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Question Number 76830 Answers: 1 Comments: 2

Question Number 76826 Answers: 1 Comments: 0

Question Number 76819 Answers: 0 Comments: 4

one of the foci of the ellipse (x^2 /9) + (y^2 /4) = 1 is A. (4,0) B. (9,0) C. (5,0) D. ((√5) , 0)

$$\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{foci}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ellipse} \\ $$$$\:\:\:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{9}}\:+\:\frac{{y}^{\mathrm{2}} }{\mathrm{4}}\:=\:\mathrm{1}\:\mathrm{is} \\ $$$$\mathrm{A}.\:\left(\mathrm{4},\mathrm{0}\right) \\ $$$$\mathrm{B}.\:\left(\mathrm{9},\mathrm{0}\right) \\ $$$$\mathrm{C}.\:\left(\mathrm{5},\mathrm{0}\right) \\ $$$$\mathrm{D}.\:\left(\sqrt{\mathrm{5}}\:,\:\mathrm{0}\right) \\ $$

Question Number 76817 Answers: 0 Comments: 2

A compound pendulum ocsillates through angles θ about its equilibrium position such that 8aθ^2 = 9g cosθ, a>0. its period is A. 2π(√((8a)/(9g))) B. ((3π)/8)(√(a/g)) C. 2π(√((9g)/(8a))) D. ((8π)/3)(√(a/g))

$$\mathrm{A}\:\mathrm{compound}\:\mathrm{pendulum}\:\mathrm{ocsillates} \\ $$$$\mathrm{through}\:\mathrm{angles}\:\theta\:\mathrm{about}\:\mathrm{its}\:\mathrm{equilibrium} \\ $$$$\mathrm{position}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{8}{a}\theta^{\mathrm{2}} \:=\:\mathrm{9}{g}\:{cos}\theta,\:{a}>\mathrm{0}.\:\mathrm{its}\:\mathrm{period}\:\mathrm{is}\: \\ $$$$\mathrm{A}.\:\mathrm{2}\pi\sqrt{\frac{\mathrm{8}{a}}{\mathrm{9}{g}}} \\ $$$$\mathrm{B}.\:\frac{\mathrm{3}\pi}{\mathrm{8}}\sqrt{\frac{{a}}{{g}}} \\ $$$$\mathrm{C}.\:\mathrm{2}\pi\sqrt{\frac{\mathrm{9}{g}}{\mathrm{8}{a}}} \\ $$$$\mathrm{D}.\:\frac{\mathrm{8}\pi}{\mathrm{3}}\sqrt{\frac{{a}}{{g}}} \\ $$

Question Number 76815 Answers: 1 Comments: 2

Which one of the following series is Not convergent? A. Σ_(r=1) ^∞ (1/r^3 ) B. Σ_(r=1) ^∞ (1/r^2 ) C. Σ_(r=1) ^∞ (1/(r^2 −4)) D. Σ_(r=1) ^∞ (1/(5r))

$$\mathrm{Which}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{series}\:\mathrm{is}\:\boldsymbol{\mathrm{N}}\mathrm{o}\boldsymbol{\mathrm{t}}\:\mathrm{convergent}? \\ $$$$\mathrm{A}.\:\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{3}} } \\ $$$$\mathrm{B}.\:\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{2}} } \\ $$$$\mathrm{C}.\:\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{2}} −\mathrm{4}} \\ $$$$\mathrm{D}.\:\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{5r}} \\ $$

Question Number 76813 Answers: 1 Comments: 7

Σ_(k=1) ^(2n) (−1)^k = A. ∞ B. 1 C. −1 D. 0

$$\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{2n}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}} \:=\: \\ $$$$\mathrm{A}.\:\infty \\ $$$$\mathrm{B}.\:\mathrm{1} \\ $$$$\mathrm{C}.\:−\mathrm{1} \\ $$$$\mathrm{D}.\:\mathrm{0} \\ $$

Question Number 76812 Answers: 1 Comments: 0

Given that A and B are 3 × 3 invertible matrices, then (A^(−1) B)^(−1) = A. AB^(−1) B.B^(−1) A C. B^(−1) A^(−1) D. BA^(−1)

