NTA JEE Main 26th July 2022 Shift 1

Let $$O$$ be the origin and $$A$$ be the point $$z_1 = 1 + 2i$$. If $$B$$ is the point $$z_2$$, $$\text{Re}(z_2) < 0$$, such that $$OAB$$ is a right angled isosceles triangle with $$OB$$ as hypotenuse, then which of the following is NOT true?

Consider two G.P.s $$2, 2^2, 2^3, \ldots$$ and $$4, 4^2, 4^3, \ldots$$ of $$60$$ and $$n$$ terms respectively. If the geometric mean of all the $$60 + n$$ terms is $$(2)^{\frac{225}{8}}$$, then $$\displaystyle\sum_{k=1}^{n} k(n-k)$$ is equal to:

Let $$S = \{\theta \in [0, 2\pi] : 8^{2\sin^2\theta} + 8^{2\cos^2\theta} = 16\}$$. Then $$n(S) + \displaystyle\sum_{\theta \in S} \left(\sec\left(\dfrac{\pi}{4} + 2\theta\right) \csc\left(\dfrac{\pi}{4} + 2\theta\right)\right)$$ is equal to:

A point $$P$$ moves so that the sum of squares of its distances from the points $$(1, 2)$$ and $$(-2, 1)$$ is $$14$$. Let $$f(x, y) = 0$$ be the locus of $$P$$, which intersects the $$x$$-axis at the points $$A, B$$ and the $$y$$-axis at the point $$C, D$$. Then the area of the quadrilateral $$ACBD$$ is equal to

Let the tangent drawn to the parabola $$y^2 = 24x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2x + 2y = 5$$. Then the normal to the hyperbola $$\dfrac{x^2}{\alpha^2} - \dfrac{y^2}{\beta^2} = 1$$ at the point $$(\alpha + 4, \beta + 4)$$ does NOT pass through the point:

The statement $$(\sim(p \Leftrightarrow \sim q)) \wedge q$$ is:

Let $$A$$ be a $$2 \times 2$$ matrix with $$\det(A) = -1$$ and $$\det((A + I)(\text{Adj}(A) + I)) = 4$$. Then the sum of the diagonal elements of $$A$$ can be:

If the system of linear equations
$$8x + y + 4z = -2$$
$$x + y + z = 0$$
$$\lambda x - 3y = \mu$$
has infinitely many solutions, then the distance of the point $$(\lambda, \mu, -\dfrac{1}{2})$$ from the plane $$8x + y + 4z + 2 = 0$$ is:

$$\tan\left(2\tan^{-1}\dfrac{1}{5} + \sec^{-1}\dfrac{\sqrt{5}}{2} + 2\tan^{-1}\dfrac{1}{8}\right)$$ is equal to:

Let $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to: