NTA JEE Main 2nd April 2017 Offline

If, for a positive integer $$n$$, the quadratic equation,
$$x(x+1) + (x+1)(x+2) + \ldots + (x+\overline{n-1})(x+n) = 10n$$
has two consecutive integral solutions, then $$n$$ is equal to:

Let $$\omega$$ be a complex number such that $$2\omega + 1 = z$$ where $$z = \sqrt{-3}$$. If
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7} \end{vmatrix} = 3k$$,
then $$k$$ can be equal to:

A man $$X$$ has 7 friends, 4 of them are ladies and 3 are men. His wife $$Y$$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $$X$$ and $$Y$$ have no common friends. Then the total number of ways in which $$X$$ and $$Y$$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $$X$$ and $$Y$$ are in this party is:

For any three positive real numbers $$a$$, $$b$$ and $$c$$. If $$9(25a^{2} + b^{2}) + 25(c^{2} - 3ac) = 15b(3a + c)$$. Then

The value of $$(^{21}C_{1} - ^{10}C_{1}) + (^{21}C_{2} - ^{10}C_{2}) + \dots + (^{21}C_{10} - ^{10}C_{10})$$ is:

If $$5\tan^{2}x - \cos^{2}x = 2\cos 2x + 9$$, then the value of $$\cos 4x$$ is:

Let $$k$$ be an integer such that the triangle with vertices $$(k, -3k)$$, $$(5, k)$$ and $$(-k, 2)$$ has area 28 sq. units. Then the orthocenter of this triangle is at the point:

The radius of a circle, having minimum area, which touches the curve $$y = 4 - x^{2}$$ and the lines, $$y = |x|$$ is:

The eccentricity of an ellipse whose centre is at the origin is $$\frac{1}{2}$$. If one of its directrices is $$x = -4$$, then the equation of the normal to it at $$\left(1, \frac{3}{2}\right)$$ is:

A hyperbola passes through the point $$P(\sqrt{2}, \sqrt{3})$$ and has foci at $$( \pm 2, 0)$$. Then the tangent to this hyperbola at $$P$$ also passes through the point