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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

Question Number 18871 Answers: 0 Comments: 1

Question Number 18866 Answers: 1 Comments: 0

Question Number 18864 Answers: 2 Comments: 1

Question Number 18859 Answers: 1 Comments: 0

If f(x+2)=f(x)+7 and f(1)=2; f(2)=5, then the ratio of f(150) to f(75) is

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}+\mathrm{2}\right)=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{7}\:\mathrm{and}\:\mathrm{f}\left(\mathrm{1}\right)=\mathrm{2};\:\mathrm{f}\left(\mathrm{2}\right)=\mathrm{5}, \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{150}\right)\:\mathrm{to}\:\mathrm{f}\left(\mathrm{75}\right)\:\mathrm{is} \\ $$

Question Number 18854 Answers: 0 Comments: 0

lim_(x→0) sin x_x =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}sin}\:\underset{\mathrm{x}} {\mathrm{x}}\:\:\:\:=? \\ $$

Question Number 18847 Answers: 1 Comments: 0

In an equilateral triangle with usual notations the value of ((27r^2 R)/(r_1 r_2 r_3 )) is equal to

$$\mathrm{In}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{with}\:\mathrm{usual} \\ $$$$\mathrm{notations}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{27}{r}^{\mathrm{2}} {R}}{{r}_{\mathrm{1}} {r}_{\mathrm{2}} {r}_{\mathrm{3}} }\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 18846 Answers: 0 Comments: 0

There are two crystalline forms of PbO ; one is yellow and the other is red. The standard enthalpies of formation of these two forms are −217.3 and −219 kJ per mole respectively. Calculate the enthalpy change for the solid-solid phase transition. PbO (yellow) → PbO (red)

$$\mathrm{There}\:\mathrm{are}\:\mathrm{two}\:\mathrm{crystalline}\:\mathrm{forms}\:\mathrm{of}\:\mathrm{PbO}\:; \\ $$$$\mathrm{one}\:\mathrm{is}\:\mathrm{yellow}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{is}\:\mathrm{red}.\:\mathrm{The} \\ $$$$\mathrm{standard}\:\mathrm{enthalpies}\:\mathrm{of}\:\mathrm{formation}\:\mathrm{of} \\ $$$$\mathrm{these}\:\mathrm{two}\:\mathrm{forms}\:\mathrm{are}\:−\mathrm{217}.\mathrm{3}\:\mathrm{and}\:−\mathrm{219} \\ $$$$\mathrm{kJ}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{respectively}.\:\mathrm{Calculate}\:\mathrm{the} \\ $$$$\mathrm{enthalpy}\:\mathrm{change}\:\mathrm{for}\:\mathrm{the}\:\mathrm{solid}-\mathrm{solid} \\ $$$$\mathrm{phase}\:\mathrm{transition}. \\ $$$$\mathrm{PbO}\:\left(\mathrm{yellow}\right)\:\rightarrow\:\mathrm{PbO}\:\left(\mathrm{red}\right) \\ $$

Question Number 18885 Answers: 0 Comments: 0

Question Number 18886 Answers: 0 Comments: 0

Question Number 18841 Answers: 1 Comments: 0

Question Number 18830 Answers: 1 Comments: 2

If sin 2x = a find the simplest expression for cos x

$$\mathrm{If}\:\mathrm{sin}\:\mathrm{2}{x}\:=\:{a} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{simplest}\:\mathrm{expression}\:\mathrm{for}\:\mathrm{cos}\:{x} \\ $$

Question Number 18827 Answers: 0 Comments: 3

Question Number 18851 Answers: 0 Comments: 0

Question Number 18823 Answers: 0 Comments: 2

If P≡(2,1) and A and B lie on x−axis and y=x respectively such that PA+PB+AB is minimum find A and B.

$$\mathrm{If}\:\mathrm{P}\equiv\left(\mathrm{2},\mathrm{1}\right)\:\mathrm{and}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{lie}\:\mathrm{on}\:\mathrm{x}−\mathrm{axis} \\ $$$$\mathrm{and}\:\mathrm{y}=\mathrm{x}\:\mathrm{respectively}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{PA}+\mathrm{PB}+\mathrm{AB}\:\mathrm{is}\:\mathrm{minimum}\:\mathrm{find} \\ $$$$\mathrm{A}\:\mathrm{and}\:\mathrm{B}. \\ $$

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

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

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