NTA JEE Mains 1st Feb 2024 Shift 1

Let $$P = \{z \in \mathbb{C} : |z + 2 - 3i| \leq 1\}$$ and $$Q = \{z \in \mathbb{C} : z(1+i) + \bar{z}(1-i) \leq -8\}$$. Let in $$P \cap Q$$, $$|z - 3 + 2i|$$ be maximum and minimum at $$z_1$$ and $$z_2$$ respectively. If $$|z_1|^2 + 2|z_2|^2 = \alpha + \beta\sqrt{2}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ equals:

Let $$3, 7, 11, 15, \ldots, 403$$ and $$2, 5, 8, 11, \ldots, 404$$ be two arithmetic progressions. Then the sum of the common terms in them is equal to:

If the coefficient of $$x^{30}$$ in the expansion of $$\left(1 + \frac{1}{x}\right)^6 (1+x^2)^7 (1-x^3)^8$$; $$x \neq 0$$ is $$\alpha$$, then $$|\alpha|$$ equals:

Let the line $$L: \sqrt{2}x + y = \alpha$$ pass through the point of the intersection $$P$$ (in the first quadrant) of the circle $$x^2 + y^2 = 3$$ and the parabola $$x^2 = 2y$$. Let the line $$L$$ touch two circles $$C_1$$ and $$C_2$$ of equal radius $$2\sqrt{3}$$. If the centres $$Q_1$$ and $$Q_2$$ of the circles $$C_1$$ and $$C_2$$ lie on the $$y$$-axis, then the square of the area of the triangle $$PQ_1Q_2$$ is equal to:

Let $$\{x\}$$ denote the fractional part of $$x$$ and $$f(x) = \frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\} - \{x\}^3}$$, $$x \neq 0$$. If $$L$$ and $$R$$ respectively denotes the left hand limit and the right hand limit of $$f(x)$$ at $$x = 0$$, then $$\frac{32}{\pi^2}(L^2 + R^2)$$ is equal to:

The number of elements in the set $$S = \{(x,y,z) : x,y,z \in \mathbb{Z}, x + 2y + 3z = 42, x,y,z \geq 0\}$$ equals:

Let $$A = \{1, 2, 3, \ldots, 20\}$$. Let $$R_1$$ and $$R_2$$ be two relations on $$A$$ such that $$R_1 = \{(a,b) : b \text{ is divisible by } a\}$$ and $$R_2 = \{(a,b) : a \text{ is an integral multiple of } b\}$$. Then, number of elements in $$R_1 - R_2$$ is equal to:

If $$\int_{-\pi/2}^{\pi/2} \frac{8\sqrt{2}\cos x\,dx}{(1+e^{\sin x})(1+\sin^4 x)} = \alpha\pi + \beta\log_e(3+2\sqrt{2})$$, where $$\alpha, \beta$$ are integers, then $$\alpha^2 + \beta^2$$ equals:

If $$x = x(t)$$ is the solution of the differential equation $$(t+1)dx = (2x + (t+1)^4)dt$$, $$x(0) = 2$$, then $$x(1)$$ equals:

Let the line of the shortest distance between the lines $$L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$$ and $$L_2: \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k})$$ intersect $$L_1$$ and $$L_2$$ at $$P$$ and $$Q$$ respectively. If $$(\alpha, \beta, \gamma)$$ is the midpoint of the line segment $$PQ$$, then $$2(\alpha + \beta + \gamma)$$ is equal to: