NTA JEE Main 16th March 2021 Shift 2

The least value of $$|z|$$ where $$z$$ is complex number which satisfies the inequality $$e^{\left(\frac{(|z|+3)(|z|-1)}{||z|+1|}\log_e 2\right)} \geq \log_{\sqrt{2}}|5\sqrt{7} + 9i|$$, $$i = \sqrt{-1}$$, is equal to:

Consider a rectangle $$ABCD$$ having 5, 6, 7, 9 points in the interior of the line segments $$AB$$, $$BC$$, $$CD$$, $$DA$$ respectively. Let $$\alpha$$ be the number of triangles having these points from different sides as vertices and $$\beta$$ be the number of quadrilaterals having these points from different sides as vertices. Then $$(\beta - \alpha)$$ is equal to:

Let $$A(-1, 1)$$, $$B(3, 4)$$ and $$C(2, 0)$$ be given three points. A line $$y = mx$$, $$m > 0$$, intersects lines $$AC$$ and $$BC$$ at point $$P$$ and $$Q$$ respectively. Let $$A_1$$ and $$A_2$$ be the areas of $$\triangle ABC$$ and $$\triangle PQC$$ respectively, such that $$A_1 = 3A_2$$, then the value of $$m$$ is equal to:

Let the lengths of intercepts on $$x$$-axis and $$y$$-axis made by the circle $$x^2 + y^2 + ax + 2ay + c = 0$$, $$(a < 0)$$ be $$2\sqrt{2}$$ and $$2\sqrt{5}$$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $$x + 2y = 0$$, is equal to:

Let $$C$$ be the locus of the mirror image of a point on the parabola $$y^2 = 4x$$ with respect to the line $$y = x$$. Then the equation of tangent to $$C$$ at $$P(2, 1)$$ is:

If the points of intersection of the ellipse $$\frac{x^2}{16} + \frac{y^2}{b^2} = 1$$ and the circle $$x^2 + y^2 = 4b$$, $$b > 4$$ lie on the curve $$y^2 = 3x^2$$, then $$b$$ is equal to:

Let $$A = \{2, 3, 4, 5, \ldots, 30\}$$ and '$$\sim$$' be an equivalence relation on $$A \times A$$, defined by $$(a, b) \sim (c, d)$$, if and only if $$ad = bc$$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $$(4, 3)$$ is equal to:

The maximum value of $$f(x) = \begin{vmatrix} \sin^2 x & 1 + \cos^2 x & \cos 2x \\ 1 + \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 x & \cos^2 x & \sin 2x \end{vmatrix}$$, $$x \in R$$ is:

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $$x$$ which satisfy $$\sin^{-1}\left(\frac{3x}{5}\right) + \sin^{-1}\left(\frac{4x}{5}\right) = \sin^{-1} x$$ is equal to:

Let $$\alpha \in R$$ be such that the function $$f(x) = \begin{cases} \frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, & x \neq 0 \\ \alpha, & x = 0 \end{cases}$$ is continuous at $$x = 0$$, where $$\{x\} = x - [x]$$, $$[x]$$ is the greatest integer less than or equal to $$x$$. Then: