1
MHT CET 2023 10th May Morning Shift
+2
-0

The population $$\mathrm{P}=\mathrm{P}(\mathrm{t})$$ at time $$\mathrm{t}$$ of certain species follows the differential equation $$\frac{d P}{d t}=0.5 P-450$$. If $$P(0)=850$$, then the time at which population becomes zero is

A
$$2 \log 18$$
B
$$\log 9$$
C
$$\frac{1}{2} \log 18$$
D
$$\log 18$$
2
MHT CET 2023 10th May Morning Shift
+2
-0

The differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}$$ determines a family of circles with

A
variable radii and fixed centre at $$(0,1)$$.
B
variable radii and fixed centre at $$(0,-1)$$.
C
fixed radius of 1 unit and variable centre along the $$\mathrm{Y}$$-axis.
D
fixed radius of 1 unit and variable centre along the $$\mathrm{X}$$-axis.
3
MHT CET 2023 10th May Morning Shift
+2
-0

General solution of the differential equation $$\cos x(1+\cos y) \mathrm{d} x-\sin y(1+\sin x) \mathrm{d} y=0$$ is

A
$$(1+\cos x)(1+\sin y)=c$$
B
$$1+\sin x+\cos y=c$$
C
$$(1+\sin x)(1+\cos y)=c$$
D
$$1+\sin x \cdot \cos y=c$$
4
MHT CET 2023 9th May Evening Shift
+2
-0

General solution of the differential equation $$\log \left(\frac{d y}{d x}\right)=a x+b y$$ is

A
$$a e^{b y}+b e^{a x}=c_1$$, where $$c_1$$ is a constant.
B
$$a e^{-b y}+b^{-a x}=c_1$$, where $$c_1$$ is a constant.
C
$$a e^{-b y}+b e^{a x}=c_1$$, where $$c_1$$ is a constant.
D
$$a e^{b y}+b e^{-a x}=c_1$$, where $$c_1$$ is a constant.
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