NTA JEE Main 26th August 2021 Shift 2

Let $$\lambda \neq 0$$ be in $$R$$. If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 - x + 2\lambda = 0$$, and $$\alpha$$ and $$\gamma$$ are the roots of the equation $$3x^2 - 10x + 27\lambda = 0$$, then $$\frac{\beta\gamma}{\lambda}$$ is equal to _________

The least positive integer $$n$$ such that $$\frac{(2i)^n}{(1-i)^{n-2}}$$, $$i = \sqrt{-1}$$, is a positive integer, is _________

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit 1 and they all are multiple of 11, is _________

Let $$a_1, a_2, \ldots, a_{10}$$ be an A.P. with common difference $$-3$$ and $$b_1, b_2, \ldots, b_{10}$$ be a G.P. with common ratio 2. Let $$c_k = a_k + b_k$$, $$k = 1, 2, \ldots, 10$$. If $$c_2 = 12$$ and $$c_3 = 13$$, then $$\sum_{k=1}^{10} c_k$$ is equal to _________

Let $$\binom{n}{k}$$ denote $$^nC_k$$ and $$\left[\frac{n}{k}\right] = \begin{cases} \binom{n}{k}, & \text{if } 0 \leq k \leq n \\ 0, & \text{otherwise} \end{cases}$$. If $$A_k = \sum_{i=0}^{9} \binom{9}{i} \left[\binom{12}{12-k+i}\right] + \sum_{i=0}^{8} \binom{8}{i} \left[\binom{13}{13-k+i}\right]$$ and $$A_4 - A_3 = 190p$$, then $$p$$ is equal to _________

Let the mean and variance of four numbers 3, 7, $$x$$ and $$y$$ ($$x > y$$) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, $$x + y$$ and $$x - y$$ is _________

Let $$A$$ be a $$3 \times 3$$ real matrix. If $$\det(2\text{Adj}(2\text{Adj}(\text{Adj}(2A)))) = 2^{41}$$, then the value of $$\det(A^2)$$ equals _________

Let $$a$$ and $$b$$ respectively be the points of local maximum and local minimum of the function $$f(x) = 2x^3 - 3x^2 - 12x$$. If $$A$$ is the total area of the region bounded by $$y = f(x)$$, the $$x$$-axis and the lines $$x = a$$ and $$x = b$$, then 4A is equal to _________

If the projection of the vector $$\hat{i} + 2\hat{j} + \hat{k}$$ on the sum of the two vectors $$2\hat{i} + 4\hat{j} - 5\hat{k}$$ and $$-\lambda\hat{i} + 2\hat{j} + 3\hat{k}$$ is 1, then $$\lambda$$ is equal to _________

Let $$Q$$ be the foot of the perpendicular from the point $$P(7, -2, 13)$$ on the plane containing the lines $$\frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8}$$ and $$\frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}$$. Then $$(PQ)^2$$ is equal to _________