JEE (Advanced) 2015 Paper-1

In $$R^3$$, consider the planes $$P_1 : y = 0$$ and $$P_2 : x + z = 1$$. Let $$P_3$$ be a plane, different from $$P_1$$ and $$P_2$$, which passes through the intersection of $$P_1$$ and $$P_2$$. If the distance of the point (0,1,0) from $$P_3$$ is 1 and the distance of a point $$(\alpha, \beta, \gamma)$$ from $$P_3$$ is 2, then which of the following relationsis (are) true?

In $$R^3$$ let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes $$P_1 : x + 2y - z + 1 = 0$$ and $$P_2 : 2x - y + z - 1 = 0.$$ Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane $$P_1$$. Which of the following points lie(s) on M ?

Let P and Q be distinct points on the parabola $$y^2 = 2x$$ such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle $$\triangle OPQ$$ is $$3 \sqrt 2$$, then which of the following is (are) the coordinates of P ?

Let y(x) be a solution of the differential equation $$(1 + e^x)y' + ye^x = 1$$. If y(0) = 2, then which of the following statements is (are) true?

Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented by the differential equation $$Py'' + Qy' + 1 = 0,$$ where P, Q are functions of x, y and y' (here $$y' = \frac{dy}{dx}, y'' = \frac{d^2y}{dx^2}$$), then which of the following statements is (are) true?

Let g: $$R \rightarrow R$$ be a differentiable function with g(0) = 0, g'(0) = 0 and $$g’(1) \neq 0$$. Let
$$f(x) = \begin{cases}\frac{x}{|x|}g(x) & x \neq 0\\0 & x = 0\end{cases}$$
and $$h(x) = e^{|x|}$$ for all $$x \in R$$. Let (f o h) (x) denote f(h(x)) and (h o f) (x) denote h(f(x)). Then which of the following is (are) true?

Let $$f(x) = \sin \left(\frac{\pi}{6}\sin\left(\frac{\pi}{2}\sin x\right)\right)$$ for all $$x \in R$$ and $$g(x) = \frac{\pi}{2} \sin x$$ for all $$x \in R$$. Let (f o g)(x) denote f(g(x)) and (g o f)(x) denote g(f(x)). Then which of the following is (are) true?

Let $$\triangle PQR$$ be a triangle. Let $$\overrightarrow{a} = \overrightarrow{QR}, \overrightarrow{b} = \overrightarrow{RP}$$ and $$\overrightarrow{c} = \overrightarrow{PQ}$$. If $$|\overrightarrow{a}| = 12, |\overrightarrow{b}| = 4 \sqrt 3$$ and $$\overrightarrow{b}. \overrightarrow{c}$$ then which of the following is (are) true?