NTA JEE Main 26th February 2021 Shift 2

Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha + \beta = 1$$ and $$\alpha\beta = -1$$. Let $$p_n = (\alpha)^n + (\beta)^n$$, $$p_{n-1} = 11$$ and $$p_{n+1} = 29$$ for some integer $$n \geq 1$$. Then, the value of $$p_n^2$$ is ______.

Let $$z$$ be those complex numbers which satisfy $$|z + 5| \leq 4$$ and $$z(1 + i) + \bar{z}(1 - i) \geq -10$$, $$i = \sqrt{-1}$$. If the maximum value of $$|z + 1|^2$$ is $$\alpha + \beta\sqrt{2}$$, then the value of $$(\alpha + \beta)$$ is

The total number of 4-digit numbers whose greatest common divisor with 18 is 3 is ______.

If the arithmetic mean and the geometric mean of the $$p^{th}$$ and $$q^{th}$$ terms of the sequence $$-16, 8, -4, 2, \ldots$$ satisfy the equation $$4x^2 - 9x + 5 = 0$$, then $$p + q$$ is equal to ______.

Let $$L$$ be a common tangent line to the curves $$4x^2 + 9y^2 = 36$$ and $$(2x)^2 + (2y)^2 = 31$$. Then the square of the slope of the line $$L$$ is ______.

Let $$X_1, X_2, \ldots, X_{18}$$ be eighteen observations such that $$\sum_{i=1}^{18}(X_i - \alpha) = 36$$ and $$\sum_{i=1}^{18}(X_i - \beta)^2 = 90$$, where $$\alpha$$ and $$\beta$$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of $$|\alpha - \beta|$$ is ______.

If the matrix $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}$$ satisfies the equation $$A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ for some real numbers $$\alpha$$ and $$\beta$$, then $$\beta - \alpha$$ is equal to ______.

Let the normals at all the points on a given curve pass through a fixed point $$(a, b)$$. If the curve passes through $$(3, -3)$$ and $$(4, -2\sqrt{2})$$, given that $$a - 2\sqrt{2}b = 3$$, then $$(a^2 + b^2 + ab)$$ is equal to ______.

Let $$a$$ be an integer such that all the real roots of the polynomial $$2x^5 + 5x^4 + 10x^3 + 10x^2 + 10x + 10$$ lie in the interval $$(a, a+1)$$. Then, $$|a|$$ is equal to ______.

If $$I_{m,n} = \int_0^1 x^{m-1}(1-x)^{n-1}dx$$, for $$m, n \geq 1$$, and $$\int_0^1 \frac{x^{m-1} + x^{n-1}}{(1+x)^{m+n}} dx = \alpha I_{m,n}$$, $$\alpha \in R$$, then $$\alpha$$ equals ______.