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AllQuestion and Answers: Page 1851

Question Number 18818    Answers: 2   Comments: 0

Question Number 18808    Answers: 1   Comments: 0

((5+2(√5)))^(1/3) + ((5−2(√5)))^(1/3) =? help me

$$\sqrt[{\mathrm{3}}]{\mathrm{5}+\mathrm{2}\sqrt{\mathrm{5}}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{5}−\mathrm{2}\sqrt{\mathrm{5}}}\:=? \\ $$$$\boldsymbol{{help}}\:\boldsymbol{{me}} \\ $$

Question Number 18801    Answers: 0   Comments: 1

Question Number 18800    Answers: 0   Comments: 0

Solve: x sin(y)dx + x^2 + y cos(y)dy = 0, subject to y(1)

$$\mathrm{Solve}:\:\:\mathrm{x}\:\mathrm{sin}\left(\mathrm{y}\right)\mathrm{dx}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}\:\mathrm{cos}\left(\mathrm{y}\right)\mathrm{dy}\:=\:\mathrm{0},\:\:\mathrm{subject}\:\mathrm{to}\:\mathrm{y}\left(\mathrm{1}\right) \\ $$

Question Number 18799    Answers: 1   Comments: 0

Question Number 18798    Answers: 1   Comments: 0

Question Number 18797    Answers: 1   Comments: 0

If A, B, C are the angles of a triangle, then 2sin(A/2)cosec(B/2)sin(C/2) − sinAcot(B/2) − cos A is (1) Independent of A, B, C (2) Function of A, B, C (3) Function of A, B (4) Function of B, C

$$\mathrm{If}\:{A},\:{B},\:{C}\:\mathrm{are}\:\mathrm{the}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}, \\ $$$$\mathrm{then}\:\mathrm{2sin}\frac{{A}}{\mathrm{2}}\mathrm{cosec}\frac{{B}}{\mathrm{2}}\mathrm{sin}\frac{{C}}{\mathrm{2}}\:−\:\mathrm{sin}{A}\mathrm{cot}\frac{{B}}{\mathrm{2}} \\ $$$$−\:\mathrm{cos}\:{A}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Independent}\:\mathrm{of}\:{A},\:{B},\:{C} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Function}\:\mathrm{of}\:{A},\:{B},\:{C} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Function}\:\mathrm{of}\:{A},\:{B} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Function}\:\mathrm{of}\:{B},\:{C} \\ $$

Question Number 18834    Answers: 1   Comments: 0

Question Number 18783    Answers: 1   Comments: 0

∫sin x

$$ \\ $$$$\int\mathrm{sin}\:{x} \\ $$

Question Number 18779    Answers: 0   Comments: 2

If (1/(2x))+(1/2)((1/(2x))+(1/2)((1/(2x))+(1/2)((1/(2x))+......=y what does x equals? a)1/2 b)2/4 c)1 d)1/4

$$\mathrm{If}\:\:\frac{\mathrm{1}}{\mathrm{2x}}+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2x}}+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2x}}+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2x}}+......=\mathrm{y}\right.\right.\right. \\ $$$$\mathrm{what}\:\mathrm{does}\:\mathrm{x}\:\mathrm{equals}? \\ $$$$ \\ $$$$\left.\mathrm{a}\right)\mathrm{1}/\mathrm{2} \\ $$$$\left.\mathrm{b}\right)\mathrm{2}/\mathrm{4} \\ $$$$\left.\mathrm{c}\right)\mathrm{1} \\ $$$$\left.\mathrm{d}\right)\mathrm{1}/\mathrm{4} \\ $$

Question Number 18763    Answers: 0   Comments: 3

Question Number 18762    Answers: 1   Comments: 0

If (1+x)^n =C_0 +C_1 x+C_2 x^2 +...+C_n x^n , then C_1 ^2 +C_2 ^2 +....+C_n ^2 is equal to

$$\mathrm{If}\:\left(\mathrm{1}+{x}\right)^{{n}} ={C}_{\mathrm{0}} +{C}_{\mathrm{1}} {x}+{C}_{\mathrm{2}} {x}^{\mathrm{2}} +...+{C}_{{n}} {x}^{{n}} , \\ $$$$\mathrm{then}\:{C}_{\mathrm{1}} \:^{\mathrm{2}} +{C}_{\mathrm{2}} \:^{\mathrm{2}} +....+{C}_{{n}} \:^{\mathrm{2}} \:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 18761    Answers: 0   Comments: 1

If x ∈ R, then the expression 9^x −3^x +1 assumes

$$\mathrm{If}\:\:{x}\:\in\:{R},\:\:\:\mathrm{then}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{9}^{{x}} −\mathrm{3}^{{x}} +\mathrm{1} \\ $$$$\mathrm{assumes}\: \\ $$

