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Question Number 21111 Answers: 0 Comments: 0

STATEMENT-1 : z_1 ^2 + z_2 ^2 + z_3 ^2 + z_4 ^2 = 0 where z_1 , z_2 , z_3 and z_4 are the fourth roots of unity. and STATEMENT-2 : (1)^(1/4) = (cos0° + i sin0°)^(1/4) .

$$\mathrm{STATEMENT}-\mathrm{1}\::\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{4}} ^{\mathrm{2}} \:= \\ $$$$\mathrm{0}\:\mathrm{where}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{and}\:{z}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{fourth} \\ $$$$\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}. \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\left(\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:=\:\left(\mathrm{cos0}°\:+\right. \\ $$$$\left.{i}\:\mathrm{sin0}°\right)^{\frac{\mathrm{1}}{\mathrm{4}}} . \\ $$

Question Number 21109 Answers: 0 Comments: 2

STATEMENT-1 : The locus of z, if arg(((z − 1)/(z + 1))) = (π/2) is a circle. and STATEMENT-2 : ∣((z − 2)/(z + 2))∣ = (π/2), then the locus of z is a circle.

$$\mathrm{STATEMENT}-\mathrm{1}\::\:\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:{z},\:\mathrm{if} \\ $$$$\mathrm{arg}\left(\frac{{z}\:−\:\mathrm{1}}{{z}\:+\:\mathrm{1}}\right)\:=\:\frac{\pi}{\mathrm{2}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\mid\frac{{z}\:−\:\mathrm{2}}{{z}\:+\:\mathrm{2}}\mid\:=\:\frac{\pi}{\mathrm{2}},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$

Question Number 21108 Answers: 1 Comments: 0

Let A, B, C be three sets of complex numbers as defined below A = {z : Im z ≥ 1} B = {z : ∣z − 2 − i∣ = 3} C = {z : Re((1 − i)z) = (√2)}. Let z be any point in A ∩ B ∩ C and let w be any point satisfying ∣w − 2 − i∣ < 3. Then, ∣z∣ − ∣w∣ + 3 lies between (1) −6 and 3 (2) −3 and 6 (3) −6 and 6 (4) −3 and 9

$$\mathrm{Let}\:{A},\:{B},\:{C}\:\mathrm{be}\:\mathrm{three}\:\mathrm{sets}\:\mathrm{of}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{as}\:\mathrm{defined}\:\mathrm{below} \\ $$$${A}\:=\:\left\{{z}\::\:\mathrm{Im}\:{z}\:\geqslant\:\mathrm{1}\right\} \\ $$$${B}\:=\:\left\{{z}\::\:\mid{z}\:−\:\mathrm{2}\:−\:{i}\mid\:=\:\mathrm{3}\right\} \\ $$$${C}\:=\:\left\{{z}\::\:\mathrm{Re}\left(\left(\mathrm{1}\:−\:{i}\right){z}\right)\:=\:\sqrt{\mathrm{2}}\right\}. \\ $$$$\mathrm{Let}\:{z}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:{A}\:\cap\:{B}\:\cap\:{C}\:\mathrm{and}\:\mathrm{let} \\ $$$${w}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{satisfying}\:\mid{w}\:−\:\mathrm{2}\:−\:{i}\mid\:< \\ $$$$\mathrm{3}.\:\mathrm{Then},\:\mid{z}\mid\:−\:\mid{w}\mid\:+\:\mathrm{3}\:\mathrm{lies}\:\mathrm{between} \\ $$$$\left(\mathrm{1}\right)\:−\mathrm{6}\:\mathrm{and}\:\mathrm{3} \\ $$$$\left(\mathrm{2}\right)\:−\mathrm{3}\:\mathrm{and}\:\mathrm{6} \\ $$$$\left(\mathrm{3}\right)\:−\mathrm{6}\:\mathrm{and}\:\mathrm{6} \\ $$$$\left(\mathrm{4}\right)\:−\mathrm{3}\:\mathrm{and}\:\mathrm{9} \\ $$

Question Number 21107 Answers: 0 Comments: 0

Find out the value of nth derivative of y=e^(msin^(−1) x ) at x=0

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{nth}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{y}=\mathrm{e}^{\mathrm{msin}^{−\mathrm{1}} \mathrm{x}\:} \:\mathrm{at}\:\mathrm{x}=\mathrm{0} \\ $$

Question Number 21200 Answers: 1 Comments: 1

Suppose an integer x, a natural number n and a prime number p satisfy the equation 7x^2 − 44x + 12 = p^n . Find the largest value of p.

$$\mathrm{Suppose}\:\mathrm{an}\:\mathrm{integer}\:{x},\:\mathrm{a}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\mathrm{and}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:{p} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{44}{x}\:+\:\mathrm{12}\:=\:{p}^{{n}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{p}. \\ $$

Question Number 21100 Answers: 0 Comments: 1

if 3^(2n+3) =m,find 3^(−n)

$$\mathrm{if}\:\mathrm{3}^{\mathrm{2n}+\mathrm{3}} =\mathrm{m},\mathrm{find}\:\mathrm{3}^{−\mathrm{n}} \\ $$

Question Number 21083 Answers: 0 Comments: 0

4^(sin^2 x) + 4^(cos^2 x) =

$$\mathrm{4}^{\mathrm{sin}^{\mathrm{2}} {x}} +\:\mathrm{4}^{\mathrm{cos}^{\mathrm{2}} {x}} \:= \\ $$

Question Number 21082 Answers: 1 Comments: 0

Solve for x, log_(0.2) (x+5) >0

$$\mathrm{Solve}\:\mathrm{for}\:{x},\:\:\:\mathrm{log}_{\mathrm{0}.\mathrm{2}} \left({x}+\mathrm{5}\right)\:>\mathrm{0} \\ $$

Question Number 21077 Answers: 1 Comments: 0

integrate with respect to x ∫(dx/(9x^2 +6x+10))

$${integrate}\:{with}\:{respect}\:{to}\:{x} \\ $$$$\int\frac{{dx}}{\mathrm{9}{x}^{\mathrm{2}} \:+\mathrm{6}{x}+\mathrm{10}} \\ $$

Question Number 21097 Answers: 1 Comments: 0

Suppose in the plane 10 pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?

$$\mathrm{Suppose}\:\mathrm{in}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{10}\:\mathrm{pairwise} \\ $$$$\mathrm{nonparallel}\:\mathrm{lines}\:\mathrm{intersect}\:\mathrm{one}\:\mathrm{another}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{polygons}\:\left(\mathrm{with}\:\mathrm{finite}\:\mathrm{areas}\right)\:\mathrm{that}\:\mathrm{can} \\ $$$$\mathrm{be}\:\mathrm{formed}? \\ $$

Question Number 21071 Answers: 2 Comments: 0

The values of ′k′ for which the equation ∣x∣^2 (∣x∣^2 − 2k + 1) = 1 − k^2 , has repeated roots, when k belongs to (1) {1, −1} (2) {0, 1} (3) {0, −1} (4) {2, 3}

$$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:'{k}'\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mid{x}\mid^{\mathrm{2}} \left(\mid{x}\mid^{\mathrm{2}} \:−\:\mathrm{2}{k}\:+\:\mathrm{1}\right)\:=\:\mathrm{1}\:−\:{k}^{\mathrm{2}} ,\:\mathrm{has} \\ $$$$\mathrm{repeated}\:\mathrm{roots},\:\mathrm{when}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:\left\{\mathrm{1},\:−\mathrm{1}\right\} \\ $$$$\left(\mathrm{2}\right)\:\left\{\mathrm{0},\:\mathrm{1}\right\} \\ $$$$\left(\mathrm{3}\right)\:\left\{\mathrm{0},\:−\mathrm{1}\right\} \\ $$$$\left(\mathrm{4}\right)\:\left\{\mathrm{2},\:\mathrm{3}\right\} \\ $$

Question Number 21070 Answers: 1 Comments: 2

Let us consider an equation f(x) = x^3 − 3x + k = 0. Then the values of k for which the equation has 1. Exactly one root which is positive, then k belongs to 2. Exactly one root which is negative, then k belongs to 3. One negative and two positive root if k belongs to

$$\mathrm{Let}\:\mathrm{us}\:\mathrm{consider}\:\mathrm{an}\:\mathrm{equation}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \\ $$$$−\:\mathrm{3}{x}\:+\:{k}\:=\:\mathrm{0}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:{k}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has} \\ $$$$\mathrm{1}.\:\mathrm{Exactly}\:\mathrm{one}\:\mathrm{root}\:\mathrm{which}\:\mathrm{is}\:\mathrm{positive}, \\ $$$$\mathrm{then}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$$$\mathrm{2}.\:\mathrm{Exactly}\:\mathrm{one}\:\mathrm{root}\:\mathrm{which}\:\mathrm{is}\:\mathrm{negative}, \\ $$$$\mathrm{then}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$$$\mathrm{3}.\:\mathrm{One}\:\mathrm{negative}\:\mathrm{and}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{root} \\ $$$$\mathrm{if}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$

Question Number 21067 Answers: 2 Comments: 0

∀n∈N, prove 9∣[n^3 +(n+1)^3 +(n+2)^3 ]

$$\forall{n}\in\mathbb{N},\:{prove}\:\mathrm{9}\mid\left[{n}^{\mathrm{3}} +\left({n}+\mathrm{1}\right)^{\mathrm{3}} +\left({n}+\mathrm{2}\right)^{\mathrm{3}} \right] \\ $$

Question Number 21076 Answers: 1 Comments: 0

integrate with respect to x ∫x^(sinx)

$${integrate}\:{with}\:{respect}\:{to}\:{x} \\ $$$$\int{x}^{{sinx}} \\ $$

Question Number 21060 Answers: 0 Comments: 8

write sin 1° in surd form please show workings.

$$\mathrm{write}\:\mathrm{sin}\:\mathrm{1}°\:\mathrm{in}\:\mathrm{surd}\:\mathrm{form} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{workings}. \\ $$

Question Number 21140 Answers: 1 Comments: 1

if tan β=((2sin αsin γ)/(sin (α+γ))) so proof cot γ+cot α=2cot β

$${if}\:\mathrm{tan}\:\beta=\frac{\mathrm{2sin}\:\alpha\mathrm{sin}\:\gamma}{\mathrm{sin}\:\left(\alpha+\gamma\right)} \\ $$$${so}\:{proof}\:\mathrm{cot}\:\gamma+\mathrm{cot}\:\alpha=\mathrm{2cot}\:\beta \\ $$

Question Number 21053 Answers: 0 Comments: 0

Question Number 21050 Answers: 1 Comments: 0

The most general solution of the equation sinx + cosx = min_(a∈R) {1, a^2 − 4a + 6} is

$$\mathrm{The}\:\mathrm{most}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{sin}{x}\:+\:\mathrm{cos}{x}\:=\:\underset{{a}\in{R}} {\mathrm{min}}\left\{\mathrm{1},\:{a}^{\mathrm{2}} \:−\:\mathrm{4}{a}\:+\:\mathrm{6}\right\} \\ $$$$\mathrm{is} \\ $$

Question Number 21048 Answers: 0 Comments: 0

If the equation 2cos2x − (a + 7)cosx + 3a − 13 = 0 possesses atleast one real solution, then the maximum integral value of ′a′ can be

$$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{2cos2}{x}\:−\:\left({a}\:+\:\mathrm{7}\right)\mathrm{cos}{x}\:+ \\ $$$$\mathrm{3}{a}\:−\:\mathrm{13}\:=\:\mathrm{0}\:\mathrm{possesses}\:\mathrm{atleast}\:\mathrm{one}\:\mathrm{real} \\ $$$$\mathrm{solution},\:\mathrm{then}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{integral} \\ $$$$\mathrm{value}\:\mathrm{of}\:'{a}'\:\mathrm{can}\:\mathrm{be} \\ $$

Question Number 21038 Answers: 1 Comments: 1

Question Number 21031 Answers: 0 Comments: 0

if :∀ε>0, ∀(a,b)∈R^2 ,a<b+ε prove: a≤b

$${if}\::\forall\epsilon>\mathrm{0},\:\forall\left({a},{b}\right)\in\mathbb{R}^{\mathrm{2}} ,{a}<{b}+\epsilon \\ $$$${prove}:\:{a}\leqslant{b} \\ $$

Question Number 21029 Answers: 0 Comments: 0

Question Number 21021 Answers: 1 Comments: 2

Question Number 21015 Answers: 0 Comments: 0

Question Number 21013 Answers: 1 Comments: 1

Question Number 21012 Answers: 1 Comments: 3

A spring with one end attached to a mass and the other to a rigid support is stretched and released. (a) Magnitude of acceleration, when just released is maximum. (b) Magnitude of acceleration, when at equilibrium position, is maximum. (c) Speed is maximum when mass is at equilibrium position. (d) Magnitude of displacement is always maximum whenever speed is minimum.

$$\mathrm{A}\:\mathrm{spring}\:\mathrm{with}\:\mathrm{one}\:\mathrm{end}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{a} \\ $$$$\mathrm{mass}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{to}\:\mathrm{a}\:\mathrm{rigid}\:\mathrm{support}\:\mathrm{is} \\ $$$$\mathrm{stretched}\:\mathrm{and}\:\mathrm{released}. \\ $$$$\left({a}\right)\:\mathrm{Magnitude}\:\mathrm{of}\:\mathrm{acceleration},\:\mathrm{when} \\ $$$$\mathrm{just}\:\mathrm{released}\:\mathrm{is}\:\mathrm{maximum}. \\ $$$$\left({b}\right)\:\mathrm{Magnitude}\:\mathrm{of}\:\mathrm{acceleration},\:\mathrm{when} \\ $$$$\mathrm{at}\:\mathrm{equilibrium}\:\mathrm{position},\:\mathrm{is}\:\mathrm{maximum}. \\ $$$$\left({c}\right)\:\mathrm{Speed}\:\mathrm{is}\:\mathrm{maximum}\:\mathrm{when}\:\mathrm{mass}\:\mathrm{is}\:\mathrm{at} \\ $$$$\mathrm{equilibrium}\:\mathrm{position}. \\ $$$$\left({d}\right)\:\mathrm{Magnitude}\:\mathrm{of}\:\mathrm{displacement}\:\mathrm{is} \\ $$$$\mathrm{always}\:\mathrm{maximum}\:\mathrm{whenever}\:\mathrm{speed}\:\mathrm{is} \\ $$$$\mathrm{minimum}. \\ $$

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