JEE Main 30 June 2022 Shift 1 Solved Paper

Given,
S1={x∈[0,12π],sin‌5x+cos5x=1}
S2={x∈[0,8π],sin‌7x+cos7x=1}
(1) sin‌5x+cos5x=1
This satisfies when sin‌x=1 and cos‌x=0
∴x=‌

π

2

,‌

5π

2

,‌

9π

2

,‌

13π

2

,‌

17π

2

,‌

21π

2

It also satisfies when sin‌x=0 and cos‌x=1
∴x=0,2π,4π,6π,8π,10π,12π
∴‌ Accepted values of ‌x‌ in ‌[0,12π]‌ is ‌=13
∴n(S1)=13
(2) sin‌7x+cos7x=1
This satisfies when sin‌x=1 and cos‌x=0
For x∈[0,8π], possible values
x=‌

π

2

,‌

5π

2

,‌

9π

2

,‌

13π

2

It also satisfies when sin‌x=0 and cos‌x=1
x∈[0,8π], possible values
x=0,2π,4π,6π,8π
∴ Total accepted values of x in [0,8π] is =9
∴n(S2)=9
∴n(S1)−n(S2)=13−9=4