NTA JEE Main 10th April 2019 Shift 1

If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation $$x^2 + x\sin\theta - 2\sin\theta = 0$$, $$\theta \in \left(0, \frac{\pi}{2}\right)$$, then $$\frac{\alpha^{12} + \beta^{12}}{(\alpha^{-12} + \beta^{-12})\cdot(\alpha - \beta)^{24}}$$ is equal to:

If $$a > 0$$ and $$z = \frac{(1+i)^2}{a-i}$$, has magnitude $$\sqrt{\frac{2}{5}}$$, then $$\bar{z}$$ is equal to:

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is:

If $$a_1, a_2, a_3, \ldots, a_n$$ are in A.P. and $$a_1 + a_4 + a_7 + \ldots + a_{16} = 114$$, then $$a_1 + a_6 + a_{11} + a_{16}$$ is equal to:

The sum $$\frac{3 \times 1^3}{1^2} + \frac{5 \times (1^3 + 2^3)}{1^2 + 2^2} + \frac{7 \times (1^3 + 2^3 + 3^3)}{1^2 + 2^2 + 3^2} + \ldots$$ upto 10$$^{th}$$ term is

If the coefficients of $$x^2$$ and $$x^3$$ are both zero, in the expansion of the expression $$(1 + ax + bx^2)(1 - 3x)^{15}$$, in powers of x, then the ordered pair (a, b) is equal to

All the pairs (x, y), that satisfy the inequality $$2^{\sqrt{\sin^2 x - 2\sin x + 5}} \cdot \frac{1}{4^{\sin^2 y}} \leq 1$$ also satisfy the equation:

The line $$x = y$$ touches a circle at the point (1, 1). If the circle also passes through the point (1, -3), then its radius is

If the circles $$x^2 + y^2 + 5Kx + 2y + K = 0$$ and $$2(x^2 + y^2) + 2Kx + 3y - 1 = 0$$, (K ∈ R), intersect at the points P and Q, then the line $$4x + 5y - K = 0$$, passes through P and Q, for:

If the line $$x - 2y = 12$$ is a tangent to the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ at the point $$\left(3, -\frac{9}{2}\right)$$, then the length of the latus rectum of the ellipse is