NTA JEE Main 8th January 2020 Shift 2

Let $$S$$, be the set of all real roots of the equation, $$3^x(3^x - 1) + 2 = |3^x - 1| + |3^x - 2|$$, then S

Let $$\alpha = \frac{-1+i\sqrt{3}}{2}$$. If $$a = (1 + \alpha)\sum_{k=0}^{100} \alpha^{2k}$$ and $$b = \sum_{k=0}^{100} \alpha^{3k}$$, then $$a$$ and $$b$$ are the roots of the quadratic equation.

If the 10$$^{th}$$ term of an A.P. is $$\frac{1}{20}$$, and its 20$$^{th}$$ term is $$\frac{1}{10}$$, then the sum of its first 200 terms is.

If $$\alpha$$ and $$\beta$$, be the coefficients of $$x^4$$ and $$x^2$$, respectively in the expansion of $$\left(x + \sqrt{x^2 - 1}\right)^6 + \left(x - \sqrt{x^2 - 1}\right)^6$$, then

If a line $$y = mx + c$$, is a tangent to the circle $$(x - 3)^2 + y^2 = 1$$, and it is perpendicular to a line $$L_1$$, where $$L_1$$ is the tangent to the circle $$x^2 + y^2 = 1$$, at the point $$\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$$, then

If a hyperbola passes through the point P(10, 16), and it has vertices at ($$\pm$$6, 0), then the equation of the normal to it at P, is.

Which of the following statement is a tautology?

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11, then the correct variance is

If $$A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix}$$ and $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$, then $$10 A^{-1}$$ is equal to.

The system of linear equations
$$\lambda x + 2y + 2z = 5$$
$$2\lambda x + 3y + 5z = 8$$
$$4x + \lambda y + 6z = 10$$ has