Question
If f(x) is a polynomial function satisfying the condition f(x). f(1/x) = f(x) + f(1/x) and f(2) = 9 then -
- 2 f(4) = 3f(6)
- 14 f(1) = f(3)
- 9 f(3) = f(5)
- f(10)= f(11)
Hint:
The correct answer is: 14 f(1) = f(3)
To choose the correct option from the given function.
The polynomial which satisfies f(x)f(1/x)=f(x)+f(1/x) is ±![x to the power of n plus 1](data:image/png;base64,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)
Given, f(2)=9
±
=9
n=3
Hence the function is f(x)=![x cubed plus 1](data:image/png;base64,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)
f(3)=28, f(1)=2
14 x f(1) = f(3)
14 x 2 = 28
28 = 28
Given, f(2)=9
Hence the function is f(x)=
Therefore, 14 f(1) = f(3)
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If R be a relation '<' from A = {1, 2, 3, 4} to B = {1, 3, 5} i.e. (a, b)
R iff a < b, then
is ![text-end text](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAHCAYAAADTcMcaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAGKUc3qwAAABJJREFUeNpjYECA/0TgUUAxAAAFGAj4Gd15qgAAAE90RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXRleHQ+LTwvbXRleHQ+PC9tYXRoPs3uNooAAAAASUVORK5CYII=)
Values of are {(3, 3), (3, 5), (5, 3), (5, 5)}
If R be a relation '<' from A = {1, 2, 3, 4} to B = {1, 3, 5} i.e. (a, b)
R iff a < b, then
is ![text-end text](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAHCAYAAADTcMcaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAGKUc3qwAAABJJREFUeNpjYECA/0TgUUAxAAAFGAj4Gd15qgAAAE90RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXRleHQ+LTwvbXRleHQ+PC9tYXRoPs3uNooAAAAASUVORK5CYII=)
Values of are {(3, 3), (3, 5), (5, 3), (5, 5)}
Which one of the following relations on R is equivalence relation
Which one of the following relations on R is equivalence relation
Let R = {(x, y) : x, y
A, x + y = 5} where A = {1, 2, 3, 4, 5} then
Hence, the given relation is not reflexive, symmetric and not transitive
Let R = {(x, y) : x, y
A, x + y = 5} where A = {1, 2, 3, 4, 5} then
Hence, the given relation is not reflexive, symmetric and not transitive
Let
be a relation defined by
Then R is
Hence, the given relation is Reflexive, transitive but not symmetric.
Let
be a relation defined by
Then R is
Hence, the given relation is Reflexive, transitive but not symmetric.
The relation R defined in N as aRb
b is divisible by a is
Hence, the given relations is reflexive but not symmetric.
The relation R defined in N as aRb
b is divisible by a is
Hence, the given relations is reflexive but not symmetric.