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If [x] denotes the greatest integer less than or equal to x, then the value of the integral ∫ − π / 2 π / 2 [ [ x ] − s i n x ] d x  is equal to:

Option 1 -

-π

Option 2 -

0

Option 3 -

1

Option 4 -

π

0 3 Views | Posted 2 months ago
Asked by Shiksha User

  • 1 Answer

  • V

    Answered by

    Vishal Baghel | Contributor-Level 10

    2 months ago
    Correct Option - 1


    Detailed Solution:

      l = ∫ − π 2 π 2 ( [ x ] + [ − s i n x ] ) d x . (i)

    I = ∫ − π 2 π 2 ( [ − x ] + [ s i n x ] ) d x . (ii)

    by using property ( ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x )

    Adding (i) and (ii) we get 2l = ∫ − π 2 π 2 ( − 2 ) d x = − 2 π ⇒ l = − π

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l n , m = ∫ 0 1 2 x n x m − 1 d x , m , n ∈ N , n > m

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| A | = ( 1 3 2 ) 3 | B | = 1 1 0 5 . 2 1 9

If α + β + = 2π, then the system of equations

x + ( c o s γ ) y + ( c o s β ) z = 0        

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( c o s β ) x + ( c o s α ) y + z = 0

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α + β + γ = 2 π

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= 1 − c o s 2 α − c o s 2 β − c o s 2 γ + 2 c o s α . c o s β . c o s γ

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If the system of linear equations

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has infinitely many solution, then α + β - αβ is equal to__________
V
Vishal Baghel

Given 2x + y – z = 3         . (i)

x – y – z = α        . (ii)

3x + 3y + βz = 3                . (iii)

(i) x 2 – (ii) – (iii) – (1 + β) z = 3 - α

For infinite solution 1 + β = 0 = 3 - α

=> α = 3, β = -1

So, α + β - αβ = 5

Let S 1 = { z 1 ∈ C : | z 1 − 3 | = 1 2 }  and S 2 = { z 2 ∈ C : | z 2 − | z 2 + 1 | | = | z 2 + | z 2 − 1 | | } .  Then, for z 1 ∈ S 1 and s 2 ∈ S 2 ,  the least value of   | z 2 − z 1 | i s :

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Vishal Baghel

  S 1 = { z , ∈ c : ⌊ z 1 − 3 ⌋ = 1 2 } a n d     S 2 = { z 2 ∈ c : | z 2 − | z 2 + 1 | | = | z 2 + | z 2 − 1 | | }

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also | z 1 − 3 | = 1 2  lie on circle having centre 3 and radius   1 2

             

Clearly | z 1 − z 2 | m i n = 5 2 − 1 = 3 2

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