Question Number 18760    Answers: 0   Comments: 1

If the coefficient of x^7 and x^8 in (2+(x/3))^n are equal then r is equal to

$$\mathrm{If}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{7}} \mathrm{and}\:{x}^{\mathrm{8}} \:\mathrm{in}\:\left(\mathrm{2}+\frac{{x}}{\mathrm{3}}\right)^{{n}} \\ $$$$\mathrm{are}\:\mathrm{equal}\:\mathrm{then}\:{r}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 18759    Answers: 2   Comments: 0

The coefficient of the middle term in the expansion of (1+x)^(2n) is

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{middle}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}} \:\mathrm{is} \\ $$

Question Number 18758    Answers: 1   Comments: 0

If the coefficient of r^(th) and (r+1)^(th) terms in the expansion of (3+7x)^(29) are equal, then r equals

$$\mathrm{If}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\:{r}^{\mathrm{th}} \:\mathrm{and}\:\left({r}+\mathrm{1}\right)^{\mathrm{th}} \:\mathrm{terms} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{3}+\mathrm{7}{x}\right)^{\mathrm{29}} \:\mathrm{are}\:\mathrm{equal}, \\ $$$$\mathrm{then}\:\:{r}\:\mathrm{equals} \\ $$

Question Number 18757    Answers: 0   Comments: 0

If a_1 , a_2 , a_3 , a_4 are the coefficient of any four four consecutive terms in the expansion of (1+x)^n , then (a_1 /(a_1 +a_2 )) + (a_3 /(a_3 +a_4 )) is equal to

$$\mathrm{If}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:{a}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{any} \\ $$$$\mathrm{four}\:\mathrm{four}\:\mathrm{consecutive}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{{n}} ,\:\mathrm{then}\:\frac{{a}_{\mathrm{1}} }{{a}_{\mathrm{1}} +{a}_{\mathrm{2}} }\:+\:\frac{{a}_{\mathrm{3}} }{{a}_{\mathrm{3}} +{a}_{\mathrm{4}} }\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 18755    Answers: 1   Comments: 0

In a △ABC, if c = 2, A = 120°, a = (√6) , then C =

$$\mathrm{In}\:\mathrm{a}\:\bigtriangleup{ABC},\:\mathrm{if}\:{c}\:=\:\mathrm{2},\:{A}\:=\:\mathrm{120}°,\:{a}\:=\:\sqrt{\mathrm{6}}\:, \\ $$$$\mathrm{then}\:\:{C}\:= \\ $$

Question Number 18803    Answers: 1   Comments: 1

Question Number 18752    Answers: 0   Comments: 0

$$ \\ $$

Question Number 18749    Answers: 1   Comments: 1

Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates A^→ = A_x i^∧ + A_y j^∧ where i^∧ and j^∧ are unit vector along x and y directions, respectively and A_x and A_y are corresponding components of A^→ (Figure). Motion can also be studied by expressing vectors in circular polar co-ordinates as A^→ = A_r r^∧ + A_θ θ^∧ where r^∧ = (r^→ /r) = cos θ i^∧ + sin θ j^∧ and θ^∧ = −sin θ i^∧ + cos θ j^∧ are unit vectors along direction in which ′r′ and ′θ′ are increasing. (a) Express i^∧ and j^∧ in terms of r^∧ and θ^∧ (b) Show that both r^∧ and θ^∧ are unit vectors and are perpendicular to each other. (c) Show that (d/dt)(r^∧ ) = ωθ^∧ where ω = (dθ/dt) and (d/dt)(θ^∧ ) = −ωr^∧ (d) For a particle moving along a spiral given by r^→ = αθr^∧ , where α = 1 (unit), find dimensions of ′α′. (e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.

