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Question Number 111001 Answers: 0 Comments: 1

Two numbers a and b are chosen at random from the set of first 30 natural numbers. The probability that a^2 −b^2 is divisible by 3 is

$$\mathrm{Two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{chosen}\:\mathrm{at} \\ $$$$\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{first}\:\mathrm{30}\:\mathrm{natural} \\ $$$$\mathrm{numbers}.\:\mathrm{The}\:\mathrm{probability}\:\mathrm{that}\:{a}^{\mathrm{2}} −{b}^{\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}\:\mathrm{is} \\ $$

Question Number 110993 Answers: 1 Comments: 1

Question Number 111092 Answers: 2 Comments: 4

(√(bemath)) lim_(x→0) ((arctan x)/(arc sin x−x))

$$\:\:\sqrt{\mathrm{bemath}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{arctan}\:\mathrm{x}}{\mathrm{arc}\:\mathrm{sin}\:\mathrm{x}−\mathrm{x}} \\ $$

Question Number 110988 Answers: 3 Comments: 0

Evaluate without using L′hopital′s rule lim_(x→4) (((√x)−2)/(x−4))

$$\:\mathrm{Evaluate}\:\mathrm{without}\:\mathrm{using}\:\mathrm{L}'\mathrm{hopital}'\mathrm{s}\:\mathrm{rule} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\:\frac{\sqrt{{x}}−\mathrm{2}}{{x}−\mathrm{4}} \\ $$

Question Number 110984 Answers: 1 Comments: 3

GCD of two unequal numbers can′t exceed their absolute difference. Prove.

$$\mathrm{GCD}\:{of}\:{two}\:{unequal}\:\:{numbers}\:{can}'{t}\: \\ $$$${exceed}\:{their}\:{absolute} \\ $$$${difference}.\:\:{Prove}. \\ $$

Question Number 110980 Answers: 2 Comments: 2

Question Number 110964 Answers: 1 Comments: 0

solve ∫_0 ^1 ((x^2 lnx)/((1+x^2 )^3 ))dx

$${solve}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} \mathrm{ln}{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }{dx} \\ $$

Question Number 111017 Answers: 2 Comments: 0

(√(bemath)) ∫ (dx/( ((4−((3−2x))^(1/(3 )) ))^(1/(4 )) )) ?

$$\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\int\:\frac{\mathrm{dx}}{\:\sqrt[{\mathrm{4}\:}]{\mathrm{4}−\sqrt[{\mathrm{3}\:}]{\mathrm{3}−\mathrm{2x}}}}\:? \\ $$

Question Number 110954 Answers: 1 Comments: 0

verify the formulae Σ_(n=−∞) ^(+∞) (1/((na +1)^p )) =−(π/a^n ) lim_(z→−(1/a)) (1/((p−1)!)){cotan(πz)}^((p−1)) inthis case 1) a =1 and p=2 2) a=2 and p=2 3)a=2 and p=3 4) a=3 and p=2

$$\mathrm{verify}\:\mathrm{the}\:\mathrm{formulae} \\ $$$$\sum_{\mathrm{n}=−\infty} ^{+\infty} \:\frac{\mathrm{1}}{\left(\mathrm{na}\:+\mathrm{1}\right)^{\mathrm{p}} }\:=−\frac{\pi}{\mathrm{a}^{\mathrm{n}} }\:\mathrm{lim}_{\mathrm{z}\rightarrow−\frac{\mathrm{1}}{\mathrm{a}}} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{p}−\mathrm{1}\right)!}\left\{\mathrm{cotan}\left(\pi\mathrm{z}\right)\right\}^{\left(\mathrm{p}−\mathrm{1}\right)} \\ $$$$\left.\mathrm{inthis}\:\mathrm{case}\:\:\mathrm{1}\right)\:\:\mathrm{a}\:=\mathrm{1}\:\mathrm{and}\:\mathrm{p}=\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}=\mathrm{2}\:\:\:\mathrm{and}\:\mathrm{p}=\mathrm{2} \\ $$$$\left.\mathrm{3}\right)\mathrm{a}=\mathrm{2}\:\mathrm{and}\:\mathrm{p}=\mathrm{3} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{a}=\mathrm{3}\:\mathrm{and}\:\mathrm{p}=\mathrm{2} \\ $$

