List of practice Questions

The expression $\frac{\tan x - 1 + \sec x}{\tan x - \sec x + 1}$ is equal to:}

If $\tan \theta + \sec \theta = \sqrt{3}$ then the principal value of $\theta$ in $[0, 2\pi]$ is

If $\begin{bmatrix}x+y & 2 \\ 1& x-y\end{bmatrix}=\begin{bmatrix}4& 2 \\ 1& 2\end{bmatrix}$, then the values of $x$ and $y$ are:}

Which of the following systems has non-trivial solution?

If $A$ is a square matrix of order 3 and $|A|=5$, then the value of $|2A^{T}|$ is

If $\begin{vmatrix}2x& 5 \\ 8 & x\end{vmatrix}=\begin{vmatrix}6 &-2 \\ 7& 3\end{vmatrix}$ then the value of $x$ is

In the matrix $A=\begin{bmatrix}1& 2& 3 \\ 4& 5 & 6 \\ 7& 4 & 9\end{bmatrix}$, the minor $M_{23}$ of the $a_{23}$ is

The differential equation is \[ \frac{dy}{dx}+y\tan x=\sec x \] and \(y(0)=1\). Then the value of \(y\left(\frac{\pi}{4}\right)\) is

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 \[ \frac{dy}{dx}= \]