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Question Number 18962 Answers: 1 Comments: 0

The number of solutions of the equation sin^5 θ + (1/(sin θ)) = (1/(cos θ)) + cos^5 θ where θ ∈ (0, (π/2)) , is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{sin}^{\mathrm{5}} \:\theta\:+\:\frac{\mathrm{1}}{\mathrm{sin}\:\theta}\:=\:\frac{\mathrm{1}}{\mathrm{cos}\:\theta}\:+\:\mathrm{cos}^{\mathrm{5}} \:\theta\:\mathrm{where} \\ $$$$\theta\:\in\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right)\:,\:\mathrm{is} \\ $$

Question Number 18961 Answers: 1 Comments: 0

Find arg(z), z = i^i^i .

$$\mathrm{Find}\:\mathrm{arg}\left({z}\right),\:{z}\:=\:{i}^{{i}^{{i}} } . \\ $$

Question Number 18951 Answers: 0 Comments: 0

Question Number 18949 Answers: 1 Comments: 1

Find the number of numbers ≤ 10^8 which are neither perfect squares, nor perfect cubes, nor perfect fifth powers.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{numbers}\:\leqslant\:\mathrm{10}^{\mathrm{8}} \\ $$$$\mathrm{which}\:\mathrm{are}\:\mathrm{neither}\:\mathrm{perfect}\:\mathrm{squares},\:\mathrm{nor} \\ $$$$\mathrm{perfect}\:\mathrm{cubes},\:\mathrm{nor}\:\mathrm{perfect}\:\mathrm{fifth}\:\mathrm{powers}. \\ $$

Question Number 18945 Answers: 0 Comments: 1

A point ′A′ is randomly chosen in a square of side length 1 unit. Find the probability the distance from A to the centre of the square does not exceed x.

$$\mathrm{A}\:\mathrm{point}\:'\mathrm{A}'\:\mathrm{is}\:\mathrm{randomly}\:\mathrm{chosen}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{square}\:\mathrm{of}\:\mathrm{side}\:\mathrm{length}\:\mathrm{1}\:\mathrm{unit}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{probability}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{A}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exceed}\:\mathrm{x}. \\ $$

Question Number 18924 Answers: 1 Comments: 0

A spring of force constant k is cut into two pieces such that one piece is twice as long as the other. Then the longer piece will have a force constant of

$$\mathrm{A}\:\mathrm{spring}\:\mathrm{of}\:\mathrm{force}\:\mathrm{constant}\:{k}\:\mathrm{is}\:\mathrm{cut}\:\mathrm{into} \\ $$$$\mathrm{two}\:\mathrm{pieces}\:\mathrm{such}\:\mathrm{that}\:\mathrm{one}\:\mathrm{piece}\:\mathrm{is}\:\mathrm{twice} \\ $$$$\mathrm{as}\:\mathrm{long}\:\mathrm{as}\:\mathrm{the}\:\mathrm{other}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{longer} \\ $$$$\mathrm{piece}\:\mathrm{will}\:\mathrm{have}\:\mathrm{a}\:\mathrm{force}\:\mathrm{constant}\:\mathrm{of} \\ $$

Question Number 18922 Answers: 0 Comments: 3

Consider a block sliding over a smooth inclined surface of inclination θ. Relating to Newton′s second law applied on the block, select the incorrect alternative. (1) ΣF_(y′) ≠ 0 (2) ΣF_y = 0 (3) ΣF_x = −mg sin θ (4) ΣF_(x′) < 0

$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{block}\:\mathrm{sliding}\:\mathrm{over}\:\mathrm{a}\:\mathrm{smooth} \\ $$$$\mathrm{inclined}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{inclination}\:\theta.\:\mathrm{Relating} \\ $$$$\mathrm{to}\:\mathrm{Newton}'\mathrm{s}\:\mathrm{second}\:\mathrm{law}\:\mathrm{applied}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{block},\:\mathrm{select}\:\mathrm{the}\:\mathrm{incorrect}\:\mathrm{alternative}. \\ $$$$\left(\mathrm{1}\right)\:\Sigma{F}_{{y}'} \:\neq\:\mathrm{0} \\ $$$$\left(\mathrm{2}\right)\:\Sigma{F}_{{y}} \:=\:\mathrm{0} \\ $$$$\left(\mathrm{3}\right)\:\Sigma{F}_{{x}} \:=\:−{mg}\:\mathrm{sin}\:\theta \\ $$$$\left(\mathrm{4}\right)\:\Sigma{F}_{{x}'} \:<\:\mathrm{0} \\ $$