Question Number 110953 Answers: 0 Comments: 1

Question Number 110951 Answers: 2 Comments: 0

Question Number 110948 Answers: 1 Comments: 6

(√(bemath)) If each point on the line 3x+4y=2 is transformed by matrix M= (((2 0)),((0 1)) ) , the image is a line ___

$$\:\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\mathrm{If}\:\mathrm{each}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:\mathrm{3x}+\mathrm{4y}=\mathrm{2} \\ $$$$\mathrm{is}\:\mathrm{transformed}\:\mathrm{by}\:\mathrm{matrix}\:\mathrm{M}=\begin{pmatrix}{\mathrm{2}\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\mathrm{1}}\end{pmatrix}\:,\:\mathrm{the} \\ $$$$\mathrm{image}\:\mathrm{is}\:\mathrm{a}\:\mathrm{line}\:\_\_\_ \\ $$

Question Number 110944 Answers: 5 Comments: 3

■(√(bemath))★ (1)If (√a) −(√b) = 20 , a,b∈R , find maximum value of a−5b ? (2)lim_(x→4) (((√x)−(√(3(√x)−2)))/(x^2 −16)) ? (3)∫ ((tan (ln x) tan (ln ((x/2))))/x) dx (4)((((√(3x−7)))^2 −2)/(x−3)) ≤ ((3−((√x))^2 )/(x−3))

$$\:\:\:\blacksquare\sqrt{\mathrm{bemath}}\bigstar \\ $$$$\left(\mathrm{1}\right)\mathrm{If}\:\sqrt{\mathrm{a}}\:−\sqrt{\mathrm{b}}\:=\:\mathrm{20}\:,\:\mathrm{a},\mathrm{b}\in\mathbb{R}\:,\:\mathrm{find}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{a}−\mathrm{5b}\:? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{3}\sqrt{\mathrm{x}}−\mathrm{2}}}{\mathrm{x}^{\mathrm{2}} −\mathrm{16}}\:? \\ $$$$\left(\mathrm{3}\right)\int\:\frac{\mathrm{tan}\:\left(\mathrm{ln}\:\mathrm{x}\right)\:\mathrm{tan}\:\left(\mathrm{ln}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{x}}\:\mathrm{dx} \\ $$$$\left(\mathrm{4}\right)\frac{\left(\sqrt{\mathrm{3x}−\mathrm{7}}\right)^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}−\mathrm{3}}\:\leqslant\:\frac{\mathrm{3}−\left(\sqrt{\mathrm{x}}\right)^{\mathrm{2}} }{\mathrm{x}−\mathrm{3}} \\ $$

Question Number 110939 Answers: 2 Comments: 1

Question Number 110926 Answers: 1 Comments: 0

A 2000kg car start from rest and accelerated to a final velocity of 20m/s in 16 seconds. Assuming a constant air resistance of 500N, find (i) the average power developed by the engine of the car. (ii) the instantaneous power developed by the engine when the car reaches its final speed.

$$\mathrm{A}\:\mathrm{2000kg}\:\mathrm{car}\:\mathrm{start}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{and} \\ $$$$\mathrm{accelerated}\:\mathrm{to}\:\mathrm{a}\:\mathrm{final}\:\mathrm{velocity}\:\mathrm{of} \\ $$$$\mathrm{20m}/\mathrm{s}\:\mathrm{in}\:\mathrm{16}\:\mathrm{seconds}.\:\mathrm{Assuming}\:\mathrm{a} \\ $$$$\mathrm{constant}\:\mathrm{air}\:\mathrm{resistance}\:\mathrm{of}\:\mathrm{500N},\:\mathrm{find} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{the}\:\mathrm{average}\:\mathrm{power}\:\mathrm{developed}\:\mathrm{by} \\ $$$$\mathrm{the}\:\mathrm{engine}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{the}\:\mathrm{instantaneous}\:\mathrm{power} \\ $$$$\mathrm{developed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{engine}\:\mathrm{when}\:\mathrm{the}\:\mathrm{car} \\ $$$$\mathrm{reaches}\:\mathrm{its}\:\mathrm{final}\:\mathrm{speed}. \\ $$

