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AllQuestion and Answers: Page 1921

Question Number 16409    Answers: 3   Comments: 9

Question Number 16406    Answers: 0   Comments: 0

D={(x,y)∣∣x∣+∣y∣≤1,∣x∣+∣y∣≥0.5} ∫∫_(D) ln (x^2 +y^2 )dxdy a)≥0; b)≤0; c)=0; d)non-existent.

$$\mathrm{D}=\left\{\left({x},\mathrm{y}\right)\mid\mid{x}\mid+\mid\mathrm{y}\mid\leqslant\mathrm{1},\mid{x}\mid+\mid{y}\mid\geqslant\mathrm{0}.\mathrm{5}\right\} \\ $$$$\underset{\mathrm{D}} {\int\int}\mathrm{ln}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$$$\left.\mathrm{a}\right)\geqslant\mathrm{0}; \\ $$$$\left.\mathrm{b}\right)\leqslant\mathrm{0}; \\ $$$$\left.\mathrm{c}\right)=\mathrm{0}; \\ $$$$\left.\mathrm{d}\right)\mathrm{non}-\mathrm{existent}. \\ $$

Question Number 16401    Answers: 1   Comments: 0

v=2i+2j+5k r=i+9j−8k Find 𝛚 I can do ((r×v)/r^2 )=𝛚 and i get 𝛚= ((61i−21j−16k)/(146)) but i dont get w×r=v. why?

$$\boldsymbol{{v}}=\mathrm{2}\boldsymbol{{i}}+\mathrm{2}\boldsymbol{{j}}+\mathrm{5}\boldsymbol{{k}} \\ $$$$\boldsymbol{{r}}=\boldsymbol{{i}}+\mathrm{9}\boldsymbol{{j}}−\mathrm{8}\boldsymbol{{k}} \\ $$$$\mathrm{Find}\:\boldsymbol{\omega} \\ $$$$\mathrm{I}\:\mathrm{can}\:\mathrm{do}\:\frac{\boldsymbol{{r}}×\boldsymbol{{v}}}{{r}^{\mathrm{2}} }=\boldsymbol{\omega} \\ $$$$\mathrm{and}\:\mathrm{i}\:\mathrm{get}\:\boldsymbol{\omega}=\:\frac{\mathrm{61}\boldsymbol{{i}}−\mathrm{21}\boldsymbol{{j}}−\mathrm{16}\boldsymbol{{k}}}{\mathrm{146}} \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{dont}\:\mathrm{get}\:\boldsymbol{{w}}×\boldsymbol{{r}}=\boldsymbol{{v}}. \\ $$$${why}? \\ $$

Question Number 16392    Answers: 3   Comments: 0

Question Number 16387    Answers: 0   Comments: 0

Why sinx is a power series?

$${Why}\:{sinx}\:{is}\:{a}\:{power}\:{series}? \\ $$

Question Number 16383    Answers: 2   Comments: 0

If P : Q : R = 2 : 3 : 4 and P^2 +Q^2 +R^2 =11600, then find (P+Q−R).

$$\mathrm{If}\:\mathrm{P}\::\:\mathrm{Q}\::\:\mathrm{R}\:=\:\mathrm{2}\::\:\mathrm{3}\::\:\mathrm{4}\:\mathrm{and}\:\mathrm{P}^{\mathrm{2}} +\mathrm{Q}^{\mathrm{2}} +\mathrm{R}^{\mathrm{2}} =\mathrm{11600}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\left(\mathrm{P}+\mathrm{Q}−\mathrm{R}\right). \\ $$

Question Number 16373    Answers: 1   Comments: 1

if Σ_(k=0) ^(200) i^k +Π_(p=1) ^(50) i^p =x+iy then..(x,y)is... a. (0,1) b. (1,−1) c. (2,3) d. (4,8)

$${if}\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{200}} {\sum}}{i}^{{k}} +\underset{{p}=\mathrm{1}} {\overset{\mathrm{50}} {\prod}}{i}^{{p}} ={x}+{iy}\:{then}..\left({x},{y}\right){is}... \\ $$$${a}.\:\left(\mathrm{0},\mathrm{1}\right) \\ $$$${b}.\:\left(\mathrm{1},−\mathrm{1}\right) \\ $$$${c}.\:\left(\mathrm{2},\mathrm{3}\right) \\ $$$${d}.\:\left(\mathrm{4},\mathrm{8}\right) \\ $$

Question Number 16364    Answers: 1   Comments: 0

Question Number 16363    Answers: 1   Comments: 3

A car drives due north at 50 km/hr. Wind blows due North-West at 50(√2) km/hr. In what direction, a flag hoisted on the roof of the car points?

