NTA JEE Main 17th March 2021 Shift 2 - Mathematics

Let $$S_1, S_2$$ and $$S_3$$ be three sets defined as
$$S_1 = \{z \in \mathbb{C} : |z-1| \leq \sqrt{2}\}$$,
$$S_2 = \{z \in \mathbb{C} : Re((1-i)z) \geq 1\}$$ and
$$S_3 = \{z \in \mathbb{C} : Im(z) \leq 1\}$$.
Then, the set $$S_1 \cap S_2 \cap S_3$$:

If the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

The value of $$\sum_{r=0}^{6} {^{6} C_{r}} \cdot {^{6} C_{6-r}}$$ is equal to:

The number of solutions of the equation $$x + 2\tan x = \frac{\pi}{2}$$ in the interval $$[0, 2\pi]$$ is:

Two tangents are drawn from a point $$P$$ to the circle $$x^2 + y^2 - 2x - 4y + 4 = 0$$, such that the angle between these tangents is $$\tan^{-1}\left(\frac{12}{5}\right)$$, where $$\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)$$. If the centre of the circle is denoted by $$C$$ and these tangents touch the circle at points $$A$$ and $$B$$, then the ratio of the areas of $$\triangle PAB$$ and $$\triangle CAB$$ is:

Let the tangent to the circle $$x^2 + y^2 = 25$$ at the point $$R(3, 4)$$ meet $$x$$-axis and $$y$$-axis at point $$P$$ and $$Q$$, respectively. If $$r$$ is the radius of the circle passing through the origin $$O$$ and having centre at the incentre of the triangle $$OPQ$$, then $$r^2$$ is equal to:

Let $$L$$ be a tangent line to the parabola $$y^2 = 4x - 20$$ at $$(6, 2)$$. If $$L$$ is also a tangent to the ellipse $$\frac{x^2}{2} + \frac{y^2}{b} = 1$$, then the value of $$b$$ is equal to:

The value of $$\lim_{n \to \infty} \frac{[r] + [2r] + \ldots + [nr]}{n^2}$$, where $$r$$ is non-zero real number and $$[r]$$ denotes the greatest integer less than or equal to $$r$$, is equal to:

The value of the limit $$\lim_{\theta \to 0} \frac{\tan(\pi\cos^2\theta)}{\sin(2\pi\sin^2\theta)}$$ is equal to:

If the Boolean expression $$(p \wedge q) \circledast (p \otimes q)$$ is a tautology, then $$\circledast$$ and $$\otimes$$ are respectively given by:

If $$x, y, z$$ are in arithmetic progression with common difference $$d$$, $$x \neq 3d$$, and the determinant of the matrix $$\begin{bmatrix} 3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z \end{bmatrix}$$ is zero, then the value of $$k^2$$ is:

The number of solutions of the equation $$\sin^{-1}\left[x^2 + \frac{1}{3}\right] + \cos^{-1}\left[x^2 - \frac{2}{3}\right] = x^2$$ for $$x \in [-1, 1]$$, and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is:

Consider the function $$f : R \to R$$ defined by $$f(x) = \begin{cases} \left(2 - \sin\left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$$. Then $$f$$ is:

Let $$f : R \to R$$ be defined as $$f(x) = e^{-x}\sin x$$. If $$F : [0, 1] \to R$$ is a differentiable function such that $$F(x) = \int_0^x f(t)dt$$, then the value of $$\int_0^1 (F'(x) + f(x))e^x dx$$ lies in the interval:

If the integral $$\int_0^{10} \frac{|\sin 2\pi x|}{e^{[x]}}dx = \alpha e^{-1} + \beta e^{-\frac{1}{2}} + \gamma$$, where $$\alpha, \beta, \gamma$$ are integers and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, then the value of $$\alpha + \beta + \gamma$$ is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx$$, $$0 \leq x \leq \frac{\pi}{2}$$, $$y(0) = 0$$. Then, $$y\left(\frac{\pi}{3}\right)$$ is equal to:

If the curve $$y = y(x)$$ is the solution of the differential equation $$2(x^2 + x^{5/4})dy - y(x + x^{1/4})dx = 2x^{9/4}dx$$, $$x > 0$$ which passes through the point $$\left(1, 1 - \frac{4}{3}\log_e 2\right)$$, then the value of $$y(16)$$ is equal to:

