NTA JEE Main 16th March 2021 Shift 1

Let $$z$$ and $$w$$ be two complex numbers such that $$w = z\bar{z} - 2z + 2$$, $$\left|\frac{z+i}{z-3i}\right| = 1$$ and $$\text{Re}(w)$$ has minimum value. Then, the minimum value of $$n \in N$$ for which $$w^n$$ is real, is equal to ________.

Consider an arithmetic series and a geometric series having four initial terms from the set $$\{11, 8, 21, 16, 26, 32, 4\}$$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ________.

Let $$ABCD$$ be a square of side of unit length. Let a circle $$C_1$$ centered at $$A$$ with unit radius is drawn. Another circle $$C_2$$ which touches $$C_1$$ and the lines $$AD$$ and $$AB$$ are tangent to it, is also drawn. Let a tangent line from the point $$C$$ to the circle $$C_2$$ meet the side $$AB$$ at $$E$$. If the length of $$EB$$ is $$\alpha + \sqrt{3}\beta$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to ________.

If $$\lim_{x \to 0} \frac{ae^x - b\cos x + ce^{-x}}{x \sin x} = 2$$, then $$a + b + c$$ is equal to ________.

Let $$P = \begin{bmatrix} -30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}$$ and $$A = \begin{bmatrix} 2 & 7 & \omega^2 \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1 \end{bmatrix}$$ where $$\omega = \frac{-1+i\sqrt{3}}{2}$$, and $$I_3$$ be the identity matrix of order 3. If the determinant of the matrix $$\left(P^{-1}AP - I_3\right)^2$$ is $$\alpha\omega^2$$, then the value of $$\alpha$$ is equal to ________.

The total number of $$3 \times 3$$ matrices $$A$$ having entries from the set $$\{0, 1, 2, 3\}$$ such that the sum of all the diagonal entries of $$AA^T$$ is 9, is equal to ________.

Let $$f : (0, 2) \to R$$ be defined as $$f(x) = \log_2\left(1 + \tan\left(\frac{\pi x}{4}\right)\right)$$. Then, $$\lim_{n \to \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \ldots + f(1)\right)$$ is equal to ________.

If the normal to the curve $$y(x) = \int_0^x (2t^2 - 15t + 10)\,dt$$ at a point $$(a, b)$$ is parallel to the line $$x + 3y = -5, a > 1$$, then the value of $$|a + 6b|$$ is equal to ________.

Let $$f : R \to R$$ be a continuous function such that $$f(x) + f(x+1) = 2$$ for all $$x \in R$$. If $$I_1 = \int_0^8 f(x)\,dx$$ and $$I_2 = \int_{-1}^3 f(x)\,dx$$, then the value of $$I_1 + 2I_2$$ is equal to ________.

Let the curve $$y = y(x)$$ be the solution of the differential equation, $$\frac{dy}{dx} = 2(x+1)$$. If the numerical value of area bounded by the curve $$y = y(x)$$ and $$x$$-axis is $$\frac{4\sqrt{8}}{3}$$, then the value of $$y(1)$$ is equal to ________.