NTA JEE Mains 31st Jan 2024 Shift 2

The number of solutions, of the equation $$e^{\sin x} - 2e^{-\sin x} = 2$$ is

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1 + z_2 = 5$$ and $$z_1^3 + z_2^3 = 20 + 15i$$. Then $$z_1^4 + z_2^4$$ equals

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

Let 2$$^{nd}$$, 8$$^{th}$$ and 44$$^{th}$$, terms of a non-constant A.P. be respectively the 1$$^{st}$$, 2$$^{nd}$$ and 3$$^{rd}$$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to

If for some n; $${}^{6}C_{m}+2^{6}C_{m+1}+{}^{6}C_{m+2}>{}^{8}C_{3}$$ and $$^{n-1}P_3 : ^nP_4 = 1:8$$, then $$^nP_{m+1} + ^{n+1}C_m$$ is equal to

Let $$A(a, b)$$, $$B(3, 4)$$ and $$(-6, -8)$$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $$P(2a + 3, 7b + 5)$$ from the line $$2x + 3y - 4 = 0$$ measured parallel to the line $$x - 2y - 1 = 0$$ is

Let a variable line passing through the centre of the circle $$x^2 + y^2 - 16x - 4y = 0$$, meet the positive coordinate axes at the point $$A$$ and $$B$$. Then the minimum value of $$OA + OB$$, where $$O$$ is the origin, is equal to

Let $$P$$ be a parabola with vertex $$(2, 3)$$ and directrix $$2x + y = 6$$. Let an ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$ of eccentricity $$\frac{1}{\sqrt{2}}$$ pass through the focus of the parabola $$P$$. Then the square of the length of the latus rectum of $$E$$, is

Let $$f: \mathbb{R} \rightarrow (0, \infty)$$ be strictly increasing function such that $$\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1$$. Then, the value of $$\lim_{x \to \infty} \left[\frac{f(5x)}{f(x)} - 1\right]$$ is equal to

Let the mean and the variance of 6 observations $$a, b, 68, 44, 48, 60$$ be 55 and 194, respectively. If $$a > b$$, then $$a + 3b$$ is