NTA JEE Mains 30th Jan 2024 Shift 2

If $$z$$ is a complex number, then the number of common roots of the equation $$z^{1985} + z^{100} + 1 = 0$$ and $$z^3 + 2z^2 + 2z + 1 = 0$$, is equal to:

Let $$a$$ and $$b$$ be two distinct positive real numbers. Let 11th term of a GP, whose first term is $$a$$ and third term is $$b$$, is equal to $$p^{th}$$ term of another GP, whose first term is $$a$$ and fifth term is $$b$$. Then $$p$$ is equal to

Suppose $$28 - p$$, $$p$$, $$70 - \alpha$$, $$\alpha$$ are the coefficients of four consecutive terms in the expansion of $$(1 + x)^n$$. Then the value of $$2\alpha - 3p$$ equals

For $$\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$$, let $$3\sin(\alpha + \beta) = 2\sin(\alpha - \beta)$$ and a real number $$k$$ be such that $$\tan\alpha = k\tan\beta$$. Then the value of $$k$$ is equal to

If $$x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0$$ is the locus of a point, which moves such that it is always equidistant from the lines $$x + 2y + 7 = 0$$ and $$2x - y + 8 = 0$$, then the value of $$g + c + h - f$$ equals

Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$ be the points on the line $$5x + 7y = 50$$. Let the point $$P$$ divide the line segment $$AB$$ internally in the ratio $$7:3$$. Let $$3x - 25 = 0$$ be a directrix of the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and the corresponding focus be $$S$$. If from $$S$$, the perpendicular on the $$x$$-axis passes through $$P$$, then the length of the latus rectum of $$E$$ is equal to

Let $$P$$ be a point on the hyperbola $$H: \frac{x^2}{9} - \frac{y^2}{4} = 1$$, in the first quadrant such that the area of triangle formed by $$P$$ and the two foci of $$H$$ is $$2\sqrt{13}$$. Then, the square of the distance of $$P$$ from the origin is

Let $$R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$$ be a non-zero $$3 \times 3$$ matrix, where $$x\sin\theta = y\sin\left(\theta + \frac{2\pi}{3}\right) = z\sin\left(\theta + \frac{4\pi}{3}\right) \neq 0$$, $$\theta \in (0, 2\pi)$$.
For a square matrix $$M$$, let Trace($$M$$) denote the sum of all the diagonal entries of $$M$$. Then, among the statements:
(I) Trace($$R$$) = 0
(II) If Trace(adj(adj($$R$$))) = 0, then $$R$$ has exactly one non-zero entry.

Consider the system of linear equations $$x + y + z = 5$$, $$x + 2y + \lambda^2 z = 9$$ and $$x + 3y + \lambda z = \mu$$, where $$\lambda, \mu \in R$$. Then, which of the following statement is NOT correct?

If the domain of the function $$f(x) = \log_e\frac{2x+3}{4x^2+x-3} + \cos^{-1}\frac{2x-1}{x+2}$$ is $$(\alpha, \beta]$$, then the value of $$5\beta - 4\alpha$$ is equal to