NTA JEE Main 10th January 2019 Shift 1

Consider the quadratic equation $$(c-5)x^2 - 2cx + (c-4) = 0$$, $$c \neq 5$$. Let $$S$$ be the set of all integral values of $$c$$ for which one root of the equation lies in the interval $$(0, 2)$$ and its other root lies in the interval $$(2, 3)$$. Then the number of elements in $$S$$ is:

Let $$z_1$$ and $$z_2$$ be any two non-zero complex numbers such that $$3|z_1| = 4|z_2|$$. If $$z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$$ then maximum value of $$|z|$$ is:

If 5, 5$$r$$, 5$$r^2$$ are the lengths of the sides of a triangle, then $$r$$ can not be equal to:

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:

If $$\sum_{i=1}^{20} \left(\frac{^{20}C_{i-1}}{^{20}C_i + ^{20}C_{i-1}}\right)^3 = \frac{k}{21}$$, then $$k$$ equals:

If the third term in the binomial expansion of $$(1 + x^{\log_2 x})^5$$ equals 2560, then a possible value of $$x$$ is:

The sum of all values of $$\theta \in (0, \frac{\pi}{2})$$ satisfying $$\sin^2 2\theta + \cos^4 2\theta = \frac{3}{4}$$ is:

If the line $$3x + 4y - 24 = 0$$ intersects the $$x$$-axis at the point $$A$$ and the $$y$$-axis at the point $$B$$, then the incentre of the triangle $$OAB$$, where $$O$$ is the origin, is:

A point $$P$$ moves on the line $$2x - 3y + 4 = 0$$. If $$Q(1, 4)$$ and $$R(3, -2)$$ are fixed points, then the locus of the centroid of $$\triangle PQR$$ is a line:

If a circle $$C$$ passing through the point $$(4, 0)$$ touches the circle $$x^2 + y^2 + 4x - 6y = 12$$ externally at the point $$(1, -1)$$, then the radius of $$C$$ is: