NTA JEE Main 8 April 2018 Offline

Let $$S = \{x \in R : x \geq 0$$ & $$2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0\}$$. Then S:

If $$\alpha, \beta \in C$$ are the distinct roots of the equation $$x^2 - x + 1 = 0$$, then $$\alpha^{101} + \beta^{107}$$ is equal to:

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series $$1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \ldots$$ If $$B - 2A = 100\lambda$$, then $$\lambda$$ is equal to:

Let $$a_1, a_2, a_3, \ldots, a_{49}$$ be in A.P. such that $$\sum_{k=0}^{12} a_{4k+1} = 416$$ and $$a_9 + a_{43} = 66$$. If $$a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m$$, then m is equal to:

The sum of the co-efficient of all odd degree terms in the expansion of $$\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$$, $$(x > 1)$$ is:

If sum of all the solutions of the equation $$8\cos x \cdot \left(\cos\left(\frac{\pi}{6} + x\right) \cdot \cos\left(\frac{\pi}{6} - x\right) - \frac{1}{2}\right) = 1$$ in $$[0, \pi]$$ is $$k\pi$$, then k is equal to:

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:

If the tangent at (1, 7) to the curve $$x^2 = y - 6$$ touches the circle $$x^2 + y^2 + 16x + 12y + c = 0$$ then the value of c is:

Tangent and normal are drawn at P(16, 16) on the parabola $$y^2 = 16x$$, which intersect the axis of the parabola at A & B, respectively. If C is the center of the circle through the points P, A & B and $$\angle CPB = \theta$$, then a value of $$\tan \theta$$ is: