NTA JEE Mains 27th Jan 2024 Shift 1

If $$S = \{z \in \mathbb{C} : |z - i| = |z + i| = |z - 1|\}$$, then $$n(S)$$ is:

The number of common terms in the progressions $$4, 9, 14, 19, \ldots$$ up to $$25^{th}$$ term and $$3, 6, 9, 12, \ldots$$ up to $$37^{th}$$ term is :

If $$A$$ denotes the sum of all the coefficients in the expansion of $$(1 - 3x + 10x^2)^n$$ and $$B$$ denotes the sum of all the coefficients in the expansion of $$(1 + x^2)^n$$, then :

$${}^{n-1}C_r = (k^2 - 8) \; {}^{n}C_{r+1}$$ if and only if :

The portion of the line $$4x + 5y = 20$$ in the first quadrant is trisected by the lines $$L_1$$ and $$L_2$$ passing through the origin. The tangent of an angle between the lines $$L_1$$ and $$L_2$$ is :

Four distinct points $$(2k, 3k), (1, 0), (0, 1)$$ and $$(0, 0)$$ lie on a circle for $$k$$ equal to :

If the shortest distance of the parabola $$y^2 = 4x$$ from the centre of the circle $$x^2 + y^2 - 4x - 16y + 64 = 0$$ is $$d$$, then $$d^2$$ is equal to :

The length of the chord of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$, whose mid point is $$(1, \frac{2}{5})$$, is equal to:

If $$a = \lim_{x \to 0} \frac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$$ and $$b = \lim_{x \to 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$$, then the value of $$ab^3$$ is :

Let $$a_1, a_2, \ldots, a_{10}$$ be 10 observations such that $$\sum_{k=1}^{10} a_k = 50$$ and $$\sum_{\forall k < j} a_k \cdot a_j = 1100$$. Then the standard deviation of $$a_1, a_2, \ldots, a_{10}$$ is equal to :