List of practice Questions

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

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

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

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

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

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

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

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

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

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

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

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

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

The length of the latus rectum and eccentricity of the Hyperbola \(9x^2-16y^2=144\) are

The line \(y=mx+2\) is a tangent to the parabola \(y^2=8x\) if

If the parabola \(y^2=4ax\) passes through the point \((3,2)\), then the length of its latus rectum is:

The equation of a circle whose Centre is \((2,-1)\) and which passes through the point \((3,6)\) is

The equation of a circle whose Centre is \((-3,2)\) and area is \(176\) units is

The conjugate of \( (1+i)^3 \) is

If \( z_1=4i^{40}-5i^{35}+6i^{17}+2,\ z_2=-1+i \), then \( |z_1+z_2|= \)

\[ \tan^{-1}1+\tan^{-1}2+\tan^{-1}3= \]

\[ \cos 6^\circ \sin 24^\circ \cos 72^\circ= \]

If \( \cos\theta=\frac{1}{2}\left(a+\frac{1}{a}\right) \), then \(4\cos^3\theta-3\cos\theta=\)

\[ \tan 9^\circ-\tan 27^\circ-\tan 63^\circ+\tan 81^\circ= \]

\[ \frac{\tan x-1+\sec x}{\tan x-\sec x+1} = \]