JEE (Advanced) 2014 Paper-1

Let a, b, c be positive integers such that $$\frac {b}{a}$$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $$\frac{a^2 + a - 14}{a + 1}$$ is

Let $$n \geq 2$$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is

Let $$n_1 < n_2 < n_3 < n_4 < n_5$$ be positive integers such that $$n_1 + n_2 + n_3 + n_4 + n_5 = 20.$$ Then the number of such distinct arrangements $$(n_1, n_2, n_3, n_4, n_5)$$ is

Let $$f: R \rightarrow R$$ and $$g: R \rightarrow R$$ be respectively given by $$f(x) = |x| + 1$$ and $$g(x) = x^2 + 1$$. Define $$h:R \rightarrow R$$ by $$h(x) = \begin{cases}max \left\{f(x), g(x)\right\} & if x \leq 0,\\min \left\{f(x), g(x)\right\} & if x > 0\end{cases}$$
The number of points at which h(x) is not differentiable is

The value of
$$\int_{0}^{1}4x^3 \left\{\frac{d^2}{dx^2} \left(1 - x^2\right)^5\right\} dx$$ is

The slope of the tangent to the curve $$(y − x^5)^2 = x(1 + x^2)^2$$ at the point (1, 3) is

The largest value of the nonnegative integer a for which
$$\lim_{x \rightarrow 1}\left\{\frac{-ax + \sin (x - 1) + a }{x + \sin (x - 1) - 1}\right\}^{\frac{1 - x}{1 - \sqrt x}} = \frac{1}{4}$$

Let $$f: [0, 4 \pi] \rightarrow [0, \pi]$$ be defined by $$f(x) = \cos^{-1} (\cos x).$$ The number of points $$x \in [0, 4 \pi]$$ satisfying the equation $$f(x) = \frac{10 - x}{10}$$ is

For a point P in the plane, let $$d_1 (P)$$ and $$d_2 (P)$$ be the distances of the point P from the lines x − y = 0 and x + y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane and satisfying $$2 \leq d_1 (P) + d_2 (P) \leq 4,$$ is

Let $$\overrightarrow{a}, \overrightarrow{b}$$ and $$\overrightarrow{c}$$ be three non-coplanar unit vectors such that the angle between every pair of them is $$\frac{\pi}{3}$$. If $$\overrightarrow{a} \times \overrightarrow{b} + \overrightarrow{b} \times \overrightarrow{c} = p\overrightarrow{a} + q\overrightarrow{b} + r \overrightarrow{c},$$ where p, q and r are scalars, then the value of $$\frac{p^2 + 2q^2 + r^2}{q^2}$$ is