JEE (Advanced) 2008 Paper-1
For the following questions answer them individually
Let f and g be real valued functions defined on interval (-1, 1) such that $$g''(x)$$ is continuous, $$g(0) \neq 0, g'(0) = 0, g''(0) \neq 0$$, and $$f(x) = g(x)\sin x$$.
STATEMENT-1: $$\lim_{x \rightarrow 0}[g(x)\cot x - g(0) \cosec x] =f''(0).$$
STATEMENT-2: $$f'(0) = g(0)$$.
Consider three planes
$$P_1 : x - y + z = 1$$
$$P_2 : x + y - z = -1$$
$$P_3 : x - 3y + 3z = 2$$.
Let $$L_1, L_2, L_3$$ be the lines of interaction of the planes $$P_2$$ and $$P_3, P_3$$ and $$P_1, $$ and $$P_1$$ and $$P_2$$, respectively.
STATEMENT-1: At least two of the lines $$L_1, L_2$$, and $$L_3$$ are non-parallel.
and
STATEMENT-2: The three planes do not have a common point.
Consider the system of equations
$$x - 2y + 3z = -1$$
$$-x + y - 2z = k$$
$$x - 3y + 4z = 1$$.
STATEMENT-1: The system of equations has no solution for $$k \neq 3$$.
and
STATEMENT-2: The determinant
$$\neq 0$$, for $$k \neq 3$$.
Consider the system of equations
$$ax + by = 0,cx + dy = 0$$, where $$a, b, c, d \in \left\{0, 1\right\}$$.
STATEMENT-1: The probability that the system of equations has a unique solution is $$\frac{3}{8}$$
and
STATEMENT-2: The probability that the system of equations hasa solution is 1.
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation $$\sqrt{3}x + y - 6 = 0$$ and the point D is $$\left(\frac{3\sqrt{3}}{2}, \frac{3}{2}\right)$$. Further, it is given that the origin and the centre of C are on the same side of the line PQ.
Consider the functions defined implicitly by the equation $$y^2 - 3y + x = 0$$ on various intervals in the real line. If $$x \in (-\infty, -2) \bigcup (2, \infty)$$, the equation implicitly defines a unique real valued differentiable function y = f(x). If $$x \in (-2, 2)$$, the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.
The area of the region bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b, where $$-\infty < a < b < -2$$, is
