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Question Number 58253 Answers: 1 Comments: 0

There are 100, 150 and 250 students in forms one, two and three, respectively in a school. If the mean ages of tbe students in the forms are 15.6 years, 16.8 years and 18years respectively, find i. the total number of students in the forms ii. correct to one decimal place, the mean age of all the students

$$\mathrm{There}\:\mathrm{are}\:\mathrm{100},\:\mathrm{150}\:\mathrm{and}\:\mathrm{250}\:\mathrm{students}\:\mathrm{in} \\ $$$$\mathrm{forms}\:\mathrm{one},\:\mathrm{two}\:\mathrm{and}\:\mathrm{three},\:\mathrm{respectively} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{school}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{ages}\:\mathrm{of}\:\mathrm{tbe}\:\mathrm{students}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{forms}\:\mathrm{are}\:\mathrm{15}.\mathrm{6}\:\mathrm{years},\:\mathrm{16}.\mathrm{8}\:\mathrm{years}\:\mathrm{and} \\ $$$$\mathrm{18years}\:\mathrm{respectively},\:\mathrm{find} \\ $$$$\mathrm{i}.\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{students}\:\mathrm{in}\:\mathrm{the}\:\mathrm{forms} \\ $$$$\mathrm{ii}.\:\mathrm{correct}\:\mathrm{to}\:\mathrm{one}\:\mathrm{decimal}\:\mathrm{place},\:\mathrm{the}\:\mathrm{mean} \\ $$$$\mathrm{age}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{students} \\ $$

Question Number 58250 Answers: 1 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3x}^{\mathrm{3}} \:−\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{2x}\:−\:\mathrm{4}}{\sqrt{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3x}\:+\:\mathrm{2}}}\right)\:\mathrm{dx} \\ $$

Question Number 58249 Answers: 0 Comments: 0

I_n ^ =∫_0 ^(π/2) cos^n xcos(nx)dx then show that I_1 ,I_2 ,I_3 ....are in G.P

$$\overset{} {{I}}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{{n}} {xcos}\left({nx}\right){dx} \\ $$$${then}\:{show}\:{that}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,{I}_{\mathrm{3}} ....{are}\:{in}\:{G}.{P} \\ $$

Question Number 58248 Answers: 1 Comments: 0

leg A_1 ,A_2 ,...A_n and H_1 ,H_2 ,...H_n are n A.M′S and H.M′S respectively between a and b prove that A_r H_(n−r+1) =ab n≥r≥1

$${leg}\:{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} ,...{A}_{{n}} \:{and}\:{H}_{\mathrm{1}} ,{H}_{\mathrm{2}} ,...{H}_{{n}} \:{are}\:{n}\:{A}.{M}'{S}\: \\ $$$${and}\:{H}.{M}'{S}\:{respectively}\:{between}\:{a}\:{and}\:{b} \\ $$$${prove}\:{that}\:{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} ={ab} \\ $$$$\:{n}\geqslant{r}\geqslant\mathrm{1} \\ $$

Question Number 58247 Answers: 0 Comments: 0

lim_(n→∞) ((S_1 S_n +S_2 S_(n−1) +S_3 S_(n−2) +...+S_n S_1 )/(S_1 ^2 +S_2 ^2 +...+S_n ^2 )) when S_n is sum of infinite series whose first term=n and common ratio (1/(n+1)) find the value of limit

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{S}_{\mathrm{1}} {S}_{{n}} +{S}_{\mathrm{2}} {S}_{{n}−\mathrm{1}} +{S}_{\mathrm{3}} {S}_{{n}−\mathrm{2}} +...+{S}_{{n}} {S}_{\mathrm{1}} }{{S}_{\mathrm{1}} ^{\mathrm{2}} +{S}_{\mathrm{2}} ^{\mathrm{2}} +...+{S}_{{n}} ^{\mathrm{2}} } \\ $$$${when}\:{S}_{{n}} \:{is}\:{sum}\:{of}\:{infinite}\:{series}\:{whose} \\ $$$${first}\:{term}={n}\:\:\:{and}\:{common}\:{ratio}\:\frac{\mathrm{1}}{{n}+\mathrm{1}} \\ $$$${find}\:{the}\:{value}\:{of}\:{limit} \\ $$