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{3}\:×\:\mathrm{3}\:\mathrm{invertible}\:\mathrm{matrices}, \\ $$$$\:\mathrm{then}\:\left(\mathrm{A}^{−\mathrm{1}} \mathrm{B}\right)^{−\mathrm{1}} \:= \\ $$$$\mathrm{A}.\:\mathrm{AB}^{−\mathrm{1}} \\ $$$$\mathrm{B}.\mathrm{B}^{−\mathrm{1}} \mathrm{A} \\ $$$$\mathrm{C}.\:\mathrm{B}^{−\mathrm{1}} \mathrm{A}^{−\mathrm{1}} \\ $$$$\mathrm{D}.\:\mathrm{BA}^{−\mathrm{1}} \\ $$

Question Number 76811 Answers: 0 Comments: 2

The eccentricity of the hyperbola (x^2 /(64)) − (y^2 /(36)) = 1 is A. (5/4) B. (3/4) C. (4/5) D. (4/3)

$$\mathrm{The}\:\mathrm{eccentricity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hyperbola} \\ $$$$\:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{64}}\:−\:\frac{{y}^{\mathrm{2}} }{\mathrm{36}}\:=\:\mathrm{1}\:\mathrm{is}\: \\ $$$$\mathrm{A}.\:\frac{\mathrm{5}}{\mathrm{4}} \\ $$$$\mathrm{B}.\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{C}.\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\mathrm{D}.\:\frac{\mathrm{4}}{\mathrm{3}} \\ $$

Question Number 76809 Answers: 0 Comments: 3

Question Number 76808 Answers: 1 Comments: 3

∫(1/(√(x^3 +1)))dx

$$\int\frac{\mathrm{1}}{\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}}{dx} \\ $$

Question Number 76807 Answers: 0 Comments: 0

Question Number 76802 Answers: 0 Comments: 0

3)αparticle of energy 5MeV pass through an ionisation chamber at the rate of 10 pwe second . Assum that all the energy is used in producing ion pairs,calculate the current produced. (35MeV is required for producing an ion pair and e=1.6×10^(-19) C) solution: Energy of α particles=5×10^6 eV Energy required for producing one ion pair=35eV No.of ion pairs produced by one α particle =((5×10^6 )/(35))=1.426×10^5 since 10 particle enter the chamber in one second =1.426×10^5 ×10=1.426×10^6 charge on dach ion=1.6×10^(-19) C Current=(1.426×10^6 )×(1.6×10^(-19) )C/s =2.287×10^(-13) A.

$$\left.\mathrm{3}\right)\alpha\mathrm{particle}\:\mathrm{of}\:\mathrm{energy}\:\mathrm{5MeV}\:\mathrm{pass} \\ $$$$\mathrm{through}\:\mathrm{an}\:\mathrm{ionisation}\:\mathrm{chamber}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{rate}\:\mathrm{of}\:\mathrm{10}\:\mathrm{pwe}\:\mathrm{second}\:.\:\mathrm{Assum}\:\mathrm{that}\:\mathrm{all} \\ $$$$\mathrm{the}\:\mathrm{energy}\:\mathrm{is}\:\mathrm{used}\:\mathrm{in}\:\mathrm{producing}\:\mathrm{ion}\: \\ $$$$\mathrm{pairs},\mathrm{calculate}\:\mathrm{the}\:\mathrm{current}\:\mathrm{produced}. \\ $$$$\left(\mathrm{35MeV}\:\mathrm{is}\:\mathrm{required}\:\mathrm{for}\:\mathrm{producing}\:\:\mathrm{an}\:\right. \\ $$$$\left.\mathrm{ion}\:\mathrm{pair}\:\mathrm{and}\:\mathrm{e}=\mathrm{1}.\mathrm{6}×\mathrm{10}^{-\mathrm{19}} \:\mathrm{C}\right) \\ $$$$\boldsymbol{\mathrm{solution}}: \\ $$$$\:\:\:\:\mathrm{Energy}\:\mathrm{of}\:\alpha\:\mathrm{particles}=\mathrm{5}×\mathrm{10}^{\mathrm{6}} \mathrm{eV} \\ $$$$\:\:\:\:\mathrm{Energy}\:\mathrm{required}\:\mathrm{for}\:\mathrm{producing}\:\mathrm{one} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{ion}\:\mathrm{pair}=\mathrm{35eV} \\ $$$$\:\:\:\:\mathrm{No}.\mathrm{of}\:\mathrm{ion}\:\mathrm{pairs}\:\mathrm{produced}\:\mathrm{by}\:\mathrm{one}\:\alpha \\ $$$$\:\:\:\:\:\:\:\:\mathrm{particle}\:=\frac{\mathrm{5}×\mathrm{10}^{\mathrm{6}} }{\mathrm{35}}=\mathrm{1}.\mathrm{426}×\mathrm{10}^{\mathrm{5}} \\ $$$$\mathrm{since}\:\mathrm{10}\:\mathrm{particle}\:\mathrm{enter}\:\mathrm{the}\:\mathrm{chamber}\:\mathrm{in} \\ $$$$\mathrm{one}\:\mathrm{second} \\ $$$$\:\:\:\:\:\:\:\:=\mathrm{1}.\mathrm{426}×\mathrm{10}^{\mathrm{5}} ×\mathrm{10}=\mathrm{1}.\mathrm{426}×\mathrm{10}^{\mathrm{6}} \\ $$$$\mathrm{charge}\:\mathrm{on}\:\mathrm{dach}\:\mathrm{ion}=\mathrm{1}.\mathrm{6}×\mathrm{10}^{-\mathrm{19}} \mathrm{C} \\ $$$$\mathrm{Current}=\left(\mathrm{1}.\mathrm{426}×\mathrm{10}^{\mathrm{6}} \right)×\left(\mathrm{1}.\mathrm{6}×\mathrm{10}^{-\mathrm{19}} \right)\mathrm{C}/\mathrm{s} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}.\mathrm{287}×\mathrm{10}^{-\mathrm{13}} \mathrm{A}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 76794 Answers: 1 Comments: 1