Let $$O$$ be the origin. Let $$\vec{OP} = x\hat{i} + y\hat{j} - \hat{k}$$ and $$\vec{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$$, $$x, y \in R$$, $$x > 0$$, be such that $$|\vec{PQ}| = \sqrt{20}$$ and the vector $$\vec{OP}$$ is perpendicular to $$\vec{OQ}$$. If $$\vec{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$$, $$z \in R$$, is coplanar with $$\vec{OP}$$ and $$\vec{OQ}$$, then the value of $$x^2 + y^2 + z^2$$ is equal to:

If the equation of plane passing through the mirror image of a point $$(2, 3, 1)$$ with respect to line $$\frac{x+1}{2} = \frac{y-3}{1} = \frac{z+2}{-1}$$ and containing the line $$\frac{x-2}{3} = \frac{1-y}{2} = \frac{z+1}{1}$$ is $$\alpha x + \beta y + \gamma z = 24$$ then $$\alpha + \beta + \gamma$$ is equal to:

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $$\frac{1}{2}$$ and probability of occurrence of 0 at the odd place be $$\frac{1}{3}$$. Then the probability that 10 is followed by 01 is equal to:

If $$1, \log_{10}(4^x - 2)$$ and $$\log_{10}\left(4^x + \frac{18}{5}\right)$$ are in arithmetic progression for a real number $$x$$ then the value of the determinant $$\begin{vmatrix} 2(x-\frac{1}{2}) & x-1 & x^2 \\ 1 & 0 & x \\ x & 1 & 0 \end{vmatrix}$$ is equal to ________.

Let the coefficients of third, fourth and fifth terms in the expansion of $$\left(x + \frac{a}{x^2}\right)^n$$, $$x \neq 0$$, be in the ratio 12 : 8 : 3. Then the term independent of $$x$$ in the expansion, is equal to ________.

Let $$\tan\alpha$$, $$\tan\beta$$ and $$\tan\gamma$$; $$\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}$$, $$n \in N$$ be the slopes of the three line segments $$OA$$, $$OB$$ and $$OC$$, respectively, where $$O$$ is origin. If circumcentre of $$\triangle ABC$$ coincides with origin and its orthocentre lies on $$y$$-axis, then the value of $$\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos\alpha \cdot \cos\beta \cdot \cos\gamma}\right)^2$$ is equal to ________.

Consider a set of $$3n$$ numbers having variance 4. In this set, the mean of first $$2n$$ numbers is 6 and the mean of the remaining $$n$$ numbers is 3. A new set is constructed by adding 1 into each of the first $$2n$$ numbers, and subtracting 1 from each of the remaining $$n$$ numbers. If the variance of the new set is $$k$$, then $$9k$$ is equal to ________.

Let $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ and $$B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$ such that $$AB = B$$ and $$a + d = 2021$$, then the value of $$ad - bc$$ is equal to ________.

Let $$f : [-1, 1] \to R$$ be defined as $$f(x) = ax^2 + bx + c$$ for all $$x \in [-1, 1]$$, where $$a, b, c \in R$$ such that $$f(-1) = 2$$, $$f'(-1) = 1$$ and for $$x \in (-1, 1)$$ the maximum value of $$f''(x)$$ is $$\frac{1}{2}$$. If $$f(x) \leq \alpha$$, $$x \in [-1, 1]$$, then the least value of $$\alpha$$ is equal to ________.

Let $$I_n = \int_1^e x^{19}(\log|x|)^n dx$$, where $$n \in N$$. If $$(20)I_{10} = \alpha I_9 + \beta I_8$$, for natural numbers $$\alpha$$ and $$\beta$$, then $$\alpha - \beta$$ is equal to ________.

Let $$f : [-3, 1] \to R$$ be given as $$f(x) = \begin{cases} \min\{(x+6), x^2\}, & -3 \leq x \leq 0 \\ \max\{\sqrt{x}, x^2\}, & 0 \leq x \leq 1 \end{cases}$$. If the area bounded by $$y = f(x)$$ and $$x$$-axis is $$A$$ sq units, then the value of $$6A$$ is equal to ________.

Let $$\vec{x}$$ be a vector in the plane containing vectors $$\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$. If the vector $$\vec{x}$$ is perpendicular to $$(3\hat{i} + 2\hat{j} - \hat{k})$$ and its projection on $$\vec{a}$$ is $$\frac{17\sqrt{6}}{2}$$, then the value of $$|\vec{x}|^2$$ is equal to ________.

Let $$P$$ be an arbitrary point having sum of the squares of the distances from the planes $$x + y + z = 0$$, $$lx - nz = 0$$ and $$x - 2y + z = 0$$ equal to 9 units. If the locus of the point $$P$$ is $$x^2 + y^2 + z^2 = 9$$, then the value of $$l - n$$ is equal to ________.