NTA JEE Mains 27th Jan 2024 Shift 1

If $$\alpha$$ satisfies the equation $$x^2 + x + 1 = 0$$ and $$(1 + \alpha)^7 = A + B\alpha + C\alpha^2$$, $$A, B, C \geq 0$$, then $$5(3A - 2B - C)$$ is equal to _______.

If $$8 = 3 + \frac{1}{4}(3 + p) + \frac{1}{4^2}(3 + 2p) + \frac{1}{4^3}(3 + 3p) + \ldots \infty$$, then the value of $$p$$ is _______.

Let the set of all $$a \in \mathbb{R}$$ such that the equation $$\cos 2x + a \sin x = 2a - 7$$ has a solution be $$[p, q]$$ and $$r = \tan 9° - \tan 27° - \frac{1}{\cot 63°} + \tan 81°$$, then $$pqr$$ is equal to _______.

Let $$A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$$, $$B = [B_1 \; B_2 \; B_3]$$, where $$B_1, B_2, B_3$$ are column matrices, and $$AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$, $$AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$$, $$AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$. If $$\alpha = |B|$$ and $$\beta$$ is the sum of all the diagonal elements of $$B$$, then $$\alpha^3 + \beta^3$$ is equal to _______.

Let $$f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3), \; x \in \mathbb{R}$$. Then $$f'(10)$$ is equal to _______.

Let for a differentiable function $$f : (0, \infty) \rightarrow \mathbb{R}$$, $$f(x) - f(y) \geq \log_e\left(\frac{x}{y}\right) + x - y, \; \forall x, y \in (0, \infty)$$. Then $$\sum_{n=1}^{20} f'\left(\frac{1}{n^2}\right)$$ is equal to _______.

Let the area of the region $$\{(x, y) : x - 2y + 4 \geq 0, \; x + 2y^2 \geq 0, \; x + 4y^2 \leq 8, \; y \geq 0\}$$ be $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime numbers. Then $$m + n$$ is equal to _______.

If the solution of the differential equation $$(2x + 3y - 2)dx + (4x + 6y - 7)dy = 0$$, $$y(0) = 3$$, is $$\alpha x + \beta y + 3\log_e|2x + 3y - \gamma| = 6$$, then $$\alpha + 2\beta + 3\gamma$$ is equal to _______.

The least positive integral value of $$\alpha$$, for which the angle between the vectors $$\alpha\hat{i} - 2\hat{j} + 2\hat{k}$$ and $$\alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$$ is acute, is _______.

A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required and let $$a = P(X = 3)$$, $$b = P(X \geq 3)$$ and $$c = P(X \geq 6 \mid X > 3)$$. Then $$\frac{b + c}{a}$$ is equal to _______.