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Question Number 21112 Answers: 0 Comments: 0

A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground (a) the direction of motion of the ball changes every 10 seconds. (b) speed of ball changes every 10 seconds. (c) average speed of ball over any 20 second interval is fixed. (d) the acceleration of ball is the same as from the train.

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