NTA JEE Mains 1st Feb 2023 Shift 1

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____.

Let $$a_1 = 8, a_2, a_3, \ldots, a_n$$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is _____.

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is _____.

The remainder when $$19^{200} + 23^{200}$$ is divided by 49, is _____.

If $$f(x) = x^2 + g'(1)x + g''(2)$$ and $$g(x) = f(1)x^2 + xf'(x) + f''(x)$$, then the value of $$f(4) - g(4)$$ is equal to _____.

If $$\int_0^1 x^{21} + x^{14} + x^7 2x^{14} + 3x^7 + 6^{1/7} dx = \frac{1}{l}(11)^{m/n}$$ where $$l, m, n \in \mathbb{N}$$, $$m$$ and $$n$$ are co-prime then $$l + m + n$$ is equal to _____.

Let $$A$$ be the area bounded by the curve $$y = x|x-3|$$, the $$x$$-axis and the ordinates $$x = -1$$ and $$x = 2$$. Then $$12A$$ is equal to _____.

Let $$f: \mathbb{R} \to \mathbb{R}$$ be a differentiable function such that $$f'(x) + f(x) = \int_0^2 f(t) dt$$. If $$f(0) = e^{-2}$$, then $$2f(0) - f(2)$$ is equal to _____.

Let $$\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$$, $$\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$$, and $$\vec{u}$$ be a vector such that $$|\vec{u}| = \alpha > 0$$. If the minimum value of the scalar triple product $$[\vec{u} \quad \vec{v} \quad \vec{w}]$$ is $$-\alpha\sqrt{3401}$$, and $$|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$$ where $$m$$ and $$n$$ are coprime natural numbers, then $$m + n$$ is equal to _____.

$$A(2, 6, 2), B(-4, 0, \lambda), C(2, 3, -1)$$ and $$D(4, 5, 0)$$, $$\lambda \leq 5$$ are the vertices of a quadrilateral $$ABCD$$. If its area is 18 square units, then $$5 - 6\lambda$$ is equal to _____.