NTA JEE Main 31st January 2023 Shift 2

The sum $$1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$$ is ______.

If the constant term in the binomial expansion of $$\left(\dfrac{x^{5/2}}{2} - \dfrac{4}{x^l}\right)^9$$ is -84 and the coefficient of $$x^{-3l}$$ is $$2^\alpha \beta$$ where $$\beta < 0$$ is an odd number, then $$|\alpha l - \beta|$$ is equal to ______.

Let $$S$$ be the set of all $$a \in N$$ such that the area of the triangle formed by the tangent at the point $$P(b, c)$$, $$b, c \in N$$, on the parabola $$y^2 = 2ax$$ and the lines $$x = b$$, $$y = 0$$ is 16 unit$$^2$$, then $$\sum_{a \in S} a$$ is equal to

Let $$A = [a_{ij}]$$, $$a_{ij} \in Z \cap [0, 4]$$, $$1 \le i, j \le 2$$. The number of matrices $$A$$ such that the sum of all entries is a prime number $$p \in (2, 13)$$ is ______.

Let $$A$$ be a $$n \times n$$ matrix such that $$|A| = 2$$. If the determinant of the matrix $$\text{Adj}\left(2 \cdot \text{Adj}(2A^{-1})\right)$$ is $$2^{84}$$, then $$n$$ is equal to ______.

Let $$\alpha > 0$$. If $$\int_{0}^{\alpha}\frac{x}{\sqrt{x+\alpha}-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15}$$ then $$\alpha$$ is equal to :

If $$\phi(x) = \dfrac{1}{\sqrt{x}} \int_{\pi/4}^{x} \left(4\sqrt{2}\sin t - 3\phi'(t)\right) dt$$, $$x > 0$$ then $$\phi'\left(\dfrac{\pi}{4}\right)$$ is equal to ______.

Let the area of the region $$\{(x,y): |2x-1| \le y \le |x^2-x|, 0 \le x \le 1\}$$ be $$A$$. Then $$(6A + 11)^2$$ is equal to ______.

Let $$\vec{a}, \vec{b}, \vec{c}$$ be three vectors such that $$|\vec{a}| = \sqrt{31}$$, $$4|\vec{b}| = |\vec{c}| = 2$$ and $$2(\vec{a} \times \vec{b}) = 3(\vec{c} \times \vec{a})$$. If the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\dfrac{2\pi}{3}$$, then $$\left(\dfrac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$$ is equal to ______.

Let $$A$$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $$[0, 60]$$ is less than or equal to $$a$$. If $$P(A) = \dfrac{11}{36}$$, then $$a$$ is equal to ______.