Question

♥ Hello Friends !!! ♥


Answer this question ⤵⤵


$\mathsf {If \: f(x) \: = \: cos \: [\pi]x \: + \: cos \:[\pi\:x],where \:[y] \: is \: the \: greatest } \\ \mathsf{integer \: function \: of \: y, \: then \: f({ \dfrac{\pi}{2})}} \\ \\ \sf{a) \: 0 \: \: \: \: \: b) \: cos \: 3 \: \: \: \: c) \: cos \: 4 \: \: \: \: \: d) \: cos \: 2}$



❎ No Spam Answers ❎



First and best = Brainliest

Answer 1

We know that : Function f : R ==> R defined by f(x) = [x] assumes the Greatest Integer which is Less than (or) Equal to x

For Example, Consider : x = 2.9

As [x] = 2 for all 2 ≤ x ≤ 3

⇒ [2.9] = 2 (Greatest Integer less than or Equal to x)

Now, Consider the Given Question :

✿  f(x) = cos[π]x + cos[πx]

$\bf{\implies f(\frac{\pi}{2}) = cos[\pi](\frac{\pi}{2}) + cos[\pi(\frac{\pi}{2})]}$

We know that : The Value of π is 3.14

As [x] = 3 for all 3 ≤ x ≤ 4

⇒ [π] = [3.14] = 3

$\bf{\implies f(\frac{\pi}{2}) = cos(\frac{3\pi}{2}) + cos[\pi(\frac{\pi}{2})]}$

$\bf{\implies f(\frac{\pi}{2}) = cos(180 + 90) + cos[\frac{(\pi)^2}{2}]}$

$\bf{\implies f(\frac{\pi}{2}) = -cos(90) + cos[\frac{(\pi)^2}{2}]}$

We know that : Value of π² = 9.869

$\bf{\implies \frac{(\pi)^2}{2} = (\frac{9.869}{2}) = 4.934}$

As [x] = 4 for all 4 ≤ x ≤ 5

$\bf{\implies [\frac{(\pi)^2}{2}] = [4.934] = 4}$

$\bf{\implies f(\frac{\pi}{2}) = cos4}$

Answer 2

[y] is the integral part of y

We know that π is 180° but that is the angular part.

For π to be an integer we need to consider π as 3.14

But 3.14 is a decimal value.

Round it of to the nearest integer.

3.14 ................................... 3

We have to take π as 3 in the expression.

So put the values :

f ( x ) = cos [ π ] x + cos [ π x ]

Finding f ( π / 2 )

f ( π / 2 ) = cos [ π ] π / 2 + cos [ π × π/2 ]

Within the braces [] π = 3

==> f ( π / 2 ) = cos 3 π / 2 + cos [ 3.14 × 3.14/2 ]

                     = cos 3 π / 2 + cos [ 9.something /2 ]

                     = cos 3 π / 2 + cos [ 4.something ]

But cos 3 π / 2 = cos 270° = cos ( 180 ° + 90° )

                                           = - cos 90°

                                           = 0

So : f ( π /2 ) = cos [ 4 . something ]

Round again to cos 4

Hence f ( π / 2 ) = cos 4

ANSWER :

Option : C

Hope it helps :-)

___________________________________________________________________________