$$\mathrm{A}\:\mathrm{car}\:\mathrm{drives}\:\mathrm{due}\:\mathrm{north}\:\mathrm{at}\:\mathrm{50}\:\mathrm{km}/\mathrm{hr}. \\ $$$$\mathrm{Wind}\:\mathrm{blows}\:\mathrm{due}\:\mathrm{North}-\mathrm{West}\:\mathrm{at}\:\mathrm{50}\sqrt{\mathrm{2}} \\ $$$$\mathrm{km}/\mathrm{hr}.\:\mathrm{In}\:\mathrm{what}\:\mathrm{direction},\:\mathrm{a}\:\mathrm{flag} \\ $$$$\mathrm{hoisted}\:\mathrm{on}\:\mathrm{the}\:\mathrm{roof}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{points}? \\ $$

Question Number 16360    Answers: 1   Comments: 0

Question Number 16359    Answers: 1   Comments: 0

In a ΔABC if ((s − a)/(a − b)) = ((s − c)/(b − c)) , then prove that r_1 , r_2 , r_3 are in A.P. Here r_1 , r_2 and r_3 are the exradii opposite to angles A, B and C respectively.

$$\mathrm{In}\:\mathrm{a}\:\Delta{ABC}\:\mathrm{if}\:\frac{{s}\:−\:{a}}{{a}\:−\:{b}}\:=\:\frac{{s}\:−\:{c}}{{b}\:−\:{c}}\:,\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} ,\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{in}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Here}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} \:\mathrm{and}\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{exradii} \\ $$$$\mathrm{opposite}\:\mathrm{to}\:\mathrm{angles}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{respectively}. \\ $$

Question Number 16358    Answers: 1   Comments: 1

In any triangle ABC, a cot A + b cot B + c cot C is equal to (1) r + R (2) r − R (3) 2(r + R) (4) 2(r − R)

$$\mathrm{In}\:\mathrm{any}\:\mathrm{triangle}\:{ABC},\:{a}\:\mathrm{cot}\:{A}\:+\:{b}\:\mathrm{cot}\:{B} \\ $$$$+\:{c}\:\mathrm{cot}\:{C}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:{r}\:+\:{R} \\ $$$$\left(\mathrm{2}\right)\:{r}\:−\:{R} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}\left({r}\:+\:{R}\right) \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\left({r}\:−\:{R}\right) \\ $$

Question Number 16450    Answers: 0   Comments: 3

Ten balls were manufactured, nine of them have the same mass, while just one of them has a slightly higher or slightly lower mass. Given is just a beam balance and no weights. comparing the masses of balls only, with the help of the balance , and in just 3 weighings explain how to judge which is the defective one and whether it is heavier or lighter than the rest (as the case may be).

$${Ten}\:{balls}\:{were}\:{manufactured}, \\ $$$$\:{nine}\:{of}\:{them}\:{have}\:{the}\:{same} \\ $$$${mass},\:{while}\:{just}\:{one}\:{of}\:{them} \\ $$$${has}\:{a}\:{slightly}\:{higher}\:{or}\:{slightly} \\ $$$${lower}\:{mass}.\:{Given}\:{is}\:{just}\:{a}\:{beam} \\ $$$${balance}\:{and}\:{no}\:{weights}.\:{comparing} \\ $$$${the}\:{masses}\:{of}\:{balls}\:{only},\:{with}\:{the} \\ $$$${help}\:{of}\:{the}\:{balance}\:,\:{and}\:{in}\:{just} \\ $$$$\mathrm{3}\:{weighings}\:{explain}\:{how}\:{to}\:{judge} \\ $$$${which}\:{is}\:{the}\:{defective}\:{one}\:{and} \\ $$$${whether}\:{it}\:{is}\:{heavier}\:{or}\:{lighter} \\ $$$${than}\:{the}\:{rest}\:\left({as}\:{the}\:{case}\:{may}\:{be}\right). \\ $$

Question Number 16441    Answers: 1   Comments: 2

Question Number 16338    Answers: 1   Comments: 2

Question Number 16330    Answers: 0   Comments: 0

Question Number 16310    Answers: 1   Comments: 3

Question Number 16302    Answers: 1   Comments: 5

Related to Q16140 What if the three lines d_1 ,d_2 ,d_3 are not parallel, but concurrent?

$$\mathrm{Related}\:\mathrm{to}\:\mathrm{Q16140} \\ $$$$\mathrm{What}\:\mathrm{if}\:\mathrm{the}\:\mathrm{three}\:\mathrm{lines}\:\mathrm{d}_{\mathrm{1}} ,\mathrm{d}_{\mathrm{2}} ,\mathrm{d}_{\mathrm{3}} \:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{parallel},\:\mathrm{but}\:\mathrm{concurrent}? \\ $$

Question Number 16294    Answers: 1   Comments: 1

Two particles, 1 and 2, move with constant velocities v_1 ^(→) and v_2 ^(→) . At the initial moment, their position vectors are equal to r_1 ^(→) and r_2 ^(→) . How must these four vectors be interrelated for the particle to collide?

