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Question

If the equation $2 to the power of x end exponent plus 4 to the power of y end exponent equals 2 to the power of y end exponent plus 4 to the power of x end exponent$ is solved for $y$ in terms of $x comma blank$ where $x less than 0 comma$ then the sum of the solutions is

$x \log_{2} (1 - 2^{x})$
$x + \log_{2} (1 - 2^{x})$
$\log_{2} (1 - 2^{x})$
$x \log_{2} (2^{x} + 1)$

The correct answer is: $x plus log subscript 2 end subscript invisible function application open parentheses 1 minus 2 to the power of x end exponent close parentheses$

$2 to the power of 2 y end exponent minus 2 to the power of y end exponent plus 2 to the power of x end exponent open parentheses 1 minus 2 to the power of x end exponent close parentheses equals 0$
Putting $2 to the power of y end exponent equals t comma$ we get
$t to the power of 2 end exponent minus t plus 2 to the power of x end exponent open parentheses 1 minus 2 to the power of x end exponent close parentheses equals 0 blank w h e r e blank t subscript 1 end subscript equals 2 to the power of y subscript 1 end subscript end exponent blank a n d blank t subscript 2 end subscript equals 2 to the power of y subscript 2 end subscript end exponent$
$t subscript 1 end subscript t subscript 2 end subscript equals 2 to the power of x end exponent open parentheses 1 minus 2 to the power of x end exponent close parentheses$
$2 to the power of y subscript 1 end subscript plus y subscript 2 end subscript end exponent equals 2 to the power of x end exponent open parentheses 1 minus 2 to the power of x end exponent close parentheses$
$y subscript 1 end subscript plus y subscript 2 end subscript equals x plus log subscript 2 end subscript invisible function application open parentheses 1 minus 2 to the power of x end exponent close parentheses$

Related Questions to study

$I f blank A$ and $blank B$ are acute positive angles satisfying the equations $3 sin to the power of 2 end exponent invisible function application A plus 2 sin to the power of 2 end exponent invisible function application B equals 1$ and $blank 3 sin invisible function application 2 A minus 2 sin invisible function application 2 B equals 0 comma$ then $A plus 2 blank B$ is equal to

$I f blank A$ and $blank B$ are acute positive angles satisfying the equations $3 sin to the power of 2 end exponent invisible function application A plus 2 sin to the power of 2 end exponent invisible function application B equals 1$ and $blank 3 sin invisible function application 2 A minus 2 sin invisible function application 2 B equals 0 comma$ then $A plus 2 blank B$ is equal to

The value of $49 to the power of open parentheses 1 minus log subscript 7 end subscript invisible function application 2 close parentheses end exponent plus 5 to the power of negative log subscript 5 end subscript invisible function application 4 end exponent$ is

The value of $49 to the power of open parentheses 1 minus log subscript 7 end subscript invisible function application 2 close parentheses end exponent plus 5 to the power of negative log subscript 5 end subscript invisible function application 4 end exponent$ is

If $square root of log subscript 2 end subscript invisible function application x end root minus 0.5 equals log subscript 2 end subscript invisible function application square root of x comma$ then $x$ equals

If $square root of log subscript 2 end subscript invisible function application x end root minus 0.5 equals log subscript 2 end subscript invisible function application square root of x comma$ then $x$ equals

parallel

The maximum value of $open parentheses cos invisible function application alpha subscript 1 end subscript close parentheses open parentheses cos invisible function application alpha subscript 2 end subscript close parentheses horizontal ellipsis open parentheses cos invisible function application alpha subscript n end subscript close parentheses comma$ unser the restrictions $0 less or equal than alpha subscript 1 end subscript comma blank alpha subscript 2 end subscript comma horizontal ellipsis alpha subscript n end subscript less or equal than pi divided by 2$ and $open parentheses cot invisible function application alpha subscript 1 end subscript close parentheses open parentheses cot invisible function application alpha subscript 2 end subscript close parentheses horizontal ellipsis open parentheses cot invisible function application alpha subscript n end subscript close parentheses equals 1$ is

