NTA JEE Main 3rd September 2020 Shift 1 - Mathematics

Consider the two sets:
$$A = \{m \in R : \text{both the roots of } x^2 - (m+1)x + m + 4 = 0 \text{ are real}\}$$ and $$B = [-3, 5)$$
Which of the following is not true?

If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 + px + 2 = 0$$ and $$\frac{1}{\alpha}$$ and $$\frac{1}{\beta}$$ are the roots of the equation $$2x^2 + 2qx + 1 = 0$$, then $$\left(\alpha - \frac{1}{\alpha}\right)\left(\beta - \frac{1}{\beta}\right)\left(\alpha + \frac{1}{\beta}\right)\left(\beta + \frac{1}{\alpha}\right)$$ is equal to:

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is:

The value of $$(2 \cdot {}^1P_0 - 3 \cdot {}^2P_1 + 4 \cdot {}^3P_2 - \ldots$$ up to 51$$^{th}$$ term$$) + (1! - 2! + 3! - \ldots$$ up to 51$$^{th}$$ term) is equal to

If the number of integral terms in the expansion of $$\left(3^{\frac{1}{2}} + 5^{\frac{1}{8}}\right)^n$$ is exactly 33, then the least value of $$n$$ is

Let P be a point on the parabola, $$y^2 = 12x$$ and N be the foot of the perpendicular drawn from P, on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the $$y$$-intercept of the line NQ is $$\frac{4}{3}$$, then:

A hyperbola having the transverse axis of length $$\sqrt{2}$$ has the same foci as that of the ellipse, $$3x^2 + 4y^2 = 12$$ then this hyperbola does not pass through which of the following points?

Let $$[t]$$ denote the greatest integer $$\leq t$$. If $$\lambda \in R - \{0, 1\}$$, $$\lim_{x \to 0}\left|\frac{1 - x + |x|}{\lambda - x + [x]}\right| = L$$, then $$L$$ is equal to

The proposition $$p \to \sim(p \wedge \sim q)$$ is equivalent to:

For the frequency distribution: Variate $$(x)$$: $$x_1, x_2, x_3, \ldots, x_{15}$$
Frequency $$(f)$$: $$f_1, f_2, f_3, \ldots, f_{15}$$
where $$0 < x_1 < x_2 < x_3 < \ldots < x_{15} = 10$$ and $$\sum_{i=1}^{15} f_i > 0$$, the standard deviation cannot be

If $$\Delta = \begin{vmatrix} x-2 & 2x-3 & 3x-4 \\ 2x-3 & 3x-4 & 4x-5 \\ 3x-5 & 5x-8 & 10x-17 \end{vmatrix} = Ax^3 + Bx^2 + Cx + D$$, then $$B + C$$ is equal to:

$$2\pi - \left(\sin^{-1}\frac{4}{5} + \sin^{-1}\frac{5}{13} + \sin^{-1}\frac{16}{65}\right)$$ is equal to:

If $$y^2 + \log_e(\cos^2 x) = y$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$ then:

The function, $$f(x) = (3x - 7)x^{\frac{2}{3}}$$, $$x \in R$$, is increasing for all $$x$$ lying in:

$$\int_{-\pi}^{\pi} |\pi - |x|| \, dx$$ is equal to

The area (in sq. units) of the region $$\{(x, y) : 0 \leq y \leq x^2 + 1, 0 \leq y \leq x + 1, \frac{1}{2} \leq x \leq 2\}$$ is

The solution curve of the differential equation, $$(1 + e^{-x})(1 + y^2)\frac{dy}{dx} = y^2$$ which passes through the point (0, 1), is

The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, -2, 3) and (1, 1, 0) lies on the plane

The lines $$\vec{r} = (\hat{i} - \hat{j}) + l(2\hat{i} + \hat{k})$$ and $$\vec{r} = (2\hat{i} - \hat{j}) + m(\hat{i} + \hat{j} - \hat{k})$$

A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared at least once is

If $$\left(\frac{1+i}{1-i}\right)^{\frac{m}{2}} = \left(\frac{1+i}{i-1}\right)^{\frac{n}{3}} = 1$$, $$(m, n \in N)$$ then the greatest common divisor of the least values of m and n is

The value of $$0.16^{\log_{2.5}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots \infty\right)}$$ is __________

The diameter of the circle, whose centre lies on the line $$x + y = 2$$ in the first quadrant and which touches both the lines $$x = 3$$ and $$y = 2$$ is __________

If $$\lim_{x \to 0}\left\{\frac{1}{x^8}\left(1 - \cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2}\cos\frac{x^2}{4}\right)\right\} = 2^{-k}$$ then the value of k is

Let $$A = \begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}$$, $$x \in R$$ and $$A^4 = [a_{ij}]$$. If $$a_{11} = 109$$, then $$a_{22}$$ is equal to __________