1
JEE Main 2022 (Online) 29th July Evening Shift
Numerical
+4
-1
Out of Syllabus

If the tangent to the curve $$y=x^{3}-x^{2}+x$$ at the point $$(a, b)$$ is also tangent to the curve $$y = 5{x^2} + 2x - 25$$ at the point (2, $$-$$1), then $$|2a + 9b|$$ is equal to __________.

2
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $$\tan ^{-1} \frac{3}{4}$$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is ______________.

3
JEE Main 2022 (Online) 27th July Morning Shift
Numerical
+4
-1
Out of Syllabus

Let $$M$$ and $$N$$ be the number of points on the curve $$y^{5}-9 x y+2 x=0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M+N$$ equals ___________.

4
JEE Main 2022 (Online) 26th July Morning Shift
Numerical
+4
-1

Let the function $$f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0$$, be decreasing in $$(0, \mathrm{a})$$ and increasing in $$(\mathrm{a}, 4)$$. A tangent to the parabola $$y^{2}=4 a x$$ at a point $$\mathrm{P}$$ on it passes through the point $$(8 \mathrm{a}, 8 \mathrm{a}-1)$$ but does not pass through the point $$\left(-\frac{1}{a}, 0\right)$$. If the equation of the normal at $$P$$ is : $$\frac{x}{\alpha}+\frac{y}{\beta}=1$$, then $$\alpha+\beta$$ is equal to ________________.

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