NTA JEE Mains 04th April 2024 Shift 2

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____

Let $$S = \{\sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0$$ has real roots$$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3((\alpha - 2)^2 + (\beta - 1)^2)$$ equals _____

Consider a triangle $$ABC$$ having the vertices $$A(1, 2)$$, $$B(\alpha, \beta)$$ and $$C(\gamma, \delta)$$ and angles $$\angle ABC = \frac{\pi}{6}$$ and $$\angle BAC = \frac{2\pi}{3}$$. If the points $$B$$ and $$C$$ lie on the line $$y = x + 4$$, then $$\alpha^2 + \gamma^2$$ is equal to _____

Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix}$$ and the determinant of $$A$$ be $$1$$. If $$A^{-1} = \alpha A + \beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha + \beta$$ equals _____

Consider the function $$f : \mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = \frac{2x}{\sqrt{1 + 9x^2}}$$. If the composition of $$f$$, $$\underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text{ times}}(x) = \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}}$$, then the value of $$\sqrt{3\alpha + 1}$$ is equal to _____

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a thrice differentiable function such that $$f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2$$ and $$f(4) = -2$$. Then, the minimum number of zeros of $$(3f'f'' + ff''')(x)$$ is _____

If $$\int \csc^5 x \, dx = \alpha \cot x \csc x \left(\csc^2 x + \frac{3}{2}\right) + \beta \log_e \left|\tan \frac{x}{2}\right| + C$$ where $$\alpha, \beta \in \mathbb{R}$$ and $$C$$ is the constant of integration, then the value of $$8(\alpha + \beta)$$ equals _____

Let $$y = y(x)$$ be the solution of the differential equation $$(x + y + 2)^2 dx = dy$$, $$y(0) = -2$$. Let the maximum and minimum values of the function $$y = y(x)$$ in $$\left[0, \frac{\pi}{3}\right]$$ be $$\alpha$$ and $$\beta$$, respectively. If $$(3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}$$, $$\gamma, \delta \in \mathbb{Z}$$, then $$\gamma + \delta$$ equals _____

Consider a line $$L$$ passing through the points $$P(1, 2, 1)$$ and $$Q(2, 1, -1)$$. If the mirror image of the point $$A(2, 2, 2)$$ in the line $$L$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + 6\gamma$$ is equal to _____

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $$\frac{1}{3}$$ and $$\frac{2}{3}$$ respectively. Let $$x$$ be the number of matches that the team wins, and $$y$$ be the number of matches that team loses. If the probability $$P(|x - y| \leq 2)$$ is $$p$$, then $$3^9 p$$ equals _____