JEE (Advanced) 2012 Paper-2

The equation of a plane passing through the line of intersection of the planes $$x + 2y + 3z = 2$$ and $$x - y + z = 3$$ and at a distance $$\frac{2}{\sqrt{3}}$$ from the point (3, 1, -1) is

If $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ are vectors such that $$\mid \overrightarrow{a} + \overrightarrow{b} \mid = \sqrt{29}$$ and $$\overrightarrow{a} \times (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = (2 \hat{i} + 3\hat{j} + 4 \hat{k}) \times \overrightarrow{b}$$, then a possible value of $$(\overrightarrow{a} + \overrightarrow{b}).(-7 \hat{i} + 2 \hat{j} + 3 \hat{k})$$ is

Let PQR be a triangle of area $$\triangle$$ with $$a = 2, b = \frac{7}{2}$$ and $$c = \frac{5}{2}$$, where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then $$\frac{2 \sin P - \sin 2P}{2 \sin P + \sin 2P}$$ equals

Four fair dice $$D_1, D_2, D_3$$ and $$D_4$$ each having six faces numbered 1, 2, 3, 4, 5 and 6 are rolled simultaneously. The probability that $$D_4$$ shows a number appearing on one of $$D_1, D_2$$ and $$D_3$$ is

The value of the integral $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^2 + \ln \frac{\pi + x}{\pi - x}\right)\cos x dx$$ is

If P is a $$3 \times 3$$ matrix such that $$P^T = 2P + I$$, where $$P^T$$ is the transpose of P and I is the $$3 \times 3$$ identity matrix, then there exists a column matrix $$X = \begin{bmatrix}x \\y \\z\end{bmatrix} \neq \begin{bmatrix}0 \\0 \\0\end{bmatrix}$$ such that

Let $$a_1, a_2, a_3,....$$ be in harmonic progression with $$a_1 = 5$$ and $$a_{20} = 25$$. The least positive integer n for which $$a_n < 0$$ is

Let $$\alpha(a)$$ and $$\beta(a)$$ be the roots of the equation $$\left(\sqrt[3]{1 + a} - 1\right)x^2 + \left(\sqrt{1 + a} - 1\right)x + \left(\sqrt[6]{1 + a} - 1\right) = 0$$ where a > -1. Then $$\lim_{a \rightarrow 0^+}\alpha(a)$$ and $$\lim_{a \rightarrow 0^+}\beta(a)$$ are

Which of the following is true ?

Consider the statements :
P : There exists some $$x \in IR$$ such that $$f(x) + 2x = 2(1 + x^2)$$
Q : There exists some $$x \in IR$$ such that $$2f(x) + 1 = 2x(1 + x)$$ Then