NTA JEE Main 9th April 2014 Online

Let $$a$$ and $$b$$ be any two numbers satisfying $$\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4}$$. Then, the foot of perpendicular from the origin on the variable line $$\frac{x}{a} + \frac{y}{b} = 1$$ lies on:

If the point $$(1, 4)$$ lies inside the circle $$x^2 + y^2 - 6x + 10y + p = 0$$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $$p$$ is the interval:

If $$OB$$ is the semi-minor axis of an ellipse, $$F_1$$ and $$F_2$$ are its foci and the angle between $$F_1B$$ and $$F_2B$$ is a right angle, then the square of the eccentricity of the ellipse is:

If $$f(x)$$ is continuous and $$f\left(\frac{9}{2}\right) = \frac{2}{9}$$, then $$\lim_{x \to 0} f\left(\frac{1-\cos 3x}{x^2}\right)$$ equals to:

The contrapositive of the statement "I go to school if it does not rain" is:

In a set of $$2n$$ distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then, the mean of the new set of observations:

Let $$P$$ be the relation defined on the set of all real numbers such that $$P = \{(a, b) : \sec^2 a - \tan^2 b = 1\}$$. Then, $$P$$ is:

If $$B$$ is a $$3 \times 3$$ matrix such that $$B^2 = 0$$, then $$\det[(I + B)^{50} - 50B]$$ is equal to:

If $$a$$, $$b$$, $$c$$ are non-zero real numbers and if the system of equations
$$(a-1)x = y + z$$
$$(b-1)y = x + z$$
$$(c-1)z = x + y$$
has a non-trivial solution, then $$ab + bc + ca$$ equals:

If $$y = e^{nx}$$, then $$\frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2}$$ is equal to: