NTA JEE Main 11th January 2019 Shift 1

If one real root of the quadratic equation $$81x^2 + kx + 256 = 0$$ is cube of the other root, then a value of k is:

Let $$\left(-2 - \frac{1}{3}i\right)^3 = \frac{x+iy}{27}$$ $$(i = \sqrt{-1})$$, where $$x$$ and $$y$$ are real numbers then $$y - x$$ equals

Let $$a_1, a_2, \ldots, a_{10}$$ be a G.P. If $$\frac{a_3}{a_1} = 25$$, then $$\frac{a_9}{a_5}$$ equals:

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $$\frac{27}{19}$$. Then the common ratio of this series is:

The sum of the real values of $$x$$ for which the middle term in the binomial expansion of $$\left(\frac{x^3}{3} + \frac{3}{x}\right)^8$$ equals 5670 is:

The value of $$r$$ for which $${}^{20}C_r \cdot {}^{20}C_0 + {}^{20}C_{r-1} \cdot {}^{20}C_1 + {}^{20}C_{r-2} \cdot {}^{20}C_2 + \ldots + {}^{20}C_0 \cdot {}^{20}C_r$$ is maximum, is:

Let $$f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$$ for $$k = 1, 2, 3, \ldots$$. Then for all $$x \in R$$, the value of $$f_4(x) - f_6(x)$$ is equal to:

In a triangle, the sum of lengths of two sides is $$x$$ and the product of the lengths of the same two sides is $$y$$. If $$x^2 - c^2 = y$$, where $$c$$ is the length of the third side of the triangle, then the circumradius of the triangle is

A square is inscribed in the circle $$x^2 + y^2 - 6x + 8y - 103 = 0$$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:

Two circles with equal radii are intersecting at the points (0,1) and (0,-1). The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is: