NTA JEE Main 6th April 2023 Shift 1

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ______.

The coefficient of $$x^{18}$$ in the expansion of $$\left(x^4 - \dfrac{1}{x^3}\right)^{15}$$ is ______.

A circle passing through the point $$P(\alpha, \beta)$$ in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of $$\alpha\beta$$ is ______.

Let the point $$p, p+1$$ lie inside the region $$E = \{x, y: 3-x \le y \le \sqrt{9-x^2}, 0 \le x \le 3\}$$. If the set of all values of $$p$$ is the interval $$(a, b)$$, then $$b^2 + b - a^2$$ is equal to ______.

Let $$A = 1, 2, 3, 4, \ldots, 10$$ and $$B = 0, 1, 2, 3, 4$$. The number of elements in the relation $$R = \{(a, b) \in A \times A: 2a - b^2 + 3a - b \in B\}$$ is ________.

Let $$a \in \mathbb{Z}$$ and $$t$$ be the greatest integer $$\le t$$, then the number of points, where the function $$f(x) = a + 13|\sin x|$$, $$x \in (0, \pi)$$ is not differentiable, is ______.

Let the tangent to the curve $$x^2 + 2x - 4y + 9 = 0$$ at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line $$x - 3y = 6$$ meet the parabola $$y^2 = 4x$$ at B. If B lies on the line $$2x - 3y = 8$$, then $$AB^2$$ is equal to ______.

If the area of the region $$S = \{(x,y): 2y - y^2 \le x^2 \le 2y, x \ge y\}$$ is equal to $$\dfrac{n+2}{n+1} - \dfrac{\pi}{n-1}$$, then the natural number $$n$$ is equal to ______.

Let $$y = y(x)$$ be a solution of the differential equation $$(x\cos x)dy + (xy\sin x + y\cos x - 1)dx = 0$$, $$0 \lt x \lt \dfrac{\pi}{2}$$. If $$\dfrac{\pi}{3}y\left(\dfrac{\pi}{3}\right) = \sqrt{3}$$, then $$\left|\dfrac{\pi}{6}y''\left(\dfrac{\pi}{6}\right) + 2y'\left(\dfrac{\pi}{6}\right)\right|$$ is equal to ______.

Let the image of the point P(1, 2, 3) in the plane $$2x - y + z = 9$$ be Q. If the coordinates of the point R are (6, 10, 7), then the square of the area of the triangle PQR is ______.