IIT-JEE-Mathematics–Mains-2005

Mains
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1. A person goes to office either by car, scooter, bus or train probability of which being 1/7,3/7,2/7 and 1/7 respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is 2/9,1/9,4/9 and 1/9 respectively. Given that he reaches office in time, then what is the probability that he travelled by a car?

2. Find the range of values of t for which

   2 sin t = (1-2x+5x²)/(3x²-2x-1),t Î [-π/2,π/2].

3. Circles with radii 3, 4 and 5 touch each other externally if P is the point of intersection of tangents to these circles at their points of contact. Find the distance of P from the point of contact.

4. Find the equation of the plane containing the line 2x – y + z – 3 = 0,
3x + y + z = 5 and at a distance of 1/√6 from the point (2, 1 – 1).

5. If |f(x₁) – f(x₂)| < (x₁ – x₂)², for all x₁, x₂ Î R. Find the equation of tangent to

the curve y = f(x) at the point (1, 2).

6. If total number of runs scored in n matches is ((n+1)/4) (2ⁿ⁺¹ – n – 2) where n > 1, and the runs scored in the kth match are given by k. 2^n+1–k, where 1 < k < n. Find n.

7. The area of the triangle formed by the intersection of a line parallel to x-axis and passing through (h, k) with the lines y = x and x + y = 2 is 4h². Find the locus of point P.

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IIT-JEE-Mathematics–Mains-2005

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