NTA JEE Main 4th April 2015 Offline

Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $$x^2 = 8y$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the ratio 1 : 3, then the locus of $$P$$ is

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $$\frac{x^2}{9} + \frac{y^2}{5} = 1$$, is

$$\lim_{x \to 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x} =$$

The negation of $$\sim s \vee (\sim r \wedge s)$$ is equivalent to

The mean of a data set comprising of 16 observations is 16. If one of the observation value 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is

If the angles of elevation of the top of a tower from three collinear points $$A$$, $$B$$ and $$C$$ on a line leading to the foot of the tower are $$30^\circ$$, $$45^\circ$$ and $$60^\circ$$ respectively, then the ratio $$AB : BC$$, is

If $$A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$$ is a matrix satisfying the equation $$AA^T = 9I$$, where $$I$$ is $$3 \times 3$$ identity matrix, then the ordered pair $$(a, b)$$ is equal to

The set of all values of $$\lambda$$ for which the system of linear equations:
$$2x_1 - 2x_2 + x_3 = \lambda x_1$$
$$2x_1 - 3x_2 + 2x_3 = \lambda x_2$$
$$-x_1 + 2x_2 = \lambda x_3$$
has a non-trivial solution,

Let $$\tan^{-1} y = \tan^{-1} x + \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$$, where $$|x| \lt \frac{1}{\sqrt{3}}$$. Then a value of $$y$$ is

If the function $$g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$$ is differentiable, then the value of $$k + m$$ is