Question Number 58246 Answers: 1 Comments: 0

show that P=x^(9999) +x^(8888) +x^(7777) +x^(6666) +x^(5555) +x^(4444) +x^(3333) +x^(2222) +x^(1111) +1 Q=x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x+1 prove P is divisible by Q

$${show}\:{that} \\ $$$${P}={x}^{\mathrm{9999}} +{x}^{\mathrm{8888}} +{x}^{\mathrm{7777}} +{x}^{\mathrm{6666}} +{x}^{\mathrm{5555}} +{x}^{\mathrm{4444}} +{x}^{\mathrm{3333}} +{x}^{\mathrm{2222}} +{x}^{\mathrm{1111}} +\mathrm{1} \\ $$$${Q}={x}^{\mathrm{9}} +{x}^{\mathrm{8}} +{x}^{\mathrm{7}} +{x}^{\mathrm{6}} +{x}^{\mathrm{5}} +{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1} \\ $$$${prove}\:\:{P}\:\:{is}\:{divisible}\:{by}\:{Q} \\ $$

Question Number 58245 Answers: 1 Comments: 0

if log(a+b+c)=loga+logb+logc prove log(((2a)/(1−a^2 ))+((2b)/(1−b^2 ))+((2c)/(1−c^2 )))=log(((2a)/(1−a^2 )))+log(((2b)/(1−b^2 )))+log(((2c)/(1−c^2 )))

$${if}\:{log}\left({a}+{b}+{c}\right)={loga}+{logb}+{logc} \\ $$$${prove} \\ $$$${log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}−{a}^{\mathrm{2}} }+\frac{\mathrm{2}{b}}{\mathrm{1}−{b}^{\mathrm{2}} }+\frac{\mathrm{2}{c}}{\mathrm{1}−{c}^{\mathrm{2}} }\right)={log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}−{a}^{\mathrm{2}} }\right)+{log}\left(\frac{\mathrm{2}{b}}{\mathrm{1}−{b}^{\mathrm{2}} }\right)+{log}\left(\frac{\mathrm{2}{c}}{\mathrm{1}−{c}^{\mathrm{2}} }\right) \\ $$

Question Number 58240 Answers: 1 Comments: 2

i=∫dx/(ax^2 +bx+c)^(3/2)

$$\mathrm{i}=\int\mathrm{dx}/\left(\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$

Question Number 58239 Answers: 1 Comments: 0

lim_(x→0^+ ) (x^(1/x) )

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left({x}^{\frac{\mathrm{1}}{{x}}} \right) \\ $$

Question Number 58238 Answers: 0 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:+\:\mathrm{2}\boldsymbol{\mathrm{x}}\:−\:\mathrm{4}}{\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:−\:\mathrm{3}\boldsymbol{\mathrm{x}}\:+\:\mathrm{2}}}\right)\:\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 58222 Answers: 2 Comments: 4

∫_0 ^1 x^x dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{x}} {dx} \\ $$

Question Number 58220 Answers: 1 Comments: 0

find ∫ (dx/((x^2 +x)(√(−x^2 +2x +3))))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}\right)\sqrt{−{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{3}}} \\ $$$$ \\ $$

Question Number 58216 Answers: 1 Comments: 1

The molar heat capacity of a metal at low temperature varies with the temperature according to the equation C = bθ + (a/H)θ^3 where a, b and H are constant. How much heat per mole is transfered during the process in which the temperature change from 0.01H to 0.02H ?

$$\mathrm{The}\:\mathrm{molar}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{metal}\:\:\mathrm{at} \\ $$$$\mathrm{low}\:\mathrm{temperature}\:\mathrm{varies}\:\mathrm{with}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\mathrm{C}\:=\:\mathrm{b}\theta\:+\:\frac{\mathrm{a}}{\mathrm{H}}\theta^{\mathrm{3}} \\ $$$$\mathrm{where}\:\mathrm{a},\:\mathrm{b}\:\mathrm{and}\:\mathrm{H}\:\mathrm{are}\:\mathrm{constant}. \\ $$$$\mathrm{How}\:\mathrm{much}\:\mathrm{heat}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{is}\:\mathrm{transfered} \\ $$$$\mathrm{during}\:\mathrm{the}\:\mathrm{process}\:\mathrm{in}\:\mathrm{which}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{change}\:\mathrm{from}\:\mathrm{0}.\mathrm{01H}\: \\ $$$$\mathrm{to}\:\mathrm{0}.\mathrm{02H}\:? \\ $$

Question Number 58212 Answers: 0 Comments: 0

let f(x) =∫_0 ^∞ e^(−x[t]) sin(xt)dt with x>0 1) find a explicit form for f(x) 2) let U_n =nf(n) find lim_(n→+∞) U_n and study the convergence of ΣU_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} \:{sin}\left({xt}\right){dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{U}_{{n}} ={nf}\left({n}\right)\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{U}_{{n}} \\ $$

Question Number 58211 Answers: 0 Comments: 2

∫_( 0) ^(2π) ∣ cos x−sin x ∣dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\:\mid\:\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\:\mid{dx}\:= \\ $$

Question Number 58210 Answers: 0 Comments: 5

find two possible number such that 1) xy=(x/y)=x−y 2)xy=((2x)/y)=3(x−y) 3) xy=(x/y)=2(x−y).

$$\mathrm{find}\:\mathrm{two}\:\mathrm{possible}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left.\mathrm{1}\right)\:\:\mathrm{xy}=\frac{\mathrm{x}}{\mathrm{y}}=\mathrm{x}−\mathrm{y} \\ $$$$\left.\mathrm{2}\right)\mathrm{xy}=\frac{\mathrm{2x}}{\mathrm{y}}=\mathrm{3}\left(\mathrm{x}−\mathrm{y}\right) \\ $$$$\left.\mathrm{3}\right)\:\:\mathrm{xy}=\frac{\mathrm{x}}{\mathrm{y}}=\mathrm{2}\left(\mathrm{x}−\mathrm{y}\right). \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 58203 Answers: 1 Comments: 0

A battery can supply current of 1.2A and 0.4A through 4Ω and 14Ω respectively. Calculate the internal resistance of the battery

$$\mathrm{A}\:\mathrm{battery}\:\mathrm{can}\:\mathrm{supply}\:\mathrm{current}\:\mathrm{of}\:\mathrm{1}.\mathrm{2A}\: \\ $$$$\mathrm{and}\:\mathrm{0}.\mathrm{4A}\:\mathrm{through}\:\mathrm{4}\Omega\:\mathrm{and}\:\mathrm{14}\Omega\:\:\mathrm{respectively}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{internal}\:\mathrm{resistance}\:\mathrm{of}\:\mathrm{the}\:\mathrm{battery} \\ $$

Question Number 58196 Answers: 1 Comments: 0

The molar heat capacity of constant presure of a gas varies with the temperature according to the equation C_p = a + bθ −(C/θ^2 ) where a,b and C are constants. How much heat is transfered during an isobaric process in which n mole of gas undergo a temperature rise from θ_(i ) to θ_f ?

$$\mathrm{The}\:\mathrm{molar}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{of}\:\mathrm{constant} \\ $$$$\mathrm{presure}\:\mathrm{of}\:\mathrm{a}\:\mathrm{gas}\:\mathrm{varies}\:\mathrm{with}\:\mathrm{the}\:\mathrm{temperature} \\ $$$$\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{C}_{\mathrm{p}} \:=\:\:\mathrm{a}\:+\:\mathrm{b}\theta\:−\frac{\mathrm{C}}{\theta^{\mathrm{2}} } \\ $$$$\mathrm{where}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{constants}. \\ $$$$\:\:\mathrm{How}\:\mathrm{much}\:\mathrm{heat}\:\mathrm{is}\:\mathrm{transfered}\:\mathrm{during} \\ $$$$\:\:\:\mathrm{an}\:\mathrm{isobaric}\:\mathrm{process}\:\mathrm{in}\:\mathrm{which}\:\mathrm{n}\:\mathrm{mole} \\ $$$$\:\:\:\mathrm{of}\:\mathrm{gas}\:\mathrm{undergo}\:\mathrm{a}\:\mathrm{temperature}\:\mathrm{rise} \\ $$$$\:\:\:\:\mathrm{from}\:\theta_{{i}\:} \mathrm{to}\:\theta_{{f}} \:? \\ $$

Question Number 58195 Answers: 2 Comments: 0

(1/x)+(1/y)=(3/4) (x^2 /y)+(y^2 /x)=9 find the value of x and y

$$\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{x}}=\mathrm{9} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

Question Number 58193 Answers: 0 Comments: 0

If z ∈ C such that R(z^n )>0 for n∈N^+ . Show that z ∈R^+ .

$$\mathrm{If}\:{z}\:\in\:\mathbb{C}\:\:\mathrm{such}\:\mathrm{that}\:\mathfrak{R}\left({z}^{{n}} \right)>\mathrm{0}\:\mathrm{for}\:{n}\in\mathbb{N}^{+} . \\ $$$$\mathrm{Show}\:\mathrm{that}\:{z}\:\in\mathbb{R}^{+} . \\ $$

Question Number 58187 Answers: 0 Comments: 0

let f(x) =∫_1 ^3 arctan(x+(x/t))dt withx>0 1) determine a explicit form of f(x) 2) give f^′ (x) at form of integral and find its value 3) calculate ∫_1 ^3 arctan(1+(1/t))dt and ∫_1 ^3 arctan(2+(2/t))dt . 4) calculate ∫_1 ^3 (2t−1)arctan(1+(1/t))dt .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left({x}+\frac{{x}}{{t}}\right){dt}\:\:\:{withx}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{give}\:{f}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{find}\:{its}\:{value} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:\:\:{and}\:\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{2}+\frac{\mathrm{2}}{{t}}\right){dt}\:. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\left(\mathrm{2}{t}−\mathrm{1}\right){arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:. \\ $$

Question Number 58185 Answers: 0 Comments: 0

find ∫ ((xdx)/(cosx +sin(2x)))

$${find}\:\int\:\:\:\frac{{xdx}}{{cosx}\:+{sin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 58184 Answers: 0 Comments: 0

find ∫ ((xdx)/(sinx +cos(2x)))

$${find}\:\:\int\:\:\:\:\:\:\frac{{xdx}}{{sinx}\:+{cos}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 58178 Answers: 0 Comments: 0

Question Number 58177 Answers: 2 Comments: 0

cos^(−1) (1/2) + 2 sin^(−1) (1/2) =

$$\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{2}\:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\:\:= \\ $$

Question Number 58176 Answers: 1 Comments: 0

If x_1 , x_2 , x_3 , x_4 are roots of the equation x^4 −x^3 sin 2β+x^2 cos 2β−x cos β−sin β=0, then tan^(−1) x_1 +tan^(−1) x_2 +tan^(−1) x_3 +tan^(−1) x_4 =

$$\mathrm{If}\:\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} ,\:{x}_{\mathrm{4}} \:\:\mathrm{are}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{4}} −{x}^{\mathrm{3}} \mathrm{sin}\:\mathrm{2}\beta+{x}^{\mathrm{2}} \mathrm{cos}\:\mathrm{2}\beta−{x}\:\mathrm{cos}\:\beta−\mathrm{sin}\:\beta=\mathrm{0}, \\ $$$$\mathrm{then} \\ $$$$\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{1}} +\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{2}} +\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{3}} +\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{4}} = \\ $$

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