NTA JEE Main 10th April 2023 Shift 1

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$$^2$$) is equal to

If $$Ix = \int e^{\sin^2 x} \cos x \sin 2x \cdot \sin x \, dx$$ and $$I(0) = 1$$, then $$I\left(\frac{\pi}{3}\right)$$ is equal to

Let $$f$$ be a differentiable function such that $$x^2 f(x) - x = 4\int_0^x tf(t) \, dt$$, $$f(1) = \frac{2}{3}$$. Then $$18f(3)$$ is equal to

The slope of tangent at any point $$(x, y)$$ on a curve $$y = y(x)$$ is $$\frac{x^2 + y^2}{2xy}$$, $$x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ is

An arc $$PQ$$ of a circle subtends a right angle at its centre $$O$$. The mid point of the arc $$PQ$$ is $$R$$. If $$\overrightarrow{OP} = \vec{u}$$, $$\overrightarrow{OR} = \vec{v}$$ and $$\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$$, then $$\alpha$$, $$\beta^2$$, are the roots of the equation

Let $$O$$ be the origin and the position vector of the point $$P$$ be $$-\hat{i} - 2\hat{j} + 3\hat{k}$$. If the position vectors of the points $$A$$, $$B$$ and $$C$$ are $$-2\hat{i} + \hat{j} - 3\hat{k}$$, $$2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$-4\hat{i} + 2\hat{j} - \hat{k}$$ respectively, then the projection of the vector $$\overrightarrow{OP}$$ on a vector perpendicular to the vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{AC}$$ is

Let two vertices of a triangle $$ABC$$ be $$(2, 4, 6)$$ and $$(0, -2, -5)$$, and its centroid be $$(2, 1, -1)$$. If the image of the third vertex in the plane $$x + 2y + 4z = 11$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha\beta + \beta\gamma + \gamma\alpha$$ is equal to

The shortest distance between the lines $$\frac{x+2}{1} = \frac{y}{-2} = \frac{z-5}{2}$$ and $$\frac{x-4}{1} = \frac{y-1}{2} = \frac{z+3}{0}$$ is

Let $$P$$ be the point of intersection of the line $$\frac{x+3}{3} = \frac{y+2}{1} = \frac{1-z}{2}$$ and the plane $$x + y + z = 2$$. If the distance of the point $$P$$ from the plane $$3x - 4y + 12z = 32$$ is $$q$$, then $$q$$ and $$2q$$ are the roots of the equation

Let $$N$$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $$2^N < N!$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, $$4m - 3n$$ is equal to