Question Number 18918 Answers: 0 Comments: 17

A bird sits on a stretched telegraph wire. The additional tension produced in the wire is (1) Zero (2) Greater than weight of the bird (3) Less than the weight of the bird (4) Equal to the weight of the bird

$$\mathrm{A}\:\mathrm{bird}\:\mathrm{sits}\:\mathrm{on}\:\mathrm{a}\:\mathrm{stretched}\:\mathrm{telegraph} \\ $$$$\mathrm{wire}.\:\mathrm{The}\:\mathrm{additional}\:\mathrm{tension}\:\mathrm{produced} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Zero} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Greater}\:\mathrm{than}\:\mathrm{weight}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bird} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Less}\:\mathrm{than}\:\mathrm{the}\:\mathrm{weight}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bird} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{weight}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bird} \\ $$

Question Number 18940 Answers: 0 Comments: 3

A light rope is passed over a pulley such that at its one end a block is attached, and on the other end a boy is climbing up with acceleration (g/2) relative to rope. Mass of the block is 30 kg and that of the boy is 40 kg. Find the tension and acceleration of the rope.

$$\mathrm{A}\:\mathrm{light}\:\mathrm{rope}\:\mathrm{is}\:\mathrm{passed}\:\mathrm{over}\:\mathrm{a}\:\mathrm{pulley}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{at}\:\mathrm{its}\:\mathrm{one}\:\mathrm{end}\:\mathrm{a}\:\mathrm{block}\:\mathrm{is}\:\mathrm{attached}, \\ $$$$\mathrm{and}\:\mathrm{on}\:\mathrm{the}\:\mathrm{other}\:\mathrm{end}\:\mathrm{a}\:\mathrm{boy}\:\mathrm{is}\:\mathrm{climbing} \\ $$$$\mathrm{up}\:\mathrm{with}\:\mathrm{acceleration}\:\frac{{g}}{\mathrm{2}}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{rope}. \\ $$$$\mathrm{Mass}\:\mathrm{of}\:\mathrm{the}\:\mathrm{block}\:\mathrm{is}\:\mathrm{30}\:\mathrm{kg}\:\mathrm{and}\:\mathrm{that}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{boy}\:\mathrm{is}\:\mathrm{40}\:\mathrm{kg}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{tension}\:\mathrm{and} \\ $$$$\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rope}. \\ $$

Question Number 18916 Answers: 0 Comments: 0

x^2 −7x+12<mod(x−4)

$${x}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{12}<{mod}\left({x}−\mathrm{4}\right) \\ $$

Question Number 18909 Answers: 0 Comments: 1

What does a_ .b_ and a_ ×b_ means ? someone please explain it.

$${What}\:{does}\:\underset{} {{a}}.\underset{} {{b}}\:\:{and}\:\underset{} {{a}}×\underset{} {{b}}\:{means}\:? \\ $$$${someone}\:{please}\:{explain}\:{it}. \\ $$

Question Number 18908 Answers: 0 Comments: 5

Why Does ∣A^→ ×B^→ ∣=ABSinθ ?

$${Why}\:{Does} \\ $$$$\mid\overset{\rightarrow} {{A}}×\overset{\rightarrow} {{B}}\mid={ABSin}\theta\:? \\ $$

Question Number 18907 Answers: 0 Comments: 3

Why Does A^→ .B^→ =ABCosθ?

$${Why}\:{Does}\: \\ $$$$\overset{\rightarrow} {{A}}.\overset{\rightarrow} {{B}}={ABCos}\theta?\: \\ $$

Question Number 18906 Answers: 1 Comments: 0

calculate (dy/dx) where y=cos^(−1) ((a+b cosx)/(b+a cosx)) (b>a)

$$\mathrm{calculate}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:\:\mathrm{where}\: \\ $$$$\mathrm{y}=\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{a}+\mathrm{b}\:\mathrm{cosx}}{\mathrm{b}+\mathrm{a}\:\mathrm{cosx}}\:\left(\mathrm{b}>\mathrm{a}\right) \\ $$

