NTA JEE Main 10th April 2015 Online

Let the tangents drawn to the circle, $$x^2 + y^2 = 16$$ from the point $$P(0, h)$$ meet the x-axis at points $$A$$ and $$B$$. If the area of $$\Delta APB$$ is minimum, then positive value of $$h$$ is:

If the tangent to the conic, $$y - 6 = x^2$$ at $$(2, 10)$$ touches the circle, $$x^2 + y^2 + 8x - 2y = k$$ (for some fixed $$k$$) at a point $$(\alpha, \beta)$$; then $$(\alpha, \beta)$$ is

An ellipse passes through the foci of the hyperbola, $$9x^2 - 4y^2 = 36$$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $$\frac{1}{2}$$, then which of the following points does not lie on the ellipse?

$$\lim_{x \to 0} \frac{e^{x^2} - \cos x}{\sin^2 x}$$ is equal to

The contrapositive of the statement "If it is raining, then I will not come", is

A factory is operating in two shifts, day and night, with 70 and 30 workers, respectively. If per day mean wage of the day shift workers is Rs. 54 and per day mean wage of all the workers is Rs. 60, then per day mean wage of the night shift workers (in Rs.) is:

In a certain town, 25% of the families own a phone and 15% own a car; 65% families own neither a phone nor a car and 2000 families own both a car and a phone. Consider the following three statements:
(i) 5% families own both a car and a phone.
(ii) 35% families own either a car or a phone.
(iii) 40000 families live in the town.
Then,

If $$A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$, then which one of the following statements is not correct?

The least value of the product $$xyz$$ (such that $$x$$, $$y$$ and $$z$$ are positive real numbers) for which the determinant $$\begin{vmatrix} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{vmatrix}$$ is non-negative is

If $$f(x) = 2\tan^{-1} x + \sin^{-1}\left(\frac{2x}{1+x^2}\right)$$, $$x > 1$$, then $$f(5)$$ is equal to