For the following questions answer them individually
Let the locus of the mid-point of the chord through the origin O of the parabola $$y^{2}= 4x$$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 :1, is:
If $$\lim_{x \rightarrow 0} \frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x \cos x-\log_{e}{(1+x)}} =2$$, then $$a^{2}+b^{2}+c^{2}$$ is equal to :
Let f and g be functions satisfying f(x+ y) =f(x)f(y), f (l) =7 and g(x+ y) = g(xy), g(1) =1, for all $$x,y \epsilon N$$. If $$\sum_{x=1}^n \left(\frac{f(x)}{g(x)}\right) = 19607$$, then n is equal to:
Let n be the number obtained on rolling a fair die. If the probability that the system
x - ny + z = 6
x + (n - 2)y + (n + l)z = 8
(n - l)y + z = l
has a unique solution is $$\frac{k}{6}$$, then the sum of k and all possible values of n is:
Let $$P(10, 2\sqrt{5})$$ be a point on the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$, whose foci are Sand S'. if the length of its latus rectum is 8, then the square of the area of $$\Delta PSS'$$ is equal to:
Let $$S= \left\{z \in \mathbb{C}: 4z^{2}+ \overline{z}=0 \right\}$$. Then $$\sum_{z\in S} |z|^{2}$$ is equal to:
Among the statements
(S1) : If A(5, -1) and B(-2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (- 4,- 7) and
(S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines ax + by + c = 0 are concurrent at (2,-2),
lf the mean deviation about the median of the numbers k, 2k, 3k, ..... , 1000k is 500, then $$k^{2}$$ is equal to :
let $$\alpha, \beta$$ be the roots of the quadratic equation $$12x^{2}-20x+3\lambda=0, \lambda\in Z$$. If $$\frac{1}{2}\leq |\beta-\alpha|\leq\frac{3}{2}$$, then the sum of all possible values of $$\lambda$$ is :
Let the domain of the function f(x) = $$\log_{3}\log_{5}(7-\log_{2}(x^{2}-10x+85))+\sin^{-1}\left(|\frac{3x-7}{17-x}|\right)$$ be $$(\alpha, \beta)$$. Then $$\alpha + \beta$$ is equal to :