$$\mathrm{Two}\:\mathrm{particles},\:\mathrm{1}\:\mathrm{and}\:\mathrm{2},\:\mathrm{move}\:\mathrm{with} \\ $$$$\mathrm{constant}\:\mathrm{velocities}\:\overset{\rightarrow} {{v}_{\mathrm{1}} }\:\mathrm{and}\:\overset{\rightarrow} {{v}_{\mathrm{2}} }.\:\mathrm{At}\:\mathrm{the} \\ $$$$\mathrm{initial}\:\mathrm{moment},\:\mathrm{their}\:\mathrm{position}\:\mathrm{vectors} \\ $$$$\mathrm{are}\:\mathrm{equal}\:\mathrm{to}\:\overset{\rightarrow} {{r}_{\mathrm{1}} }\:\mathrm{and}\:\overset{\rightarrow} {{r}_{\mathrm{2}} }.\:\mathrm{How}\:\mathrm{must}\:\mathrm{these} \\ $$$$\mathrm{four}\:\mathrm{vectors}\:\mathrm{be}\:\mathrm{interrelated}\:\mathrm{for}\:\mathrm{the} \\ $$$$\mathrm{particle}\:\mathrm{to}\:\mathrm{collide}? \\ $$

Question Number 16281    Answers: 0   Comments: 0

A plane moves in windy weather due east while the pilot points the plane somewhat south of east. The wind is blowing at 50 km/hr directed 30° east of north, while the plane moves at 200 km/hr relative to the wind. What is the velocity of the plane relative to the ground and what is the direction in which the pilot points the plane?

$$\mathrm{A}\:\mathrm{plane}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{windy}\:\mathrm{weather}\:\mathrm{due} \\ $$$$\mathrm{east}\:\mathrm{while}\:\mathrm{the}\:\mathrm{pilot}\:\mathrm{points}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{somewhat}\:\mathrm{south}\:\mathrm{of}\:\mathrm{east}.\:\mathrm{The}\:\mathrm{wind}\:\mathrm{is} \\ $$$$\mathrm{blowing}\:\mathrm{at}\:\mathrm{50}\:\mathrm{km}/\mathrm{hr}\:\mathrm{directed}\:\mathrm{30}°\:\mathrm{east} \\ $$$$\mathrm{of}\:\mathrm{north},\:\mathrm{while}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{moves}\:\mathrm{at}\:\mathrm{200} \\ $$$$\mathrm{km}/\mathrm{hr}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{the}\:\mathrm{wind}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{ground}\:\mathrm{and}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{direction}\:\mathrm{in} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{pilot}\:\mathrm{points}\:\mathrm{the}\:\mathrm{plane}? \\ $$

Question Number 16277    Answers: 3   Comments: 1

Question Number 16273    Answers: 2   Comments: 4

If in ΔABC r_1 = r_2 + r_3 + r, prove that triangle is right angled.

$$\mathrm{If}\:\mathrm{in}\:\Delta{ABC}\:{r}_{\mathrm{1}} \:=\:{r}_{\mathrm{2}} \:+\:{r}_{\mathrm{3}} \:+\:{r},\:\mathrm{prove} \\ $$$$\mathrm{that}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{right}\:\mathrm{angled}. \\ $$

Question Number 16271    Answers: 1   Comments: 0

The unemployment rate among workers under 25 in a state went from 8.2% to 7.5% in one year. Assume an average of 1340200 workers and estimate the decrease in the number unemployed.

$${The}\:{unemployment}\:{rate}\:{among}\:{workers} \\ $$$${under}\:\mathrm{25}\:{in}\:{a}\:{state}\:{went}\:{from}\:\mathrm{8}.\mathrm{2\%} \\ $$$${to}\:\mathrm{7}.\mathrm{5\%}\:{in}\:{one}\:{year}.\:{Assume}\:{an}\:{average} \\ $$$${of}\:\mathrm{1340200}\:{workers}\:{and}\:{estimate} \\ $$$${the}\:{decrease}\:{in}\:{the}\:{number}\:{unemployed}. \\ $$

Question Number 16269    Answers: 1   Comments: 0

2^(nd) part of Q. 16214: Prove that r_1 = s tan ((A/2)), r_2 = s tan ((B/2)), r_3 = s tan ((C/2)).

$$\mathrm{2}^{\mathrm{nd}} \:\mathrm{part}\:\mathrm{of}\:\mathrm{Q}.\:\mathrm{16214}:\:\mathrm{Prove}\:\mathrm{that} \\ $$$${r}_{\mathrm{1}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{A}}{\mathrm{2}}\right),\:{r}_{\mathrm{2}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{B}}{\mathrm{2}}\right), \\ $$$${r}_{\mathrm{3}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{C}}{\mathrm{2}}\right). \\ $$

Question Number 16240    Answers: 2   Comments: 1

Question Number 16238    Answers: 0   Comments: 0

we have a^5 +b^5 =1 and u^5 +v^5 =1 find value a^3 u^5 +b^3 v^5 =?

$${we}\:{have}\:{a}^{\mathrm{5}} +{b}^{\mathrm{5}} =\mathrm{1}\:{and}\:{u}^{\mathrm{5}} +{v}^{\mathrm{5}} =\mathrm{1} \\ $$$${find}\:{value}\:\:{a}^{\mathrm{3}} {u}^{\mathrm{5}} +{b}^{\mathrm{3}} {v}^{\mathrm{5}} =? \\ $$

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