NTA JEE Main 18th March 2021 Shift 2

If $$f(x)$$ and $$g(x)$$ are two polynomials such that the polynomial $$P(x) = f(x^3) + xg(x^3)$$ is divisible by $$x^2 + x + 1$$, then $$P(1)$$ is equal to ___.

If $$\sum_{r=1}^{10} r!(r^3 + 6r^2 + 2r + 5) = \alpha(11!)$$, then the value of $$\alpha$$ is equal to ___.

The term independent of $$x$$ in the expansion of $$\left[\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right]^{10}$$, $$x \neq 1$$, is equal to ___.

Let $${}^nC_r$$ denote the binomial coefficient of $$x^r$$ in the expansion of $$(1+x)^n$$. If $$\sum_{k=0}^{10} (2^2 + 3k) {}^{n}C_k = \alpha \cdot 3^{10} + \beta \cdot 2^{10}$$, $$\alpha, \beta \in R$$, then $$\alpha + \beta$$ is equal to ___.

Let $$I$$ be an identity matrix of order $$2 \times 2$$ and $$P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}$$. Then the value of $$n \in N$$ for which $$P^n = 5I - 8P$$ is equal to ___.

Let $$f : R \to R$$ satisfy the equation $$f(x+y) = f(x) \cdot f(y)$$ for all $$x, y \in R$$ and $$f(x) \neq 0$$ for any $$x \in R$$. If the function $$f$$ is differentiable at $$x = 0$$ and $$f'(0) = 3$$, then $$\lim_{h \to 0} \frac{1}{h}(f(h) - 1)$$ is equal to ___.

Let $$P(x)$$ be a real polynomial of degree 3 which vanishes at $$x = -3$$. Let $$P(x)$$ have local minima at $$x = -1$$ and $$\int_{-1}^{1} P(x)dx = 18$$, then the sum of all the coefficients of the polynomial $$P(x)$$ is equal to ___.

Let $$y = y(x)$$ be the solution of the differential equation $$xdy - ydx = \sqrt{(x^2 - y^2)}dx$$, $$x \ge 1$$, with $$y(1) = 0$$. If the area bounded by the line $$x = 1$$, $$x = e^\pi$$, $$y = 0$$ and $$y = y(x)$$ is $$\alpha e^{2\pi} + \beta$$, then the value of $$10(\alpha + \beta)$$ is equal to ___.

Let the mirror image of the point $$(1, 3, a)$$ with respect to the plane $$\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) - b = 0$$ be $$(-3, 5, 2)$$. Then the value of $$|a + b|$$ is equal to ___.

Let $$P$$ be a plane containing the line $$\frac{x-1}{3} = \frac{y+6}{4} = \frac{z+5}{2}$$ and parallel to the line $$\frac{x-3}{4} = \frac{y-2}{-3} = \frac{z+5}{7}$$. If the point $$(1, -1, \alpha)$$ lies on the plane $$P$$, then the value of $$|5\alpha|$$ is equal to ___.