MHT CET 2023 11th May Morning Shift | Differential Equations Question 70 | Mathematics

A curve passes through the point $$\left(1, \frac{\pi}{6}\right)$$. Let the slope of the curve at each point $$(x, y)$$ be $$\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0$$, then, the equation of the curve is

$$\sin \left(\frac{y}{x}\right)=\log (x)+\frac{1}{2}$$

$$\operatorname{cosec}\left(\frac{y}{x}\right)=\log (x)+2$$

$$\sec \left(\frac{2 y}{x}\right)=\log (x)+2$$

$$\cos \left(\frac{2 y}{x}\right)=\log (x)+\frac{1}{2}$$

MHT CET 2023 10th May Evening Shift

MCQ (Single Correct Answer)

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The general solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{3 x^2}{1+x^3}\right) y=\frac{1}{x^3+1}$$ is

$$y\left(1+x^3\right)=x^3+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

$$y\left(1+x^3\right)=x+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

$$y\left(1+x^3\right)=x^2+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

$$y\left(1+x^3\right)=2 x+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

MHT CET 2023 10th May Evening Shift

MCQ (Single Correct Answer)

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The differential equation of all circles, passing through the origin and having their centres on the $$\mathrm{X}$$-axis, is

$$y^2=x^2+x y \frac{\mathrm{d} y}{\mathrm{~d} x}$$

$$x^2=y^2+2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}$$

$$y^2=x^2+2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}$$

$$x^2=y^2-x y \frac{\mathrm{d} y}{\mathrm{~d} x}$$

MHT CET 2023 10th May Morning Shift

MCQ (Single Correct Answer)

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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

$$2 \log 18$$

$$\log 9$$

$$\frac{1}{2} \log 18$$

$$\log 18$$

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