$$\mathrm{Motion}\:\mathrm{in}\:\mathrm{two}\:\mathrm{dimensions},\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{studied}\:\mathrm{by}\:\mathrm{expressing}\:\mathrm{position}, \\ $$$$\mathrm{velocity}\:\mathrm{and}\:\mathrm{acceleration}\:\mathrm{as}\:\mathrm{vectors}\:\mathrm{in} \\ $$$$\mathrm{Cartesian}\:\mathrm{co}-\mathrm{ordinates}\:\overset{\rightarrow} {{A}}\:=\:{A}_{{x}} \overset{\wedge} {{i}}\:+\:{A}_{{y}} \overset{\wedge} {{j}} \\ $$$$\mathrm{where}\:\overset{\wedge} {{i}}\:\mathrm{and}\:\overset{\wedge} {{j}}\:\mathrm{are}\:\mathrm{unit}\:\mathrm{vector}\:\mathrm{along}\:{x} \\ $$$$\mathrm{and}\:{y}\:\mathrm{directions},\:\mathrm{respectively}\:\mathrm{and}\:{A}_{{x}} \\ $$$$\mathrm{and}\:{A}_{{y}} \:\mathrm{are}\:\mathrm{corresponding}\:\mathrm{components} \\ $$$$\mathrm{of}\:\overset{\rightarrow} {{A}}\:\left(\mathrm{Figure}\right).\:\mathrm{Motion}\:\mathrm{can}\:\mathrm{also}\:\mathrm{be} \\ $$$$\mathrm{studied}\:\mathrm{by}\:\mathrm{expressing}\:\mathrm{vectors}\:\mathrm{in}\:\mathrm{circular} \\ $$$$\mathrm{polar}\:\mathrm{co}-\mathrm{ordinates}\:\mathrm{as}\:\overset{\rightarrow} {{A}}\:=\:{A}_{{r}} \overset{\wedge} {{r}}\:+\:{A}_{\theta} \overset{\wedge} {\theta} \\ $$$$\mathrm{where}\:\overset{\wedge} {{r}}\:=\:\frac{\overset{\rightarrow} {{r}}}{{r}}\:=\:\mathrm{cos}\:\theta\:\overset{\wedge} {{i}}\:+\:\mathrm{sin}\:\theta\:\overset{\wedge} {{j}}\:\mathrm{and}\:\overset{\wedge} {\theta}\:= \\ $$$$−\mathrm{sin}\:\theta\:\overset{\wedge} {{i}}\:+\:\mathrm{cos}\:\theta\:\overset{\wedge} {{j}}\:\mathrm{are}\:\mathrm{unit}\:\mathrm{vectors}\:\mathrm{along} \\ $$$$\mathrm{direction}\:\mathrm{in}\:\mathrm{which}\:'{r}'\:\mathrm{and}\:'\theta'\:\mathrm{are} \\ $$$$\mathrm{increasing}. \\ $$$$\left({a}\right)\:\mathrm{Express}\:\overset{\wedge} {{i}}\:\mathrm{and}\:\overset{\wedge} {{j}}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\overset{\wedge} {{r}}\:\mathrm{and}\:\overset{\wedge} {\theta} \\ $$$$\left({b}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{both}\:\overset{\wedge} {{r}}\:\mathrm{and}\:\overset{\wedge} {\theta}\:\mathrm{are}\:\mathrm{unit} \\ $$$$\mathrm{vectors}\:\mathrm{and}\:\mathrm{are}\:\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{each} \\ $$$$\mathrm{other}. \\ $$$$\left({c}\right)\:\mathrm{Show}\:\mathrm{that}\:\frac{{d}}{{dt}}\left(\overset{\wedge} {{r}}\right)\:=\:\omega\overset{\wedge} {\theta}\:\mathrm{where} \\ $$$$\omega\:=\:\frac{{d}\theta}{{dt}}\:\mathrm{and}\:\frac{{d}}{{dt}}\left(\overset{\wedge} {\theta}\right)\:=\:−\omega\overset{\wedge} {{r}} \\ $$$$\left({d}\right)\:\mathrm{For}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{moving}\:\mathrm{along}\:\mathrm{a}\:\mathrm{spiral} \\ $$$$\mathrm{given}\:\mathrm{by}\:\overset{\rightarrow} {{r}}\:=\:\alpha\theta\overset{\wedge} {{r}},\:\mathrm{where}\:\alpha\:=\:\mathrm{1}\:\left(\mathrm{unit}\right), \\ $$$$\mathrm{find}\:\mathrm{dimensions}\:\mathrm{of}\:'\alpha'. \\ $$$$\left({e}\right)\:\mathrm{Find}\:\mathrm{velocity}\:\mathrm{and}\:\mathrm{acceleration}\:\mathrm{in} \\ $$$$\mathrm{polar}\:\mathrm{vector}\:\mathrm{representation}\:\mathrm{for}\:\mathrm{particle} \\ $$$$\mathrm{moving}\:\mathrm{along}\:\mathrm{spiral}\:\mathrm{described}\:\mathrm{in}\:\left({d}\right) \\ $$$$\mathrm{above}. \\ $$

Question Number 18748    Answers: 1   Comments: 0

Question Number 18747    Answers: 2   Comments: 0

Question Number 18746    Answers: 2   Comments: 0

Question Number 18744    Answers: 0   Comments: 1

Question Number 18742    Answers: 1   Comments: 0

The value of cot16°cot44° + cot44°cot76° − cot76°cot16° is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{cot16}°\mathrm{cot44}°\:+\:\mathrm{cot44}°\mathrm{cot76}° \\ $$$$−\:\mathrm{cot76}°\mathrm{cot16}°\:\mathrm{is} \\ $$

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