NTA JEE Main 8th January 2020 Shift 2 - Mathematics

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

Let $$f : (1, 3) \rightarrow R$$, be a function defined by $$f(x) = \frac{x[x]}{1+x^2}$$, where $$[x]$$ denotes the greatest integer $$\le x$$. Then the range of $$f$$, is

Let $$S$$, be the set of all functions $$f : [0, 1] \rightarrow R$$, which are continuous on [0, 1], and differentiable on (0, 1). Then for every $$f$$ in $$S$$, there exists $$c \in (0, 1)$$, depending on $$f$$, such that.

The length of the perpendicular from the origin, on normal to the curve, $$x^2 + 2xy - 3y^2 = 0$$, at the point (2, 2), is.

$$\lim_{x \to 0} \frac{\int_0^x t \sin(10t) dt}{x}$$ is equal to

If $$I = \int_1^2 \frac{dx}{\sqrt{2x^3 - 9x^2 + 12x + 4}}$$, then

The area (in sq. units) of the region $$\{(x, y) \in R^2 : x^2 \le y \le 3 - 2x\}$$, is.

The differential equation of the family of curves, $$x^2 = 4b(y + b)$$, $$b \in R$$, is.

Let $$\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$$ and $$\vec{b} = \hat{i} - \hat{j} + \hat{k}$$ be two vectors. If $$\vec{c}$$ is a vector such that $$\vec{b} \times \vec{c} = \vec{b} \times \vec{a}$$ and $$\vec{c} \cdot \vec{a} = 0$$, then $$\vec{c} \cdot \vec{b}$$ is equal to.

The mirror image of the point (1, 2, 3), in a plane is $$\left(-\frac{7}{3}, -\frac{4}{3}, -\frac{1}{3}\right)$$. Which of the following points lies on this plane?

Let $$A$$ and $$B$$, be two events such that the probability that exactly one of them occurs is $$\frac{2}{5}$$, and the probability that $$A$$ or $$B$$, occurs is $$\frac{1}{2}$$, then the probability of both of them occur together is.

The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word EXAMINATION is

The sum, $$\sum_{n=1}^{7} \frac{n(n+1)(2n+1)}{4}$$, is equal to

If $$\frac{\sqrt{2}\sin\alpha}{\sqrt{1+\cos 2\alpha}} = \frac{1}{7}$$ and $$\sqrt{\frac{1-\cos 2\beta}{2}} = \frac{1}{\sqrt{10}}$$, $$\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$$, then $$\tan(\alpha + 2\beta)$$ is equal to

Let a line $$y = mx$$ $$(m \gt 0)$$, intersect the parabola, $$y^2 = x$$, at a point P, other than the origin. Let the tangent to it at P, meet the x-axis at the point Q. If area ($$\triangle OPQ$$) = 4 square unit, then m is equal to

Let $$f(x)$$, be a polynomial of degree 3, such that $$f(-1) = 10$$, $$f(1) = -6$$, $$f(x)$$, has a critical point at $$x = -1$$ and $$f'(x)$$, has a critical point at $$x = 1$$. Then $$f(x)$$, has local minima at $$x =$$