1
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1

Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha+1)$$ in the first quadrant is __________.

2
JEE Main 2023 (Online) 8th April Morning Shift
Numerical
+4
-1

If $$a_{\alpha}$$ is the greatest term in the sequence $$\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots$$, then $$\alpha$$ is equal to _____________.

3
JEE Main 2023 (Online) 6th April Evening Shift
Numerical
+4
-1

Let a curve $$y=f(x), x \in(0, \infty)$$ pass through the points $$P\left(1, \frac{3}{2}\right)$$ and $$Q\left(a, \frac{1}{2}\right)$$. If the tangent at any point $$R(b, f(b))$$ to the given curve cuts the $$\mathrm{y}$$-axis at the point $$S(0, c)$$ such that $$b c=3$$, then $$(P Q)^{2}$$ is equal to __________.

4
JEE Main 2023 (Online) 6th April Evening Shift
Numerical
+4
-1

The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.

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