1
MHT CET 2023 14th May Morning Shift
+2
-0

Let the curve be represented by $$x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)$$. Then normal at any point '$$t$$' of the curve is at a distance of ______ units from the origin.

A
1
B
0
C
2
D
4
2
MHT CET 2023 14th May Morning Shift
+2
-0

Let $$\mathrm{B} \equiv(0,3)$$ and $$\mathrm{C} \equiv(4,0)$$. The point $$\mathrm{A}$$ is moving on the line $$y=2 x$$ at the rate of 2 units/second. The area of $$\triangle \mathrm{ABC}$$ is increasing at the rate of

A
$$\frac{11}{\sqrt{5}}$$ (units)$$^2$$/ $$\mathrm{sec}$$
B
$$\frac{11}{5}$$ (units)$$^2$$/ $$\mathrm{sec}$$
C
$$\frac{13}{\sqrt{5}}$$ (units)$$^2$$/ $$\mathrm{sec}$$
D
$$\frac{13}{5}$$ (units)$$^2$$/ $$\mathrm{sec}$$
3
MHT CET 2023 14th May Morning Shift
+2
-0

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathrm{R} / x^2+30 \leq 11 x\right\}$$ is

A
122
B
$$-$$122
C
$$-$$222
D
222
4
MHT CET 2023 13th May Evening Shift
+2
-0

Slope of the tangent to the curve $$y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)$$, where $$0 \leq x \leq 2 \pi$$ is minimum at $$x=$$

A
0
B
$$\pi$$
C
$$2 \pi$$
D
1
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