JEE Main 12th January 2019 Shift 1

If $$\lambda$$ be the ratio of the roots of the quadratic equation in x, $$3m^2x^2 + m(m-4)x + 2 = 0$$, then the least value of m for which $$\lambda + \frac{1}{\lambda} = 1$$, is:

If $$\frac{z - \alpha}{z + \alpha}$$ $$(\alpha \in R)$$ is a purely imaginary number and $$|z| = 2$$, then a value of $$\alpha$$ is:

Let $$S = \{1, 2, 3, \ldots, 100\}$$, then number of non-empty subsets A of S such that the product of elements in A is even is:

Consider three boxes, each containing 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by $$n_i$$, the label of the ball drawn from the $$i^{th}$$ box, $$(i = 1, 2, 3)$$. Then, the number of ways in which the balls can be chosen such that $$n_1 < n_2 < n_3$$ is:

Let $$S_k = \frac{1+2+3+\ldots+k}{k}$$. If $$S_1^2 + S_2^2 + \ldots + S_{10}^2 = \frac{5}{12}A$$, then A is equal to:

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P., then the sum of the original three terms of the given G.P. is:

A ratio of the 5$$^{th}$$ term from the beginning to the 5$$^{th}$$ term from the end in the binomial expansion of $$\left(2^{1/3} + \frac{1}{2(3)^{1/3}}\right)^{10}$$ is

The maximum value of $$3\cos\theta + 5\sin\left(\theta - \frac{\pi}{6}\right)$$ for any real value of $$\theta$$ is:

If the straight line $$2x - 3y + 17 = 0$$ is perpendicular to the line passing through the points $$(7, 17)$$ and $$(15, \beta)$$, then $$\beta$$ equals:

Let $$C_1$$ and $$C_2$$ be the centres of the circles $$x^2 + y^2 - 2x - 2y - 2 = 0$$ and $$x^2 + y^2 - 6x - 6y + 14 = 0$$ respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $$PC_1QC_2$$ is: