JEE Main 11th January 2019 Shift 2

Let $$\alpha$$ and $$\beta$$ be the roots of the quadratic equation $$x^2 \sin\theta - x(\sin\theta \cos\theta + 1) + \cos\theta = 0$$ $$(0 < \theta < 45°)$$, and $$\alpha < \beta$$. Then $$\sum_{n=0}^{\infty}\left(\alpha^n + \frac{(-1)^n}{\beta^n}\right)$$ is equal to:

Let z be a complex number such that $$|z| + z = 3 + i$$ (where $$i = \sqrt{-1}$$). Then $$|z|$$ is equal to:

If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is:

Let $$S_n = 1 + q + q^2 + \ldots + q^n$$ and $$T_n = 1 + \left(\frac{q+1}{2}\right) + \left(\frac{q+1}{2}\right)^2 + \ldots + \left(\frac{q+1}{2}\right)^n$$ where q is a real number and $$q \neq 1$$. If $${}^{101}C_1 + {}^{101}C_2 \cdot S_1 + \ldots + {}^{101}C_{101} \cdot S_{100} = \alpha T_{100}$$, then $$\alpha$$ is equal to:

Let $$(x + 10)^{50} + (x - 10)^{50} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{50} x^{50}$$, for all $$x \in R$$; then $$\frac{a_2}{a_0}$$ is equal to:

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1,2), (3,4) and (2,5), then the equation of the diagonal AD is:

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is:

If the area of the triangle whose one vertex is at the vertex of the parabola, $$y^2 + 4(x - a^2) = 0$$ and the other two vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is:

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is: