NTA JEE Mains 29th Jan 2024 Shift 1

If $$z = \frac{1}{2} - 2i$$, is such that $$|z + 1| = \alpha z + \beta(1 + i)$$, $$i = \sqrt{-1}$$ and $$\alpha, \beta \in R$$, then $$\alpha + \beta$$ is equal to

In an A.P., the sixth term $$a_6 = 2$$. If the $$a_1 a_4 a_5$$ is the greatest, then the common difference of the A.P., is equal to

If in a G.P. of $$64$$ terms, the sum of all the terms is $$7$$ times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

If $$\alpha$$, $$-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$$ is the solution of $$4\cos\theta + 5\sin\theta = 1$$, then the value of $$\tan\alpha$$ is

Let $$(5, \frac{a}{4})$$, be the circumcenter of a triangle with vertices $$A(a, -2)$$, $$B(a, 6)$$ and $$C(\frac{a}{4}, -2)$$. Let $$\alpha$$ denote the circumradius, $$\beta$$ denote the area and $$\gamma$$ denote the perimeter of the triangle. Then $$\alpha + \beta + \gamma$$ is

In a $$\Delta ABC$$, suppose $$y = x$$ is the equation of the bisector of the angle $$B$$ and the equation of the side $$AC$$ is $$2x - y = 2$$. If $$2AB = BC$$ and the point $$A$$ and $$B$$ are respectively $$(4, 6)$$ and $$(\alpha, \beta)$$, then $$\alpha + 2\beta$$ is equal to

$$\lim_{x \to \frac{\pi}{2}} \left(\frac{1}{(x - \frac{\pi}{2})^2} \int_{x^3}^{(\frac{\pi}{2})^3} \cos\left(\frac{1}{t^3}\right) dt\right)$$ is equal to

Let $$R$$ be a relation on $$Z \times Z$$ defined by $$(a, b)R(c, d)$$ if and only if $$ad - bc$$ is divisible by $$5$$. Then $$R$$ is

Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$$ and $$|2A|^3 = 2^{21}$$ where $$\alpha, \beta \in Z$$, Then a value of $$\alpha$$ is

Let A be a square matrix such that $$AA^T = I$$. Then $$\frac{1}{2}A\left[(A + A^T)^2 + (A - A^T)^2\right]$$ is equal to