NTA JEE Main 27th July 2021 Shift 1 - Mathematics

Let $$\alpha, \beta$$ be two roots of the equation $$x^2 + (20)^{1/4}x + (5)^{1/2} = 0$$. Then $$\alpha^8 + \beta^8$$ is equal to:

Let $$C$$ be the set of all complex numbers. Let
$$S_1 = \{z \in C \mid |z - 3 - 2i|^2 = 8\}$$,
$$S_2 = \{z \in C \mid \text{Re}(z) \geq 5\}$$ and
$$S_3 = \{z \in C \mid |z - \bar{z}| \geq 8\}$$.
Then the number of elements in $$S_1 \cap S_2 \cap S_3$$ is equal to

If the coefficients of $$x^7$$ in $$\left(x^2 + \frac{1}{bx}\right)^{11}$$ and $$x^{-7}$$ in $$\left(x - \frac{1}{bx^2}\right)^{11}$$, $$b \neq 0$$, are equal, then the value of $$b$$ is equal to:

If $$\sin \theta + \cos \theta = \frac{1}{2}$$, then $$16(\sin(2\theta) + \cos(4\theta) + \sin(6\theta))$$ is equal to:

Two tangents are drawn from the point $$P(-1, 1)$$ to the circle $$x^2 + y^2 - 2x - 6y + 6 = 0$$. If these tangents touch the circle at points $$A$$ and $$B$$, and if $$D$$ is a point on the circle such that length of the segments $$AB$$ and $$AD$$ are equal, then the area of the triangle $$ABD$$ is equal to:

Let $$P$$ and $$Q$$ be two distinct points on a circle which has center at $$C(2, 3)$$ and which passes through origin $$O$$. If $$OC$$ is perpendicular to both the line segments $$CP$$ and $$CQ$$, then the set $$\{P, Q\}$$ is equal to

Let
$$A = \{(x, y) \in R \times R \mid 2x^2 + 2y^2 - 2x - 2y = 1\}$$
$$B = \{(x, y) \in R \times R \mid 4x^2 + 4y^2 - 16y + 7 = 0\}$$ and
$$C = \{(x, y) \in R \times R \mid x^2 + y^2 - 4x - 2y + 5 \leq r^2\}$$.
Then the minimum value of $$|r|$$ such that $$A \cup B \subseteq C$$ is equal to

A ray of light through $$(2, 1)$$ is reflected at a point $$P$$ on the $$y$$-axis and then passes through the point $$(5, 3)$$. If this reflected ray is the directrix of an ellipse with eccentricity $$\frac{1}{3}$$ and the distance of the nearer focus from this directrix is $$\frac{8}{\sqrt{53}}$$, then the equation of the other directrix can be:

Let $$f : R \rightarrow R$$ be a function such that $$f(2) = 4$$ and $$f'(2) = 1$$. Then, the value of $$\lim_{x \to 2} \frac{x^2 f(2) - 4f(x)}{x - 2}$$ is equal to:

The compound statement $$(P \vee Q) \wedge (\sim P) \Rightarrow Q$$ equivalent to:

If the mean and variance of the following data: 6, 10, 7, 13, $$a$$, 12, $$b$$, 12 are 9 and $$\frac{37}{4}$$ respectively, then $$(a - b)^2$$ is equal to:

Let $$A = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}$$. If $$A^{-1} = \alpha I + \beta A$$, $$\alpha, \beta \in R$$, $$I$$ is a $$2 \times 2$$ identity matrix, then $$4(\alpha - \beta)$$ is equal to:

Let $$f : \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow R$$ be defined as,
$$f(x) = \begin{cases} (1 + |\sin x|)^{\frac{3a}{|\sin x|}}, & -\frac{\pi}{4} < x < 0 \\ b, & x = 0 \\ e^{\cot 4x / \cot 2x}, & 0 < x < \frac{\pi}{4} \end{cases}$$
If $$f$$ is continuous at $$x = 0$$ then the value of $$6a + b^2$$ is equal to:

The value of $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} \frac{(2j-1) + 8n}{(2j-1) + 4n}$$ is equal to:

The value of the definite integral $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{(1 + e^{x\cos x})(\sin^4 x + \cos^4 x)}$$ is equal to:

If the area of the bounded region $$R = \{(x, y) : \max\{0, \log_e x\} \leq y \leq 2^x, \frac{1}{2} \leq x \leq 2\}$$ is, $$\alpha(\log_e 2)^{-1} + \beta(\log_e 2) + \gamma$$ then the value of $$(\alpha + \beta - 2\gamma)^2$$ is equal to:

