#mht-cet pcm
- \(\int\frac{dx}{1+e^x}=\) Answer: C. x-ln(1+e^x)+C 1/1+e^x=1-e^x/1+e^x ; integrate term by term. 1
- \(\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}\,dx=\) Answer: C. π/4 Using x→fracπ2-x , I=∫cos x/sin x+cos xdx ; adding, 2I=fracπ2 . 1
- \(\int_0^1 x(1-x)^5\,dx=\) Answer: B. 1/42 Using x→1-x : ∫₀¹(1-x)x⁵dx=1/6-1/7=frac142 . 1
- The area enclosed between the curves \(y=x^2\) and \(y=x\) is Answer: B. 1/6 sq. unit ∫₀¹(x-x²)dx=1/2-1/3=1/6 . 1
- The area of the region bounded by the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\) is Answer: C. 12π sq. units Area =π ab=π·4·3=12π . 1
- The order and degree of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{dy}{dx}\right)^4+y=0\) are respectively Answer: B. 2 and 3 Highest derivative d²y/dx² (order 2), raised to power 3 (degree 3). 1