List of practice Questions

The solution of the differential equation \[ x\frac{dy}{dx}+y=0 \] passing through the point \((1,1)\) is \(y=\)

Degree of the differential equation \[ y=x\frac{dy}{dx}+a\sqrt{1+\left(\frac{dy}{dx}\right)^2} \] is

The order of the differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is

If \(a\) and \(b\) are arbitrary constants, then the differential equation representing the family of curves \[ y=a\sin(x+b) \] is

The differential equation is \[ \frac{dy}{dx}+\frac{y}{x}=0 \] and \(y(1)=2\). Then the value of \(y(3)\) is

The general solution of the differential equation \[ \frac{dy}{dx}=e^{x-y}+x^2e^{-y} \] is

If \[ \int \frac{\sin^3x+\cos^3x}{\sin^2x\cos^2x}\,dx = A\sec x+B\cosec x+c, \] then \((A,B)\) are

The integral of \(f(x)=1+x^2+x^4\) with respect to \(x^2\) is

\[ \int_0^{\frac{\pi}{2}} \frac{\sin^{100}x}{\sin^{100}x+\cos^{100}x}\,dx = \]

\[ \int_0^1 x\sqrt{x^2+4}\,dx= \]

\[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\sin^5x\cos^3x}{x^4}\,dx= \]

\[ \int \frac{dx}{\sqrt{16-25x^2}}= \]

\[ \int \frac{dx}{\sqrt{x+1}+\sqrt{x}}= \]

If \[ f(x)= \begin{cases} 4(5^x), & x 8k+x, & x\geq 0 \end{cases} \] then \(f'(-1)=\)

If \[ 2^x+2^y=2^{x+y}, \] then \[ \frac{dy}{dx}= \]

If \[ y+\sin^{-1}(1-x^2)=e^x, \] then \[ \frac{dy}{dx}= \]

If \(y(x)=x^x,\ x>0\), then \[ y''(2)-2y'(2)= \]

If \[ z=x^2y^3+e^y\sin x, \] then \[ \frac{\partial^2 z}{\partial x\partial y}= \]

\[ \int \frac{dx}{\sin^2x\cos^2x}= \]

The equation of the ellipse with foci at \((\pm 3,0)\) and the eccentricity as \(1/3\) is:

\[ \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x= \]

\[ \lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}= \]

If \[ y=\frac{a\cos x+b\sin x+c}{\sin x}, \] then \[ \frac{dy}{dx}= \]

If \[ y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots\infty}}}, \] then \[ \frac{dy}{dx}= \]

Slope of the tangent to the curve \[ y=9x^2+7x^4+5 \] at the point \(x=1\) is