JEE (Advanced) 2016 Paper-1

The least value of $$\alpha \in R$$ for which $$4 \alpha x^2 + \frac{1}{x} \geq 1$$, for all x > 0, is

Consider a pyramid OPQRS located in the first octant $$(x \geq 0, y \geq 0, z \geq 0)$$ with O as origin, and OP and ORalongthe x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP =3. Thepoint S is directly above the mid-point T of diagonal OQ such that TS = 3. Then

Let $$f : (0, \infty) \rightarrow R$$ be a differentiable function such that $$f'(x) = 2 - \frac{f(x)}{x}$$ for all $$x \in (0, \infty)$$ and $$f(x) \neq 1$$. Then

Let $$P = \begin{bmatrix}3 & -1 & -2 \\2 & 0 & \alpha \\3 & -5 & 0 \end{bmatrix}$$ where $$\alpha \in R$$. Suppose $$Q = [q_{ij}]$$ is a matrix such that $$PQ = kI$$, where $$k \in R, k \neq 0$$ and $$I$$ is the identity matrix of order 3. If $$q_{23} = -\frac{k}{8}$$ and $$det(Q) = \frac{k^2}{2}$$, then

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and $$2s = x + y + z$$. If $$\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2}$$ andareaofincircle of the triangle XYZ is $$\frac{8 \pi}{3}$$, Then

A solution curve of the differential equation $$(x^2 + xy + 4x + 2y + 4)\frac{dy}{dx} - y^2 = 0, x > 0$$, passes through the point (1, 3). Then the solution curve

Let $$f : R \rightarrow R, g : R \rightarrow R$$ and $$h : R \rightarrow R$$ be differentiable functions such that $$f(x) = x^3 + 3x + 2, g(f(x)) = x$$ and $$h(g(g(x))) = x$$ for all $$x \in R$$. Then

The circle $$C_1: x^2 + y^2 = 3$$, with centre at O, intersects the parabola $$x^2 = 2y$$ at the point P in the first quadrant. Let the tangent to the circle $$C_1$$, at P touches other two circles $$C_2$$ and $$C_3$$ at $$R_2$$ and $$R_3$$, respectively. Suppose $$C_2$$ and $$C_3$$ have equal radii $$2\sqrt{3}$$ and centres $$Q_2$$, and $$Q_3$$, respectively. If $$Q_2$$ and $$Q_3$$ lie on the y-axis, then

Let RS be the diameter of the circle $$x^2 + y^2 = 1$$, where S is the point (1,0). Let P be a variable point (other than R and S) on thecircle and tangents to the circle at S and P meet at the point Q@. The normal to the circle at P intersects a line drawn through Q parallel to RS at point £. Then thelocus of £ passes through the point(s)

The total number of distinct $$x \in R$$ for which