Question Number 18905 Answers: 0 Comments: 0

prove that the differentiation of(((√(1+x^2 ))−(√(1−x^2 )))/((√(1+x^2 ))+(√(1−x^2 )))) with respect to (√(1−x^4 )) is ((√(1−x^4 ))/x^6 )

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{differentiation} \\ $$$$\mathrm{of}\frac{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }−\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }+\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\:\mathrm{with}\:\mathrm{respect} \\ $$$$\mathrm{to}\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }\:\:\mathrm{is}\:\:\frac{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}{\mathrm{x}^{\mathrm{6}} } \\ $$

Question Number 18904 Answers: 1 Comments: 0

Solve the triangle in which a=((√3)+1), b=((√3)−1) and ∠C=60°.

$${Solve}\:{the}\:{triangle}\:{in}\:{which}\:{a}=\left(\sqrt{\mathrm{3}}+\mathrm{1}\right),\: \\ $$$${b}=\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)\:{and}\:\angle{C}=\mathrm{60}°. \\ $$$$ \\ $$

Question Number 18893 Answers: 1 Comments: 0

Find all values of x, y and k for which the system of equations sin x cos 2y = k^4 − 2k^2 + 2 cos x sin 2y = k + 1 has a solution.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x},\:{y}\:\mathrm{and}\:{k}\:\mathrm{for}\:\mathrm{which} \\ $$$$\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\mathrm{sin}\:{x}\:\mathrm{cos}\:\mathrm{2}{y}\:=\:{k}^{\mathrm{4}} \:−\:\mathrm{2}{k}^{\mathrm{2}} \:+\:\mathrm{2} \\ $$$$\mathrm{cos}\:{x}\:\mathrm{sin}\:\mathrm{2}{y}\:=\:{k}\:+\:\mathrm{1} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{solution}. \\ $$

Question Number 18892 Answers: 1 Comments: 0

If the equation sin6x + cos4x = −2 have a family of nonnegative solutions x_k ′s, where 0 ≤ x_1 < x_2 < x_3 < .... < x_k < x_(k+1) ....., then the value of (1/π)Σ_(k=1) ^(1000) ∣x_(k+1) − x_k ∣ is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{sin6}{x}\:+\:\mathrm{cos4}{x}\:=\:−\mathrm{2}\:\mathrm{have} \\ $$$$\mathrm{a}\:\mathrm{family}\:\mathrm{of}\:\mathrm{nonnegative}\:\mathrm{solutions}\:{x}_{{k}} '\mathrm{s}, \\ $$$$\mathrm{where}\:\mathrm{0}\:\leqslant\:{x}_{\mathrm{1}} \:<\:{x}_{\mathrm{2}} \:<\:{x}_{\mathrm{3}} \:<\:....\:<\:{x}_{{k}} \:<\:{x}_{{k}+\mathrm{1}} \\ $$$$.....,\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{1}}{\pi}\underset{{k}=\mathrm{1}} {\overset{\mathrm{1000}} {\sum}}\mid{x}_{{k}+\mathrm{1}} \:−\:{x}_{{k}} \mid\:\mathrm{is} \\ $$

Question Number 18891 Answers: 2 Comments: 0

If angles A and B satisfy (√2) cos A = cos B + cos^3 B and (√2) sin A = sin B − sin^3 B, then the value of 1620sin^2 (A − B) is

$$\mathrm{If}\:\mathrm{angles}\:{A}\:\mathrm{and}\:{B}\:\mathrm{satisfy}\:\sqrt{\mathrm{2}}\:\mathrm{cos}\:{A}\:= \\ $$$$\mathrm{cos}\:{B}\:+\:\mathrm{cos}^{\mathrm{3}} \:{B}\:\mathrm{and}\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:{A}\:=\:\mathrm{sin}\:{B}\:− \\ $$$$\mathrm{sin}^{\mathrm{3}} \:{B},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{1620sin}^{\mathrm{2}} \left({A}\:−\:{B}\right) \\ $$$$\mathrm{is} \\ $$

Question Number 18889 Answers: 0 Comments: 2

∫ ((d((√(3x))))/(√((√x) + 7)))

$$\int\:\frac{{d}\left(\sqrt{\mathrm{3}{x}}\right)}{\sqrt{\sqrt{{x}}\:+\:\mathrm{7}}} \\ $$

Question Number 18872 Answers: 2 Comments: 0