JEE (Advanced) 2013 Paper-2
A box $$B_1$$ contains 1 white ball, 3 red balls and 2 black balls. Another box $$B_2$$, contains 2 white balls, 3 red balls and 4 black balls. A third box $$B_3$$ contains 3 white balls, 4 red balls and 5 black balls.
If 1 ball is drawn from each of the boxes $$B_1, B_2$$, and $$B_3$$, the probability that all 3 drawn balls are of the same colour is
If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the otherball is red, the probability that these 2 balls are drawn
from box $$B_2$$ is
Let $$f : [0, 1] \rightarrow R$$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0) = f(1) = 0 and satisfies $$f"(x) - 2f'(x) + f(x) \geq e^x, x \in [0, 1]$$.
If the function $$e^{-x} f(x)$$ assumes its minimum in the interval [0, 1] at $$x = \frac{1}{4}$$ which of the following is true ?
Let PQ be a focal chord of the parabola $$y^2 = 4ax$$. The tangents to the parabola at P and Q meet at a point lying on the line $$y = 2x + a, a > 0.$$
If chord PQ subtends an angle $$\theta$$ at the vertex of $$y^2 = 4ax$$, then $$\tan \theta =$$
For the following questions answer them individually
A line L : y = mx + 3 meets y-axis at E(0, 3) and the arc of the parabola $$y^2 = 16 x, 0 \leq y \leq 6$$ at the point $$F(x_0, y_0)$$. The tangent to the parabola at $$F(x_0, y_0)$$ intersects the y-axis at $$G(0, y_1)$$. The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List I and List II and select the correct answer using the code given below the lists:
Match List I with List II and select the correct answer using the code given below the lists:
Consider the lines $$L_1 : \frac{x - 1}{2} = \frac{y}{-1} = \frac{z + 3}{1}, L_2 : \frac{x - 4}{1} = \frac{y+3}{1} = \frac{z + 3}{2}$$ and the planes $$P_1 : 7x + y + 2z = 3, P_2 : 3x + 5y - 6z = 4$$. Let ax + by + cz = d be the equation of the plane pasing through the point of intersection of lines $$L_1$$ and $$L_2$$, and perpendicular to planes $$P_1$$ and $$P_2$$.
Match List — I with List — II and select the correct answer using the code given below the lists :
Match List - I with List - II and select the correct answer using the code given below the lists:
