JEE (Advanced) 2008 Paper-2
For the following questions answer them individually
Let a, b, c, p, q be real numbers. Suppose $$\alpha, \beta$$ are the roots of the equation $$x^2 + 2px + q = 0$$ and $$\alpha, \frac{1}{\beta}$$ are the roots of the equation $$ax^2 + 2bx + c = 0$$, where $$\beta^2 \notin \left\{-1, 0, 1\right\}$$.
STATEMENT-1 : $$\left(p^2 - q\right)\left(b^2 - ac\right)\geq 0$$
and
STATEMENT-2 : $$b \neq pa$$ or $$c \neq qa$$
Consider
$$L_1: 2x + 3y + p - 3 = 0$$
$$L_2: 2x + 3y + p + 3 = 0$$
where p is a real number, and $$C : x^2 + y^2 + 6x - 10y + 30 = 0$$.
STATEMENT-1: If line $$L_1$$ is a chord of circle C, then line $$L_2$$ is not always a diameter of circle C.
and
STATEMENT-2: If line $$L_1$$ is a diameterof circle C, then line $$L_2$$ is not a chord of circle C.
Letasolution y = y(x) of the differential equation
$$x\sqrt{x^2 - 1} dy - y\sqrt{y^2 - 1}dx = 0$$ satisfy $$y(2) = \frac{2}{\sqrt{3}}$$
STATEMENT-1: $$y(x) = \sec\left(\sec^{-1}x - \frac{\pi}{6}\right)$$
STATEMENT-2: y(x) is given by $$\frac{1}{y} = \frac{2\sqrt{3}}{x} - \sqrt{1 - \frac{1}{x^2}}$$
Consider the function $$f : (-\infty, \infty) \rightarrow (-\infty, \infty)$$ defined by
$$f(x) = \frac{x^2 - ax + 1}{x^2 + ax + 1}, 0 < a < 2$$.
Consider the lines
$$L_1:\frac{x+1}{3} = \frac{y+2}{1} = \frac{z+1}{2}$$
$$L_1:\frac{x-2}{1} = \frac{y+2}{2} = \frac{z-3}{3}$$
The distance of the point (1,1, 1) from the plane passing through the point (-1, -2, -1) and whose normalis perpendicularto both the lines $$L_1$$ and $$L_2$$ is
For the following questions answer them individually
Consider the lines given by
$$L_1: x + 3y - 5 = 0$$
$$L_2: 3x - ky - 1 = 0$$
$$L_3: 5x + 2y - 12 = 0$$
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.
789
456
123
0.-
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