NTA JEE Main 16th March 2021 Shift 2 - Mathematics

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:

Let $$f : S \to S$$ where $$S = (0, \infty)$$ be a twice differentiable function such that $$f(x+1) = xf(x)$$. If $$g : S \to R$$ be defined as $$g(x) = \log_e f(x)$$, then the value of $$|g''(5) - g''(1)|$$ is equal to:

Let $$f$$ be a real valued function, defined on $$R - \{-1, 1\}$$ and given by $$f(x) = 3\log_e\left|\frac{x-1}{x+1}\right| - \frac{2}{x-1}$$. Then in which of the following intervals, function $$f(x)$$ is increasing?

Consider the integral $$I = \int_0^{10} \frac{[x]e^{[x]}}{e^{x-1}}dx$$ where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. Then the value of $$I$$ is equal to:

Let $$P(x) = x^2 + bx + c$$ be a quadratic polynomial with real coefficients such that $$\int_0^1 P(x)dx = 1$$ and $$P(x)$$ leaves remainder 5 when it is divided by $$(x-2)$$. Then the value of $$9(b+c)$$ is equal to:

If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + (\tan x)y = \sin x$$, $$0 \leq x \leq \frac{\pi}{3}$$, with $$y(0) = 0$$, then $$y\left(\frac{\pi}{4}\right)$$ is equal to:

Let $$C_1$$ be the curve obtained by the solution of differential equation $$2xy\frac{dy}{dx} = y^2 - x^2$$, $$x > 0$$. Let the curve $$C_2$$ be the solution of $$\frac{2xy}{x^2-y^2} = \frac{dy}{dx}$$. If both the curves pass through $$(1, 1)$$, then the area (in sq. units) enclosed by the curves $$C_1$$ and $$C_2$$ is equal to:

Let $$\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$$ and $$\vec{b} = 2\hat{i} - 3\hat{j} + 5\hat{k}$$. If $$\vec{r} \times \vec{a} = \vec{b} \times \vec{r}$$, $$\vec{r} \cdot (\alpha\hat{i} + 2\hat{j} + \hat{k}) = 3$$ and $$\vec{r} \cdot (2\hat{i} + 5\hat{j} - \alpha\hat{k}) = -1$$, $$\alpha \in R$$, then the value of $$\alpha + |\vec{r}|^2$$ is equal to:

If $$(x, y, z)$$ be an arbitrary point lying on a plane $$P$$ which passes through the point $$(42, 0, 0)$$, $$(0, 42, 0)$$ and $$(0, 0, 42)$$, then the value of expression $$3 + \frac{x-11}{(y-19)^2(z-12)^2} + \frac{y-19}{(x-11)^2(z-12)^2} + \frac{z-12}{(x-11)^2(y-19)^2} - \frac{x+y+z}{14(x-11)(y-19)(z-12)}$$ is

If the foot of the perpendicular from point $$(4, 3, 8)$$ on the line $$L_1: \frac{x-a}{l} = \frac{y-3}{3} = \frac{z-b}{4}$$, $$l \neq 0$$ is $$(3, 5, 7)$$, then the shortest distance between the line $$L_1$$ and line $$L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$$ is equal to:

Let $$A$$ denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event $$A$$ is equal to:

Let $$\frac{1}{16}$$, $$a$$ and $$b$$ be in G.P. and $$\frac{1}{a}$$, $$\frac{1}{b}$$, 6 be in A.P., where $$a, b > 0$$. Then $$72(a+b)$$ is equal to ________.

Let $$S_n(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \ldots$$ up to $$n$$-terms, where $$a > 1$$. If $$S_{24}(x) = 1093$$ and $$S_{12}(2x) = 265$$, then value of $$a$$ is equal to ________.

Let $$n$$ be a positive integer. Let $$A = \sum_{k=0}^{n} (-1)^k \cdot {^nC_k}\left[\left(\frac{1}{2}\right)^k + \left(\frac{3}{4}\right)^k + \left(\frac{7}{8}\right)^k + \left(\frac{15}{16}\right)^k + \left(\frac{31}{32}\right)^k\right]$$. If $$63A = 1 - \frac{1}{2^{30}}$$, then $$n$$ is equal to ________.

Consider the statistics of two sets of observations as follows:

If the variance of the combined set of these two observations is $$\frac{17}{9}$$, then the value of $$n$$ is equal to ________.

In $$\triangle ABC$$, the lengths of sides $$AC$$ and $$AB$$ are 12 cm and 5 cm, respectively. If the area of $$\triangle ABC$$ is 30 cm$$^2$$ and $$R$$ and $$r$$ are respectively the radii of circumcircle and incircle of $$\triangle ABC$$, then the value of $$2R + r$$ (in cm) is equal to ________.

Let $$A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$$ and $$B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$$ be two $$2 \times 1$$ matrices with real entries such that $$A = XB$$, where $$X = \frac{1}{\sqrt{3}}\begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}$$, and $$k \in R$$. If $$a_1^2 + a_2^2 = \frac{2}{3}(b_1^2 + b_2^2)$$ and $$(k^2 + 1)b_2^2 \neq -2b_1 b_2$$, then the value of $$k$$ is ________.

Let $$f : R \to R$$ and $$g : R \to R$$ be defined as $$f(x) = \begin{cases} x+a, & x < 0 \\ |x-1|, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x+1, & x < 0 \\ (x-1)^2 + b, & x \geq 0 \end{cases}$$, where $$a, b$$ are non-negative real numbers. If $$g \circ f(x)$$ is continuous for all $$x \in R$$, then $$a + b$$ is equal to ________.

For real numbers $$\alpha, \beta, \gamma$$ and $$\delta$$, if $$\int \frac{(x^2-1)+\tan^{-1}\left(\frac{x^2+1}{x}\right)}{(x^4+3x^2+1)\tan^{-1}\left(\frac{x^2+1}{x}\right)}dx = \alpha\log_e\left(\tan^{-1}\left(\frac{x^2+1}{x}\right)\right) + \beta\tan^{-1}\left(\frac{\gamma(x^2-1)}{x}\right) + \delta\tan^{-1}\left(\frac{x^2+1}{x}\right) + C$$ where $$C$$ is an arbitrary constant, then the value of $$10(\alpha + \beta\gamma + \delta)$$ is equal to ________.

Let $$\vec{c}$$ be a vector perpendicular to the vectors $$\vec{a} = \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$$. If $$\vec{c} \cdot (\hat{i} + \hat{j} + 3\hat{k}) = 8$$, then the value of $$\vec{c} \cdot (\vec{a} \times \vec{b})$$ is equal to ________.

If the distance of the point $$(1, -2, 3)$$ from the plane $$x + 2y - 3z + 10 = 0$$ measured parallel to the line, $$\frac{x-1}{3} = \frac{2-y}{m} = \frac{z+3}{1}$$ is $$\sqrt{\frac{7}{2}}$$, then the value of $$|m|$$ is equal to ________.