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AllQuestion and Answers: Page 1935

Question Number 12111 Answers: 0 Comments: 0

show that: Σ_(n=0) ^∞ (((−1)^n )/(n!))=(1/e) please show your working

$$\mathrm{show}\:\mathrm{that}: \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}=\frac{\mathrm{1}}{{e}} \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{your}\:\mathrm{working} \\ $$

Question Number 12109 Answers: 0 Comments: 0

show that: Σ_(n=1) ^∞ (((−1)^(n+1) )/n)=ln(2) please show your working

$$\mathrm{show}\:\mathrm{that}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{{n}}=\mathrm{ln}\left(\mathrm{2}\right) \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{your}\:\mathrm{working} \\ $$

Question Number 12107 Answers: 1 Comments: 2

10^3 + 11^3 + 12^3 + ... + 20^3 Is there any ways to count the sum of that sequence without sum them manually?

$$\mathrm{10}^{\mathrm{3}} \:+\:\mathrm{11}^{\mathrm{3}} \:+\:\mathrm{12}^{\mathrm{3}} \:+\:...\:+\:\mathrm{20}^{\mathrm{3}} \\ $$$$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{count}\:\mathrm{the}\:\mathrm{sum}\: \\ $$$$\mathrm{of}\:\mathrm{that}\:\mathrm{sequence}\:\mathrm{without}\:\mathrm{sum}\:\mathrm{them}\:\mathrm{manually}? \\ $$

Question Number 12106 Answers: 1 Comments: 0

Question Number 12103 Answers: 0 Comments: 0

Evaluate ∫((ln(1+x))/(1+x^2 ))dx

$${Evaluate}\:\int\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 12102 Answers: 2 Comments: 0

Evaluate ∫(1/(cos^2 (x)))dx

$${Evaluate}\:\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)}{dx} \\ $$

Question Number 12101 Answers: 1 Comments: 0

Evaluate ∫_0 ^1 ((tan^(−1) (x))/( x))dx

$${Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{tan}^{−\mathrm{1}} \left({x}\right)}{\:{x}}{dx} \\ $$

Question Number 12100 Answers: 0 Comments: 0

Evaluate −∫_0 ^∞ e^(−x) ln(x)dx

$${Evaluate}\:−\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {ln}\left({x}\right){dx} \\ $$

Question Number 12098 Answers: 2 Comments: 0

Evaluate ∫(√(x^2 −a^2 ))dx

$${Evaluate}\:\int\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 12096 Answers: 1 Comments: 0

Question Number 12094 Answers: 0 Comments: 0

lim_(x→0) ((x tan 5x)/(cos 2x . cos 7x))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\:\mathrm{tan}\:\mathrm{5}{x}}{\mathrm{cos}\:\mathrm{2}{x}\:.\:\mathrm{cos}\:\mathrm{7}{x}} \\ $$

Question Number 12093 Answers: 1 Comments: 0

log_(abc) b=3 log_(abc) c=4 log_(abc) a=?

$$\mathrm{log}_{\mathrm{abc}} \mathrm{b}=\mathrm{3} \\ $$$$\mathrm{log}_{\mathrm{abc}} \mathrm{c}=\mathrm{4} \\ $$$$\mathrm{log}_{\mathrm{abc}} \mathrm{a}=? \\ $$

Question Number 12092 Answers: 0 Comments: 0

Prove that tan 70°−tan 50°+tan 10° =(√3)

$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{tan}\:\mathrm{70}°−\mathrm{tan}\:\mathrm{50}°+\mathrm{tan}\:\mathrm{10}°\:=\sqrt{\mathrm{3}} \\ $$

Question Number 12087 Answers: 1 Comments: 0

In how many ways can 10 objects be split into two groups containing 4 and 6 objects respectively ?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{10}\:\mathrm{objects}\:\mathrm{be}\:\mathrm{split}\:\mathrm{into}\:\mathrm{two}\:\:\mathrm{groups}\:\mathrm{containing}\: \\ $$$$\mathrm{4}\:\mathrm{and}\:\mathrm{6}\:\mathrm{objects}\:\mathrm{respectively}\:? \\ $$

Question Number 12085 Answers: 2 Comments: 0

Question Number 12082 Answers: 1 Comments: 0

Question Number 12080 Answers: 1 Comments: 0

Question Number 12078 Answers: 1 Comments: 0

If tan^2 α = 1+2tan^2 β then prove that cos 2β =1+2cos 2α .

$$\mathrm{If}\:\mathrm{tan}\:^{\mathrm{2}} \alpha\:=\:\mathrm{1}+\mathrm{2tan}\:^{\mathrm{2}} \beta\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\mathrm{cos}\:\mathrm{2}\beta\:=\mathrm{1}+\mathrm{2cos}\:\mathrm{2}\alpha\:. \\ $$

Question Number 12077 Answers: 1 Comments: 0

A man travels 29 km on an open road at a certain speed. In the city, he reduce his speed by 420 km/hr, and he find that he take him the same time to cover 15 km, Find his average speed. (a) On the open road (b) In the city

$$\mathrm{A}\:\mathrm{man}\:\mathrm{travels}\:\mathrm{29}\:\mathrm{km}\:\mathrm{on}\:\mathrm{an}\:\mathrm{open}\:\mathrm{road}\:\mathrm{at}\:\mathrm{a}\:\mathrm{certain}\:\mathrm{speed}.\:\mathrm{In}\:\mathrm{the}\:\mathrm{city},\:\mathrm{he}\: \\ $$$$\mathrm{reduce}\:\mathrm{his}\:\mathrm{speed}\:\mathrm{by}\:\mathrm{420}\:\mathrm{km}/\mathrm{hr},\:\mathrm{and}\:\mathrm{he}\:\mathrm{find}\:\mathrm{that}\:\mathrm{he}\:\mathrm{take}\:\mathrm{him}\:\mathrm{the}\:\mathrm{same}\:\mathrm{time} \\ $$$$\mathrm{to}\:\mathrm{cover}\:\mathrm{15}\:\mathrm{km},\:\mathrm{Find}\:\mathrm{his}\:\mathrm{average}\:\mathrm{speed}.\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{On}\:\mathrm{the}\:\mathrm{open}\:\mathrm{road} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{In}\:\mathrm{the}\:\mathrm{city} \\ $$

Question Number 12074 Answers: 1 Comments: 0

Question Number 12071 Answers: 0 Comments: 0

Question Number 12064 Answers: 0 Comments: 0

A heat pump gas C.P = 3, while the indoor temperature is 27°C and the outdoor temperature is 8°C. How much work per hour is required to pump 3.07 J of heat per hour into the individual unit of 10^5 J.

$$\mathrm{A}\:\mathrm{heat}\:\mathrm{pump}\:\mathrm{gas}\:\mathrm{C}.\mathrm{P}\:=\:\mathrm{3},\:\mathrm{while}\:\mathrm{the}\:\mathrm{indoor}\:\mathrm{temperature}\:\mathrm{is}\:\mathrm{27}°\mathrm{C}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{outdoor}\:\mathrm{temperature}\:\mathrm{is}\:\mathrm{8}°\mathrm{C}.\:\mathrm{How}\:\mathrm{much}\:\mathrm{work}\:\mathrm{per}\:\mathrm{hour}\:\mathrm{is}\:\mathrm{required}\:\mathrm{to}\:\mathrm{pump} \\ $$$$\mathrm{3}.\mathrm{07}\:\mathrm{J}\:\mathrm{of}\:\mathrm{heat}\:\mathrm{per}\:\mathrm{hour}\:\mathrm{into}\:\mathrm{the}\:\mathrm{individual}\:\mathrm{unit}\:\mathrm{of}\:\mathrm{10}^{\mathrm{5}} \:\mathrm{J}.\: \\ $$

Question Number 12063 Answers: 1 Comments: 0

Question Number 12062 Answers: 2 Comments: 0

lim_(x→y) ((x^n − y^n )/(x − y))

$$\underset{{x}\rightarrow\mathrm{y}} {\mathrm{lim}}\:\:\frac{\mathrm{x}^{\mathrm{n}} \:−\:\mathrm{y}^{\mathrm{n}} }{\mathrm{x}\:−\:\mathrm{y}} \\ $$

Question Number 12047 Answers: 1 Comments: 1

∫4x^3 (3x^2 + 2)^5 dx

$$\int\mathrm{4}{x}^{\mathrm{3}} \left(\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{2}\right)^{\mathrm{5}} \:{dx} \\ $$

Question Number 12046 Answers: 1 Comments: 0

how much matrices of integers number A= [(a,b),(c,d) ]if A^2 +A=2I, c=0, det(A)=4

$${how}\:{much}\:{matrices}\:{of}\:{integers}\:{number} \\ $$$${A}=\begin{bmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{bmatrix}{if}\:{A}^{\mathrm{2}} +{A}=\mathrm{2}{I},\:{c}=\mathrm{0},\:{det}\left({A}\right)=\mathrm{4} \\ $$

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