NTA JEE Mains 1st Feb 2024 Shift 2

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$px^2 + qx - r = 0$$, where $$p \neq 0$$. If $$p$$, $$q$$ and $$r$$ be the consecutive terms of a non-constant G.P and $$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$$, then the value of $$(\alpha - \beta)^2$$ is:

If $$z$$ is a complex number such that $$|z| \leq 1$$, then the minimum value of $$\left|z + \frac{1}{2}(3 + 4i)\right|$$ is:

Let $$S_n$$ denote the sum of the first n terms of an arithmetic progression. If $$S_{10} = 390$$ and the ratio of the tenth and the fifth terms is 15 : 7, then $$S_{15} - S_5$$ is equal to:

Let $$m$$ and $$n$$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $$\left(\frac{1}{3}x^{1/3} + \frac{1}{2x^{2/3}}\right)^{18}$$. Then $$\left(\frac{n}{m}\right)^{1/3}$$ is:

The number of solutions of the equation $$4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$$; $$x \in [-2\pi, 2\pi]$$ is:

Let the locus of the mid points of the chords of circle $$x^2 + (y-1)^2 = 1$$ drawn from the origin intersect the line $$x + y = 1$$ at P and Q. Then, the length of PQ is:

Let P be a point on the ellipse $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Let the line passing through P and parallel to y-axis meet the circle $$x^2 + y^2 = 9$$ at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that $$PR : RQ = 4 : 3$$ as P moves on the ellipse, is:

Let $$f(x) = \begin{cases} x-1, & x \text{ is even} \\ 2x, & x \text{ is odd} \end{cases}$$, $$x \in N$$. If for some $$a \in N$$, $$f(f(f(a))) = 21$$, then $$\lim_{x \to a^-} \left\lfloor \frac{x^3}{a} \right\rfloor - \left\lfloor \frac{x}{a} \right\rfloor$$, where $$\lfloor t \rfloor$$ denotes the greatest integer less than or equal to $$t$$, is equal to:

Consider 10 observations $$x_1, x_2, \ldots, x_{10}$$, such that $$\sum_{i=1}^{10}(x_i - \alpha) = 2$$ and $$\sum_{i=1}^{10}(x_i - \beta)^2 = 40$$, where $$\alpha, \beta$$ are positive integers. Let the mean and the variance of the observations be $$\frac{6}{5}$$ and $$\frac{84}{25}$$ respectively. Then $$\frac{\beta}{\alpha}$$ is equal to:

Consider the relations $$R_1$$ and $$R_2$$ defined as $$aR_1b \Leftrightarrow a^2 + b^2 = 1$$ for all $$a, b \in R$$ and $$(a,b)R_2(c,d) \Leftrightarrow a + d = b + c$$ for all $$(a,b,c,d) \in N \times N$$. Then