NTA JEE Main 8th April 2017 Online

Consider an ellipse, whose center is at the origin and its major axis is along the $$x$$-axis. If its eccentricity is $$\frac{3}{5}$$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

$$\displaystyle\lim_{x \to 3} \frac{\sqrt{3x-3}}{\sqrt{2x-4} - \sqrt{2}}$$ is equal to:

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

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is:

Let $$A$$ be any $$3 \times 3$$ invertible matrix. Then which one of the following is not always true?

The number of real values of $$\lambda$$ for which the system of linear equations, $$2x + 4y - \lambda z = 0$$, $$4x + \lambda y + 2z = 0$$ and $$\lambda x + 2y + 2z = 0$$, has infinitely many solutions, is:

If $$S = \left\{x \in [0, 2\pi] : \begin{vmatrix} 0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{vmatrix} = 0 \right\}$$, then $$\displaystyle\sum_{x \in S} \tan\left(\frac{\pi}{3} + x\right)$$ is equal to:

The value of $$\tan^{-1}\left[\frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}}\right]$$, $$|x| \lt \frac{1}{2}$$, $$x \neq 0$$, is equal to:

Let $$f(x) = 2^{10}x + 1$$ and $$g(x) = 3^{10}x - 1$$. If $$(fog)(x) = x$$, then $$x$$ is equal to:

If $$y = \left[x + \sqrt{x^2-1}\right]^{15} + \left[x - \sqrt{x^2-1}\right]^{15}$$, then $$(x^2-1)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$$ is equal to: