NTA JEE Main 11th April 2014 Online

A stair-case of length $$l$$ rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the staircase begins to slide on the floor, then the locus of P is:

Let P($$3\sec\theta, 2\tan\theta$$) and Q($$3\sec\phi, 2\tan\phi$$) where $$\theta + \phi = \frac{\pi}{2}$$, be two distinct points on the hyperbola $$\frac{x^2}{9} - \frac{y^2}{4} = 1$$. Then the ordinate of the point of intersection of the normals at P and Q is:

If $$\lim_{x \to 2} \frac{\tan(x - 2)\{x^2 + (k+2)x - 2k\}}{x^2 - 4x + 4} = 5$$, then k is equal to:

The proposition $$\sim (p \vee \sim q) \vee \sim (p \vee q)$$ is logically equivalent to:

Two ships A and B are sailing straight away from a fixed point O along routes such that $$\angle AOB$$ is always 120°. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr):

The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be $$\alpha$$. After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to $$\beta$$. Then the height (in metres) of the tower is:

Let A(2, 3, 5), B($$-1, 3, 2$$) and C($$\lambda, 5, \mu$$) be the vertices of a $$\triangle$$ABC. If the median through A is equally inclined to the coordinate axes, then:

Let A be a $$3 \times 3$$ matrix such that
$$A\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$$
Then A$$^{-1}$$ is:

Let for i = 1, 2, 3, $$p_i(x)$$ be a polynomial of degree 2 in $$x$$, $$p'_i(x)$$ and $$p''_i(x)$$ be the first and second order derivatives of $$p_i(x)$$ respectively. Let,
$$A(x) = \begin{bmatrix} p_1(x) & p'_1(x) & p''_1(x) \\ p_2(x) & p'_2(x) & p''_2(x) \\ p_3(x) & p'_3(x) & p''_3(x) \end{bmatrix}$$
and $$B(x) = [A(x)]^T A(x)$$. Then determinant of B(x):

Let f be an odd function defined on the set of real numbers such that for $$x \geq 0$$, $$f(x) = 3\sin x + 4\cos x$$. Then $$f(x)$$ at $$x = -\frac{11\pi}{6}$$ is equal to: