NTA JEE Mains 04th April 2024 Shift 1

If the domain of the function $$\sin^{-1}\left(\frac{3x-22}{2x-19}\right) + \log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$$ is $$(\alpha, \beta]$$, then $$3\alpha + 10\beta$$ is equal to:

Let the sum of the maximum and the minimum values of the function $$f(x) = \frac{2x^2-3x+8}{2x^2+3x+8}$$ be $$\frac{m}{n}$$, where gcd(m, n) = 1. Then m + n is equal to:

Let $$f : R to R$$ be a function given by $$f(x) = \begin{cases}\frac{1-\cos 2x}{x^2}, & x < 0\\ \alpha, & x = 0\\ \frac{\beta\sqrt{1-\cos x}}{x}, & x > 0\end{cases}$$, where $$\alpha, \beta \in R$$. If f is continuous at x = 0, then $$\alpha^2 + \beta^2$$ is equal to:

Let $$f(x) = x^5 + 2e^{x/4}$$ for all $$x \in R$$. Consider a function g(x) such that $$(g \circ f)(x) = x$$ for all $$x \in R$$. Then the value of $$8g'(2)$$ is:

Let $$f(x) = \begin{cases}-2, & -2 \le x \le 0\\ x-2, & 0 < x \le 2\end{cases}$$ and $$h(x) = f(|x|) + |f(x)|$$. Then $$\int_{-2}^{2}h(x)dx$$ is equal to:

One of the points of intersection of the curves $$y = 1 + 3x - 2x^2$$ and $$y = \frac{1}{x}$$ is $$\left(\frac{1}{2}, 2\right)$$. Let the area of the region enclosed by these curves be $$\frac{1}{24}(l\sqrt{5}+m) - n\log_e(1+\sqrt{5})$$, where $$l, m, n \in N$$. Then $$l + m + n$$ is equal to:

If the solution $$y = y(x)$$ of the differential equation $$(x^4 + 2x^3 + 3x^2 + 2x + 2)dy - (2x^2 + 2x + 3)dx = 0$$ satisfies $$y(-1) = -\frac{\pi}{4}$$, then y(0) is equal to:

Let a unit vector which makes an angle of 60° with $$2\hat{i} + 2\hat{j} - \hat{k}$$ and angle 45° with $$\hat{i} - \hat{k}$$ be $$\overrightarrow{C}$$. Then $$\overrightarrow{C} + \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)$$ is:

Let the point, on the line passing through the points P(1, −2, 3) and Q(5, −4, 7), farther from the origin and at distance of 9 units from the point P, be $$(\alpha, \beta, \gamma)$$. Then $$\alpha^2 + \beta^2 + \gamma^2$$ is equal to:

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is: