NTA JEE Main 27th June 2022 Shift 1

The area of the polygon, whose vertices are the non-real roots of the equation $$\bar{z} = iz^2$$ is

If $$x = \sum_{n=0}^{\infty} a^n, y = \sum_{n=0}^{\infty} b^n, z = \sum_{n=0}^{\infty} c^n$$, where $$a, b, c$$ are in A.P. and $$|a| < 1, |b| < 1, |c| < 1, abc \neq 0$$, then

The value of $$\cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right)$$ is equal to

In an isosceles triangle $$ABC$$, the vertex $$A$$ is $$(6, 1)$$ and the equation of the base $$BC$$ is $$2x + y = 4$$. Let the point $$B$$ lie on the line $$x + 3y = 7$$. If $$(\alpha, \beta)$$ is the centroid of $$\triangle ABC$$, then $$15(\alpha + \beta)$$ is equal to

Let the eccentricity of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$$, be $$\frac{1}{4}$$. If this ellipse passes through the point $$\left(-4\sqrt{\frac{2}{5}}, 3\right)$$, then $$a^2 + b^2$$ is equal to

Let $$a$$ be an integer such that $$\lim_{x \to 7} \frac{18 - [1-x]}{[x-3a]}$$ exists, where $$[t]$$ is greatest integer $$\leq t$$. Then $$a$$ is equal to

The boolean expression $$(\sim(p \wedge q)) \vee q$$ is equivalent to

Let the system of linear equations $$x + 2y + z = 2, \alpha x + 3y - z = \alpha, -\alpha x + y + 2z = -\alpha$$ be inconsistent. Then $$\alpha$$ is equal to

$$\sin^{-1}\left(\sin\frac{2\pi}{3}\right) + \cos^{-1}\left(\cos\frac{7\pi}{6}\right) + \tan^{-1}\left(\tan\frac{3\pi}{4}\right)$$ is equal to

If $$\cos^{-1}\left(\frac{y}{2}\right) = \log_e\left(\frac{x}{5}\right)^5, |y| < 2$$, then