NTA JEE Main 29th July 2022 Shift 2

Let $$\alpha, \beta$$ ($$\alpha > \beta$$) be the roots of the quadratic equation $$x^2 - x - 4 = 0$$. If $$P_n = \alpha^n - \beta^n$$, $$n \in \mathbb{N}$$, then $$\frac{P_{15}P_{16} - P_{14}P_{16} - P_{15}^2 + P_{14}P_{15}}{P_{13}P_{14}}$$ is equal to _____

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _____

If $$\sum_{k=1}^{10} K^2 (10_{C_{K}})^{2} = 22000L$$, then L is equal to _____

Let AB be a chord of length 12 of the circle $$(x-2)^2 + (y+1)^2 = \frac{169}{4}$$. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to _____

Let $$S = \{(x,y) \in \mathbb{N} \times \mathbb{N} : 9(x-3)^2 + 16(y-4)^2 \leq 144\}$$ and $$T = \{(x,y) \in \mathbb{R} \times \mathbb{R} : (x-7)^2 + (y-4)^2 \leq 36\}$$. Then $$n(S \cap T)$$ is equal to _____

Let $$X = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$$ and $$A = \begin{pmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{pmatrix}$$. For $$k \in \mathbb{N}$$, if $$X'A^kX = 33$$, then $$k$$ is equal to

If $$[t]$$ denotes the greatest integer $$\leq t$$, then number of points, at which the function $$f(x) = 4|2x+3| + 9\left[x + \frac{1}{2}\right] - 12[x+20]$$ is not differentiable in the open interval $$(-20, 20)$$, is _____

If the tangent to the curve $$y = x^3 - x^2 + x$$ at the point (a, b) is also tangent to the curve $$y = 5x^2 + 2x - 25$$ at the point (2, -1), then $$|2a + 9b|$$ is equal to _____

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}+\vec{b}|^2 = |\vec{a}|^2 + 2|\vec{b}|^2$$, $$\vec{a} \cdot \vec{b} = 3$$ and $$|\vec{a} \times \vec{b}|^2 = 75$$. Then $$|\vec{a}|^2$$ is equal to _____

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is