Which of the following are correct? (A) A statistical hypothesis is a statement about the population parameter which may be true or false. (B) In a…

The given differential equation is $\frac{dy}{dx} + y\tan x = \sec x$dxdy​+ytanx=secx

This is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x) \cdot y = Q(x)$dxdy​+P(x)⋅y=Q(x) where $P(x) = \tan x$P(x)=tanx and $Q(x) = \sec x$Q(x)=secx.

---

The integrating factor is $\displaystyle IF = e^{\int P(x)dx} = e^{\int \tan x dx}$IF=e∫P(x)dx=e∫tanxdx

$$\int \tan x \, dx = \int \frac{\sin x}{\cos x} dx$$∫tanxdx=∫cosxsinx​dx

Let $u = \cos x$u=cosx, then $du = -\sin x \, dx$du=−sinxdx

$$= -\int \frac{1}{u} du$$=−∫u1​du

$$= -\ln|u|$$=−ln∣u∣

$$= -\ln|\cos x|$$=−ln∣cosx∣

$$= \ln|\sec x|$$=ln∣secx∣

Therefore $IF = e^{\ln|\sec x|} = \sec x$IF=eln∣secx∣=secx

---

Multiplying the entire equation by $\sec x$secx:

$$\sec x \cdot \frac{dy}{dx} + y \sec x \tan x = \sec^2 x$$secx⋅dxdy​+ysecxtanx=sec2x

The left side is the derivative of $(y \sec x)$(ysecx) by the product rule.

$$\frac{d}{dx}(y\sec x) = \sec^2 x$$dxd​(ysecx)=sec2x

---

Integrating both sides:

$$\int \frac{d}{dx}(y\sec x) \, dx = \int \sec^2 x \, dx$$∫dxd​(ysecx)dx=∫sec2xdx

$$y\sec x = \tan x + c$$ysecx=tanx+c

$$y\sec x - \tan x = c$$ysecx−tanx=c

where $c$c is an arbitrary constant.