Find the sum of x + (x/(1+x)) + (x/((1+x)^2 )) + (x/((1+x)^3 ))+.... for ∣(1/(1+x))∣<1

$${Find}\:{the}\:{sum}\:{of} \\ $$$${x}\:+\:\frac{{x}}{\mathrm{1}+{x}}\:+\:\frac{{x}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }\:+\:\frac{{x}}{\left(\mathrm{1}+{x}\right)^{\mathrm{3}} }+....\:{for} \\ $$$$\mid\frac{\mathrm{1}}{\mathrm{1}+{x}}\mid<\mathrm{1} \\ $$

Question Number 76793 Answers: 2 Comments: 2

The sum of the first n terms of a series is given by: S_n =n^2 +7n+2. (i)Find a formula for the nth term (ii)write down the first 5 terms of the sequence

$${The}\:{sum}\:{of}\:{the}\:{first}\:{n}\:{terms}\:{of}\:{a}\:{series} \\ $$$${is}\:{given}\:{by}:\:{S}_{{n}} ={n}^{\mathrm{2}} +\mathrm{7}{n}+\mathrm{2}. \\ $$$$\left({i}\right){Find}\:{a}\:{formula}\:{for}\:{the}\:{nth}\:{term} \\ $$$$\left({ii}\right){write}\:{down}\:{the}\:{first}\:\mathrm{5}\:{terms}\:{of}\:{the} \\ $$$${sequence} \\ $$$$ \\ $$

Question Number 76791 Answers: 0 Comments: 0

calculate ((sin44^o +sin66^o +sin70^o )/(cos22^o ×cos33^o ×cos35^o )).

$${calculate}\:\frac{{sin}\mathrm{44}^{{o}} +{sin}\mathrm{66}^{{o}} +{sin}\mathrm{70}^{{o}} }{{cos}\mathrm{22}^{{o}} ×{cos}\mathrm{33}^{{o}} ×{cos}\mathrm{35}^{{o}} }. \\ $$

Question Number 76790 Answers: 3 Comments: 1

calculate ((sin72^o +sin148^o +sin140^o )/(sin36^o ×sin74^o ×sin70^o )).

$${calculate}\:\frac{{sin}\mathrm{72}^{{o}} +{sin}\mathrm{148}^{{o}} +{sin}\mathrm{140}^{{o}} }{{sin}\mathrm{36}^{{o}} ×{sin}\mathrm{74}^{{o}} ×{sin}\mathrm{70}^{{o}} }. \\ $$

Question Number 76783 Answers: 0 Comments: 0

let f(x)=x^3 ,2π periodic odd developp f at fourier serie

$${let}\:{f}\left({x}\right)={x}^{\mathrm{3}} \:\:\:\:,\mathrm{2}\pi\:{periodic}\:{odd}\:{developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$

Question Number 76782 Answers: 1 Comments: 0

calculate ∫_0 ^∞ e^(−x) ((sin(x^2 ))/x^2 )dx