JEE Advanced Model Paper 8 with solutions for online practice

(A) L.H.S. = 2−m

π 2

∫

0

(sin2x)m dx =

2−m

2

π

∫

0

(sint)m = 2−m

π∕2

∫

0

(sint)m dt = 2−m

π∕2

∫

0

(cost)m dt = RHS
(B) I =

π

∫

0

(π−x)sinx

1+cos2x

dx ⇒ 2I = π

π

∫

0

sinx

1+cos2x

dx r 2I = 2π

π∕2

∫

0

sinx

1+cos2x

dx
⇒ I = π

π∕2

∫

0

sinx

1+cos2x

dx = π

1

∫

0

dt

1+t2

=

π

2

1

∫

0

dt

1+t2

(C)

π∕2

∫

0

1+cos3x

1+2cosx

dx =

π∕2

∫

0

1+cos3x

2cosx−1

dx = 1
(D) I =

π∕2

∫

0

1+sin3x

1+2sinx

dx
Now , using

a

∫

0

f (a - x) dx =

a

∫

0

f(x) dx

π∕2

∫

0

1+cos3x

2cosx−1

dx =

π∕2

∫

0

(cos2x + cosx) dx = 1
∴

π∕2

∫

0

1+sin3x

1+2sinx

dx =

π∕2

∫

0

1+cos3x

2cosx−1

dx