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JEE Advanced2006MathematicsDefinite IntegrationActual

Match the following.

Options

  1. A(i) (a), (ii) (b), (iii) (d), (iv) (c)
  2. B(i) (c), (ii) (a), (iii) (b), (iv) (d)
  3. C(i) (a), (ii) (d), (iii) (b), (iv) (c)
  4. D(i) (b), (ii) (a), (iii) (c), (iv) (d)

Correct answer

C. (i) (a), (ii) (d), (iii) (b), (iv) (c)

Step-by-step solution

(i) I= ₀^ / 2 ( x)^ x ( x x- ( x)^ x ) d x= ₀^ / 2 d d x ( x)^ x d x=1 (ii) The point of intersection of -4 y^2=x and x-1=-5 y^2 is (-4,-1) and (-4,1) . Hence, required area =2 ₀^1 (1-5 y^2 ) d y- ₀^1-4 y^2 d y = 4 3 . (iii) The point intersection of y=3^ x-1 x and y=x^x-1 is (1,0) . Hence, d y d x = 3^ x-1 x +3^ x-1 3 x ( d y d x )_ (1,0) =1 For ( d y d x )_ (1,0) =1 If is angle between the curves, then =0 . =0^ (iv) aligned & d y d x = 2 x+y d x d y - x 2 = y 2 & x e^ -y / 2 = 1 2 y e^ -y / 2 d y & x+y+2=k e^ y /

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