NTA JEE Mains 31st Jan 2024 Shift 2

Let $$a, b, c$$ be the length of three sides of a triangle satisfying the condition $$(a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0$$. If the set of all possible values of $$x$$ is in the interval $$(\alpha, \beta)$$, then $$12(\alpha^2 + \beta^2)$$ is equal to

Let the coefficient of $$x^r$$ in the expansion of $$(x+3)^{n-1} + (x+3)^{n-2}(x+2) + (x+3)^{n-3}(x+2)^2 + \ldots + (x+2)^{n-1}$$ be $$\alpha_r$$. If $$\sum_{r=0}^{n}\alpha_r = \beta^n - \gamma^n$$, $$\beta, \gamma \in \mathbb{N}$$, then the value of $$\beta^2 + \gamma^2$$ equals

Let $$A(-2, -1)$$, $$B(1, 0)$$, $$C(\alpha, \beta)$$ and $$D(\gamma, \delta)$$ be the vertices of a parallelogram $$ABCD$$. If the point $$C$$ lies on $$2x - y = 5$$ and the point $$D$$ lies on $$3x - 2y = 6$$, then the value of $$|\alpha + \beta + \gamma + \delta|$$ is equal to

If $$\lim_{x \to 0} \frac{ax^2e^x - b\log_e(1+x) + cxe^{-x}}{x^2\sin x} = 1$$, then $$16(a^2 + b^2 + c^2)$$ is equal to

Let $$A = \{1, 2, 3, \ldots, 100\}$$. Let $$R$$ be a relation on $$A$$ defined by $$(x, y) \in R$$ if and only if $$2x = 3y$$. Let $$R_1$$ be a symmetric relation on $$A$$ such that $$R \subset R_1$$ and the number of elements in $$R_1$$ is $$n$$. Then the minimum value of $$n$$ is

Let $$A$$ be a $$3 \times 3$$ matrix and $$\det(A) = 2$$. If $$n = \det(\underbrace{adj(adj(\ldots adj(A)))}_{\text{2024 times}})$$, then the remainder when $$n$$ is divided by 9 is equal to

$$\frac{120}{\pi^3}\int_{0}^{\pi}\frac{x^2\sin x\cos x}{\sin^4 x + \cos^4 x}dx$$ is equal to

Let $$y = y(x)$$ be the solution of the differential equation $$\sec^2 x \, dx + e^{2y}(\tan^2 x + \tan x) \, dy = 0$$, $$0 < x < \frac{\pi}{2}$$, $$y\left(\frac{\pi}{4}\right) = 0$$. If $$y\left(\frac{\pi}{6}\right) = \alpha$$, then $$e^{8\alpha}$$ is equal to

Let $$\vec{a} = 3\hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + \vec{b}) \times \vec{c} = 2(\vec{a} \times \vec{b}) + 24\hat{j} - 6\hat{k}$$ and $$(\vec{a} - \vec{b} + \hat{i}) \cdot \vec{c} = -3$$. Then $$|\vec{c}|^2$$ is equal to

A line passes through $$A(4, -6, -2)$$ and $$B(16, -2, 4)$$. The point $$P(a, b, c)$$ where $$a, b, c$$ are non-negative integers, on the line $$AB$$ lies at a distance of 21 units, from the point $$A$$. The distance between the points $$P(a, b, c)$$ and $$Q(4, -12, 3)$$ is equal to