}}{dx} \\ $$

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