NTA JEE Main 29th June 2022 Shift 1

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + (2i - 1) = 0$$. Then, the value of $$|\alpha^8 + \beta^8|$$ is equal to

Let $$\{a_n\}_{n=0}^{\infty}$$ be a sequence such that $$a_0 = a_1 = 0$$ and $$a_{n+2} = 2a_{n+1} - a_n + 1$$ for all $$n \geq 0$$. Then, $$\sum_{n=2}^{\infty} \frac{a_n}{7^n}$$ is equal to

If the constant term in the expansion of $$\left(3x^3 - 2x^2 + \frac{5}{x^5}\right)^{10}$$ is $$2^k \cdot l$$, where $$l$$ is an odd integer, then the value of $$k$$ is equal to

The distance between the two points $$A$$ and $$A'$$ which lie on $$y = 2$$ such that both the line segments $$AB$$ and $$A'B$$ (where $$B$$ is the point $$(2, 3)$$) subtend angle $$\frac{\pi}{4}$$ at the origin, is equal to

Let the tangent to the circle $$C_1 : x^2 + y^2 = 2$$ at the point $$M(-1, 1)$$ intersect the circle $$C_2 : (x-3)^2 + (y-2)^2 = 5$$, at two distinct points $$A$$ and $$B$$. If the tangents to $$C_2$$ at the points $$A$$ and $$B$$ intersect at $$N$$, then the area of the triangle $$ANB$$ is equal to

Let $$PQ$$ be a focal chord of the parabola $$y^2 = 4x$$ such that it subtends an angle of $$\frac{\pi}{2}$$ at the point $$(3, 0)$$. Let the line segment $$PQ$$ be also a focal chord of the ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a^2 > b^2$$. If $$e$$ is the eccentricity of the ellipse $$E$$, then the value of $$\frac{1}{e^2}$$ is equal to

Let $$\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$$ be such that $$(p \wedge q)\Delta((p \vee q) \Rightarrow q)$$ is a tautology. Then $$\Delta$$ is equal to

Let the mean and the variance of 5 observations $$x_1, x_2, x_3, x_4, x_5$$ be $$\frac{24}{5}$$ and $$\frac{194}{25}$$ respectively. If the mean and variance of the first 4 observations are $$\frac{7}{2}$$ and $$a$$ respectively, then $$(4a + x_5)$$ is equal to

Let a set $$A = A_1 \cup A_2 \cup \ldots \cup A_k$$, where $$A_i \cap A_j = \phi$$ for $$i \neq j$$; $$1 \leq i, j \leq k$$. Define the relation $$R$$ from $$A$$ to $$A$$ by $$R = \{(x,y) : y \in A_i$$ if and only if $$x \in A_i, 1 \leq i \leq k\}$$. Then, $$R$$ is:

The probability that a randomly chosen $$2 \times 2$$ matrix with all the entries from the set of first 10 primes, is singular, is equal to