Let $$y = y(x)$$ be solution of the differential equation $$\log_e\left(\frac{dy}{dx}\right) = 3x + 4y$$, with $$y(0) = 0$$. If $$y\left(-\frac{2}{3}\log_e 2\right) = \alpha \log_e 2$$, then the value of $$\alpha$$ is equal to:

Let $$\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$$ and $$\vec{b} = -\hat{i} + 2\hat{j} + 3\hat{k}$$. Then the vector product $$\left(\vec{a} + \vec{b}\right) \times \left(\left(\vec{a} \times \left(\left(\vec{a} - \vec{b}\right) \times \vec{b}\right)\right) \times \vec{b}\right)$$ is equal to:

Let the plane passing through the point $$(-1, 0, -2)$$ and perpendicular to each of the planes $$2x + y - z = 2$$ and $$x - y - z = 3$$ be $$ax + by + cz + 8 = 0$$. Then the value of $$a + b + c$$ is equal to:

The probability that a randomly selected 2-digit number belongs to the set $$\{n \in N : (2^n - 2)$$ is a multiple of 3$$\}$$ is equal to

If $$\log_3 2, \log_3(2^x - 5), \log_3\left(2^x - \frac{7}{2}\right)$$ are in an arithmetic progression, then the value of $$x$$ is equal to _________.

For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations: $$x + y - z = 2$$, $$x + 2y + \alpha z = 1$$ and $$2x - y + z = \beta$$. If the system has infinite solutions, then $$\alpha + \beta$$ is equal to _________.

Let $$f(x) = \begin{vmatrix} \sin^2 x & -2 + \cos^2 x & \cos 2x \\ 2 + \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 x & \cos^2 x & 1 + \cos 2x \end{vmatrix}$$, $$x \in [0, \pi]$$. Then the maximum value of $$f(x)$$ is equal to _________.

Let the domain of the function $$f(x) = \log_4(\log_5(\log_3(18x - x^2 - 77)))$$ be $$(a, b)$$. Then the value of the integral $$\int_a^b \frac{\sin^3 x}{\sin^3 x + \sin^3(a + b - x)}$$ is equal to _________.

Let $$S = \{1, 2, 3, 4, 5, 6, 7\}$$. Then the number of possible functions $$f : S \rightarrow S$$ such that $$f(m \cdot n) = f(m) \cdot f(n)$$ for every $$m, n \in S$$ and $$m \cdot n \in S$$, is equal to _________.

Let $$f : [0, 3] \rightarrow R$$ be defined by $$f(x) = \min\{x - [x], 1 + [x] - x\}$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$. Let $$P$$ denote the set containing all $$x \in [0, 3]$$ where $$f$$ is discontinuous, and $$Q$$ denote the set containing all $$x \in (0, 3)$$ where $$f$$ is not differentiable. Then the sum of number of elements in $$P$$ and $$Q$$ is equal to _________.

Let $$F : [3, 5] \rightarrow R$$ be a twice differentiable function on $$(3, 5)$$ such that $$F(x) = e^{-x} \int_3^x (3t^2 + 2t + 4F'(t)) \, dt$$. If $$F'(4) = \frac{\alpha e^\beta - 224}{(e^\beta - 4)^2}$$, then $$\alpha + \beta$$ is equal to _________.

If $$y = y(x)$$, $$y \in \left[0, \frac{\pi}{2}\right)$$ is the solution of the differential equation $$\sec y \frac{dy}{dx} - \sin(x + y) - \sin(x - y) = 0$$, with $$y(0) = 0$$, then $$5y'\left(\frac{\pi}{2}\right)$$ is equal to _________.

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b}$$ and $$\vec{c} = \hat{j} - \hat{k}$$ be three vectors such that $$\vec{a} \times \vec{b} = \vec{c}$$ and $$\vec{a} \cdot \vec{b} = 1$$. If the length of projection vector of the vector $$\vec{b}$$ on the vector $$\vec{a} \times \vec{c}$$ is $$l$$, then the value of $$3l^2$$ is equal to _________.

Let a plane $$P$$ pass through the point $$(3, 7, -7)$$ and contain the line, $$\frac{x - 2}{-3} = \frac{y - 3}{2} = \frac{z + 2}{1}$$. If distance of the plane $$P$$ from the origin is $$d$$, then $$d^2$$ is equal to _________.