Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration
Matrix Matching ( array l Column - I & Column-II (A) ₀^ 1 (1+x) (1+x^2 ) d x is equal to & (p) 3e array l (B) If f(x)=1+x+ ₁^x ( ^2 t+2 t ) d t. Find the value of f(a) where f^ (a)=0 array & (q) 1- 1 2 array l (C) Let I₁= ₀^1 e^x 1+x d x and I₂= ₀^1 x^2 e^ x^3 (2-x^3 ) d x, then I₁ I₂ is equal to array & (r) 1+2 e ⁻¹ array l (D) ₀^ / 4 (x-[x]) d(x-[x]) is equal to (where denote greatest integer function of x) array &
Options
- A( A ) ( p ) ;( B ) ( q ) ;( C ) ( s ) ;( D ) ( r )
- B( A ) ( q ) ;( B ) ( r ) ;( C ) ( p ) ;( D ) ( s )
- C( A ) ( s ) ;( B ) ( r ) ;( C ) ( p ) ;( D ) ( q )
- D( A ) ( r ) ;( B ) ( s ) ;( C ) ( q ) ;( D ) ( p )
Correct answer
C. ( A ) ( s ) ;( B ) ( r ) ;( C ) ( p ) ;( D ) ( q )
Step-by-step solution
(A) I = ₀^ 1 (1+ x ) (1+ x ^2 ) dx put x = aligned & I = ₀^ / 2 ^2 ~d (1+ ) ^2 = ₀^ / 2 1 1+ d & I= ₀^ / 2 + d & I= ₀^ / 2 + d & 2 I= ₀^ / 2 1 d = 2 ₀^a f(x) d x= ₀^a f(a-x) d x & I= 4 aligned (B) aligned & f(x)=1+x+ ₁^x ( ^2 t+2 t ) d t & f^ (x)=1+ ^2 x+2 x & f^ (a)=0 ( a+1)^2=0 a=e⁻¹ aligned So, f(a)=f (e⁻¹ )=1+e⁻¹+ ₁^ e⁻¹ ( ^2 t+2 t ) d t Put t=y e^y=t e^y d y=d t aligned & f(a)=1+e⁻¹+ ₀⁻¹ e^y (y^2+2 y ) d y & =1+e⁻¹+ [e^y y^2 ]₀⁻¹=1+e^1+e⁻¹ & f(a)=1+2 e⁻¹ aligned (C) aligned & I₂= 1 3 ₀^1 1 e^t(2-t) d t & P u t