UPSEE Solved Model Paper 7

Given
f (x) = cos[π2]x+cos[−π2x]
As π = 3.141
⇒ π2 = 9.8 (approx)
Using definition of greatest integer function
[π2] = [9.8] = 9 and [−π2] = [- 9.8] = - 10
∴ f (x) = cos 9x + cos (- 10) x
= cos 9x + cos 10x [Since cos (- θ) = cos θ]
f(

π

4

) = cos

9π

4

+cos

10π

4

= cos (2π+

π

4

) + cos (2π+

2π

4

)
= cos

π

4

+ cos

π

2

=

1

√2

+0 =

1

√2

Similarly,
f (- π) = cos (- π) + cos (- 10π)
= cos 9π + cos 10π
= - 1 + 1 = 0
Also
f (π) = cos -π + cos 10π
= - 1 + 1 = 0
f(

π

2

) = cos

9π

2

+ cos

10π

2

= cos (4π+

π

2

) + cos 5π
= cos

π

2

- 1 = 0 - 1 = - 1
Hence, option 'D' is correct.