The maximum value of $open parentheses cos invisible function application alpha subscript 1 end subscript close parentheses open parentheses cos invisible function application alpha subscript 2 end subscript close parentheses horizontal ellipsis open parentheses cos invisible function application alpha subscript n end subscript close parentheses comma$ unser the restrictions $0 less or equal than alpha subscript 1 end subscript comma blank alpha subscript 2 end subscript comma horizontal ellipsis alpha subscript n end subscript less or equal than pi divided by 2$ and $open parentheses cot invisible function application alpha subscript 1 end subscript close parentheses open parentheses cot invisible function application alpha subscript 2 end subscript close parentheses horizontal ellipsis open parentheses cot invisible function application alpha subscript n end subscript close parentheses equals 1$ is

If both the distinct roots of the equation $open vertical bar sin invisible function application x close vertical bar to the power of 2 end exponent plus open vertical bar sin invisible function application x close vertical bar plus b equals 0 blank$ in $blank open square brackets 0 comma pi close square brackets$ are real, then the values of $b$ are

If both the distinct roots of the equation $open vertical bar sin invisible function application x close vertical bar to the power of 2 end exponent plus open vertical bar sin invisible function application x close vertical bar plus b equals 0 blank$ in $blank open square brackets 0 comma pi close square brackets$ are real, then the values of $b$ are

In triangle $A B C comma$ internal angle bisector $angle A$ makes an angle $theta$ with side $B C.$ The value of $sin invisible function application theta$ is equal to

In triangle $A B C comma$ internal angle bisector $angle A$ makes an angle $theta$ with side $B C.$ The value of $sin invisible function application theta$ is equal to

parallel

If $open parentheses 4 close parentheses to the power of log subscript 9 end subscript invisible function application 3 end exponent plus open parentheses 9 close parentheses to the power of log subscript 2 end subscript invisible function application 4 end exponent equals open parentheses 10 close parentheses to the power of log subscript x end subscript invisible function application 83 end exponent comma$ then $x$ is equal to

If $open parentheses 4 close parentheses to the power of log subscript 9 end subscript invisible function application 3 end exponent plus open parentheses 9 close parentheses to the power of log subscript 2 end subscript invisible function application 4 end exponent equals open parentheses 10 close parentheses to the power of log subscript x end subscript invisible function application 83 end exponent comma$ then $x$ is equal to

The value of $log invisible function application a b minus log invisible function application open vertical bar b close vertical bar equals$

The value of $log invisible function application a b minus log invisible function application open vertical bar b close vertical bar equals$

If $alpha comma blank beta comma blank gamma comma blank delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k comma$ then the value of $4 sin invisible function application fraction numerator alpha over denominator 2 end fraction plus 3 sin invisible function application fraction numerator beta over denominator 2 end fraction plus 2 sin invisible function application fraction numerator gamma over denominator 2 end fraction plus sin invisible function application fraction numerator delta over denominator 2 end fraction$ is equal to

If $alpha comma blank beta comma blank gamma comma blank delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k comma$ then the value of $4 sin invisible function application fraction numerator alpha over denominator 2 end fraction plus 3 sin invisible function application fraction numerator beta over denominator 2 end fraction plus 2 sin invisible function application fraction numerator gamma over denominator 2 end fraction plus sin invisible function application fraction numerator delta over denominator 2 end fraction$ is equal to

parallel

The value of $x$ satisfying $square root of 3 to the power of negative 4 plus 2 log subscript square root of 5 end subscript invisible function application x end exponent equals 1 divided by 9$ is

The value of $x$ satisfying $square root of 3 to the power of negative 4 plus 2 log subscript square root of 5 end subscript invisible function application x end exponent equals 1 divided by 9$ is

