IIT-JEE-Mathematics-Mains-2003

MAINS
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6. For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is P. If he fails in one of the exams then the probability of his passing in the next exam is P/2 otherwise it remains the same. Find the probability that he will qualify.

7. For the circle x² + y² = r², find the value of r for which the area enclosed by the tangents drawn from the point P(6, 8) to the circle and the chord of contact is maximum.

8. Prove that there exists no complex number z such that |z| < 1/3 and

∑_r=1ⁿ a_r z^r =1 where |a_r| < 2.

9. A is targeting to B, B and C are targeting to A. Probability of hitting the target by A, B and C are 2/3,1/2 and 1/3 respectively. If A is hit then find the probability that B hits the target and C does not.

10. If a function f : [–2a, 2a] --> R is an odd function such that f(x) = f(2a – x) for x Î [a, 2a[ and the left hand derivative at x = a is 0 then find the left hand derivative at x = –a.

11. Using the relation 2(1 – cos x) < x², x ¹ 0 or otherwise, prove that sin (tan x) > x ∀ x Î [0,π/4].

12. If a, b, c are in A.P., a², b², c² are in H.P., then prove that either a = b = c or a, b, -c/2 form a G.P.

13. If x² + (a – b)x + (1 – a – b) = 0 where a, b Î R then find the values of a for which equation has unequal real roots for all values of b.

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IIT-JEE-Mathematics-Mains-2003

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