IIT-JEE-Mathematics-Paper2-2008

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8.     Let

               .

        Then, for an arbitrary constant C, the value of J - I equals.

9.     Let g(x)&nbs    Let g(x)  = log f(x) where f(x) is a twice differentiable positive function on (0, ∞) such that f(x + 1) = x f(x). Then, for N = 1, 2, 3, ......,

10. Suppose four distinct positive numbers a₁, a₂, a₃, a₄ are in G.P. Let 
b₁ = a₁, b₂ = b₁ + a₂, b₃ = b₂ + a₃ and b₄ = b₃ + a₄.

        STATEMENT-1: The numbers b₁, b₂, b₃, b₄ are neither in A.P. nor in G.P.

        and

        STATEMENT-2: The numbers b₁, b₂, b₃, b₄ are in H.P.

(A)    Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.

(B)    Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.

(C)    Statement-1 is True, Statement-2 is False

(D)    Statement-1 is False, Statement-2 is True

11.    Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x² + 2px + q = 0 and α, 1/β are the roots of the equation ax² + 2bx + c = 0, where β² is not belongs to {-1, 0, 1}.

        STATEMENT-1: (p² - q)(b² - ac) > 0

        and

        STATEMENT-2: b ¹ pa or c ¹ qa

(A)    Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.

(B)    Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.

(C)    Statement-1 is True, Statement-2 is False

(D)    Statement-1 is False, Statement-2 is True

12.    Consider

                L₁ : 2x + 3y + p - 3 = 0

                L₂ : 2x + 3y + p + 3 = 0

        where p is a real number, and C : x² + y² + 6x + 10y + 30 = 0

STATEMENT-1: If line L₁ is a chord of circle C, then line L₂ is not always a diameter of circle C.

and

STATEMENT-2: If line L₁ is a diameter of circle C, then line L₂ is not a chord of circle C.

(A)    Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.

(B)    Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.

(C)    Statement-1 is True, Statement-2 is False

(D)    Statement-1 is False, Statement-2 is True

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