Maths-
General
Easy
Question
If the mapping
maps [-1,1] onto [0,2] then for all values of
,
is such that
Hint:
In the question we will simplify the equation to make it a quadratic after that we can easily find the minimum and maximum value.
The correct answer is: ![f open parentheses 1 fourth close parentheses less or equal than A less or equal than f left parenthesis 0 right parenthesis](data:image/png;base64,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)
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=
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d then the minimum value of
is equal to (where is variable)
If
d then the minimum value of
is equal to (where is variable)
Maths-General
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and
then
=
and
then
=
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If
then
=
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then
=
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The maximum value of the expression ![open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar](data:image/png;base64,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)
where and x are real numbers is
The maximum value of the expression ![open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar](data:image/png;base64,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)
where and x are real numbers is
Maths-General
Maths-
If
and
then ![open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared](data:image/png;base64,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)
If
and
then ![open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared](data:image/png;base64,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)
Maths-General
Maths-
The maximum value of
under the restrictions
and
is
The maximum value of
under the restrictions
and
is
Maths-General
Maths-
Let
be two sets. Then
Let
be two sets. Then
Maths-General
Maths-
If
then
is
If
then
is
Maths-General