JEE (Advanced) 2007 Paper-1

The number of distinct real values of $$\lambda$$, for which the vectors $$-\lambda^2 \hat{i} + \hat{j} + \hat{k}, \hat{i} - \lambda^2 \hat{j} + \hat{k}$$ and $$\hat{i} + \hat{j} - \lambda^2 \hat{k}$$ are coplanar, is

A man walks a distance of 3 units from the origin towards the north-east $$(N 45^\circ E)$$ direction. From there, he walks a distance of 4 units towards the north-west $$(N 45^\circ W)$$ direction to reach a point P. Then the position of P in the Argand plane is

The number of solutions of the pair of equations
$$2 \sin^2 \theta - \cos 2 \theta = 0$$
$$2 \cos^2 \theta - 3 \sin \theta = 0$$
in the interval $$[0, 2 \pi]$$ is

Let $$H_1, H_2, ...., H_n$$ be mutually exclusive and exhaustive events with $$P(H_i) > 0, i = 1, 2, 3, ...., n$$. Let E be any other event with $$0 < P(E) < 1$$.
STATEMENT-1: $$P(H_i \mid E) > P(E \mid H_i).P(H_i)$$ for i = 1, 2, ..., n.
because
STATEMENT-2: $$\sum_{i=1}^n P(H_i) = 1$$

Tangents are drawn from the point (17, 7) to the circle $$x^2 + y^2 = 169$$.
STATEMENT-1: The tangents are mutually perpendicular.
because
STATEMENT-2: The locus of the points from which mutually perpendicular tangents can be drawn to the givencircle is $$x^2 + y^2 = 338$$.

Let the vectors $$\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RS}, \overrightarrow{ST}, \overrightarrow{TU}$$ and $$\overrightarrow{UP}$$ represent the sides of a regular
hexagon.
STATEMENT-1: $$\overrightarrow{PQ}\times \left(\overrightarrow{RS} + \overrightarrow{ST}\right) \neq \overrightarrow{0}.$$
because
STATEMENT-2: $$\overrightarrow{PQ}\times \overrightarrow{RS} = \overrightarrow{0}$$ and $$\overrightarrow{PQ}\times \overrightarrow{ST} \neq \overrightarrow{0}$$

Let $$F(x)$$ be an indefinite integral of $$\sin^2 x$$.
STATEMENT-1 : The function $$F(x)$$ satisfies $$F(x + \pi) = F(x)$$ for all real x.
because
STATEMENT-2: $$\sin^2 (x + \pi) = \sin^2 x$$ for all real x.

The sum $$V_1 + V_2 + ... + V_n$$ is

$$T_r$$ is always

Which one of the following is a correct statement?