In $blank increment A B C$ , if $b to the power of 2 end exponent plus c to the power of 2 end exponent equals 2 a to the power of 2 end exponent comma$ then value of $fraction numerator cot invisible function application A over denominator cot invisible function application B plus cot invisible function application C end fraction$ is

In $blank increment A B C$ , if $b to the power of 2 end exponent plus c to the power of 2 end exponent equals 2 a to the power of 2 end exponent comma$ then value of $fraction numerator cot invisible function application A over denominator cot invisible function application B plus cot invisible function application C end fraction$ is

If $log subscript 10 end subscript invisible function application open square brackets fraction numerator 1 over denominator 2 to the power of x end exponent plus blank x minus 1 end fraction close square brackets equals x open square brackets log subscript 10 end subscript invisible function application 5 minus 1 close square brackets comma blank t h e n blank x equals$

If $log subscript 10 end subscript invisible function application open square brackets fraction numerator 1 over denominator 2 to the power of x end exponent plus blank x minus 1 end fraction close square brackets equals x open square brackets log subscript 10 end subscript invisible function application 5 minus 1 close square brackets comma blank t h e n blank x equals$

parallel

r triangle $A B C comma blank R equals 5 divided by 2 blank$ and $blank r equals 1.$ Let $I$ be the incentre of the triangle and $D comma blank E$ and $blank F$ be the feet of the perpendiculars from $I blank$ to $blank B C comma blank C A blank$ and $blank A B comma$ respectively. The value of $fraction numerator I D cross times I E cross times I F over denominator I A cross times I B cross times I C end fraction blank$ is equal to

r triangle $A B C comma blank R equals 5 divided by 2 blank$ and $blank r equals 1.$ Let $I$ be the incentre of the triangle and $D comma blank E$ and $blank F$ be the feet of the perpendiculars from $I blank$ to $blank B C comma blank C A blank$ and $blank A B comma$ respectively. The value of $fraction numerator I D cross times I E cross times I F over denominator I A cross times I B cross times I C end fraction blank$ is equal to

any $increment A B C comma$ the value of $a open parentheses b to the power of 2 end exponent plus c to the power of 2 end exponent close parentheses cos invisible function application A plus b left parenthesis c to the power of 2 end exponent plus a to the power of 2 end exponent right parenthesis cos invisible function application B plus c left parenthesis a to the power of 2 end exponent plus b to the power of 2 end exponent right parenthesis cos invisible function application C equals$

any $increment A B C comma$ the value of $a open parentheses b to the power of 2 end exponent plus c to the power of 2 end exponent close parentheses cos invisible function application A plus b left parenthesis c to the power of 2 end exponent plus a to the power of 2 end exponent right parenthesis cos invisible function application B plus c left parenthesis a to the power of 2 end exponent plus b to the power of 2 end exponent right parenthesis cos invisible function application C equals$

$fraction numerator sin invisible function application x over denominator sin invisible function application y end fraction equals fraction numerator 1 over denominator 2 end fraction comma fraction numerator cos invisible function application x over denominator cos invisible function application y end fraction equals fraction numerator 3 over denominator 2 end fraction blank w h e r e blank x comma blank y blank element of open parentheses 0 comma fraction numerator pi over denominator 2 end fraction close parentheses comma blank t h e n blank t h e blank v a l u e blank o f blank left parenthesis x plus y right parenthesis blank i s blank e q u a l blank t o$

$fraction numerator sin invisible function application x over denominator sin invisible function application y end fraction equals fraction numerator 1 over denominator 2 end fraction comma fraction numerator cos invisible function application x over denominator cos invisible function application y end fraction equals fraction numerator 3 over denominator 2 end fraction blank w h e r e blank x comma blank y blank element of open parentheses 0 comma fraction numerator pi over denominator 2 end fraction close parentheses comma blank t h e n blank t h e blank v a l u e blank o f blank left parenthesis x plus y right parenthesis blank i s blank e q u a l blank t o$

parallel

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