Question Number 110970 Answers: 2 Comments: 1

Question Number 110911 Answers: 0 Comments: 0

Four teachers A, B, C, and D each proposed two exercises, one on algebra and another on analyses, to form an exam. The students have to choose two exercises at random. 1. Calculate the probability P(a) of a student to choose two exercises on algebra. a\ P(a)=(3/(16)) , b\P(a)=(3/(14)) , c\P(a)=(1/4) , d\None 2. Calculate the probability P(b) of choosing two exercises proposed by the same teacher. a\P(b)=(1/(10)) , b\P(b)=(1/(60)) , c\P(b)=(1/7) , d\None 3. Calculate the probability P(c) of choosing two exercises proposed by teacher A. a\P(c)=(1/3) , b\P(c)=(1/4) , c\P(c)=(1/(28)) , d\None

Question Number 110910 Answers: 2 Comments: 0

mr M.N july 1970 the question you posted earlier here goes the solution ∫_0 ^(1/2) ((ln^2 (1−x))/x)dx

$${mr}\:{M}.{N}\:{july}\:\mathrm{1970}\:{the}\:{question}\:{you}\:{posted}\:{earlier}\:{here}\:{goes}\:{the}\:{solution} \\ $$$$\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$ \\ $$

Question Number 110897 Answers: 3 Comments: 0

(1)4x−4 ≤ ∣x^2 −3x+2 ∣ find the solution set (2) ((1+cos ((α/2))−sin ((α/2)))/(1−cos ((α/2))−sin ((α/2))))=?

$$\left(\mathrm{1}\right)\mathrm{4x}−\mathrm{4}\:\leqslant\:\mid\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{2}\:\mid\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\: \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}+\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}{\mathrm{1}−\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}=? \\ $$

Question Number 110895 Answers: 3 Comments: 0

How many ways can 2018 be expressed as the sum of two squares?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{2018}\:\mathrm{be} \\ $$$$\mathrm{expressed}\:\mathrm{as}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{squares}? \\ $$

Question Number 110879 Answers: 0 Comments: 1

Question Number 110875 Answers: 4 Comments: 0

(1)∫_e ^e^e ((ln (x).ln (ln (x)))/x) dx ? (2)lim_(x→π/4) ((cosec^2 x−2)/(cot x−1)) (3) Given { ((xy=((16y−9x)/(45)))),(((4/( (√x)))−(3/( (√y))) = 5)) :} ⇒find 9(√(xy))

$$\left(\mathrm{1}\right)\underset{\mathrm{e}} {\overset{\mathrm{e}^{\mathrm{e}} } {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{x}\right).\mathrm{ln}\:\left(\mathrm{ln}\:\left(\mathrm{x}\right)\right)}{\mathrm{x}}\:\mathrm{dx}\:? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\:\frac{\mathrm{cosec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{2}}{\mathrm{cot}\:\mathrm{x}−\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Given}\:\begin{cases}{\mathrm{xy}=\frac{\mathrm{16y}−\mathrm{9x}}{\mathrm{45}}}\\{\frac{\mathrm{4}}{\:\sqrt{\mathrm{x}}}−\frac{\mathrm{3}}{\:\sqrt{\mathrm{y}}}\:=\:\mathrm{5}}\end{cases} \\ $$$$\Rightarrow\mathrm{find}\:\mathrm{9}\sqrt{\mathrm{xy}} \\ $$

Question Number 110869 Answers: 1 Comments: 0

Question Number 110868 Answers: 0 Comments: 0

x^2 +y^2 =z Level sets and surface plot using Geogebra

$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={z} \\ $$$${Level}\:{sets}\:{and}\:{surface}\:{plot} \\ $$$${using}\:{Geogebra} \\ $$$$ \\ $$

Question Number 110865 Answers: 0 Comments: 0

Plotting of x^2

$${Plotting}\:{of} \\ $$$${x}^{\mathrm{2}} \\ $$

Question Number 110861 Answers: 2 Comments: 0

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