Instructions

Let a, r, s, t be nonzero real numbers. Let $$P(at^2 , 2at), Q, R(ar^2 , 2ar)$$ and $$S(as^2 , 2as)$$ be distinct points on the parabola $$y^2 = 4ax$$. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0).

Question 51

Question 52

# If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

Instructions

Given that for each $$a \in (0, 1)$$
$$\lim_{h \rightarrow 0^+} \int_{h}^{1 - h}t^{-a}(1 - t)^{a - 1}dt$$
exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).

Question 53

Question 54

# The value of $$g^{'} \left(\frac{1}{2}\right)$$ is

Instructions

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $$x _i$$ be the number on the card drawn from the $$i^{th}$$ box, i = 1,2,3.

Question 55

Question 56

# The probability that $$x_1, x_2, x_3$$ are in an arithmetic progression, is

Instructions

For the following questions answer them individually

Question 57

Question 60

OR