The general solution of the differential equation \(ydx + xdy = 0\) is:

If \(\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\), then number of solution(s) of the given equation is :

If \(\left(t_1, y_1\right),\left(t_2, y_2\right),\left(t_3, y_3\right), \ldots,\left(t_n, y_n\right)\) denote the time series and \(y_t\) are the trend values of the variable \(y\), then \(\sum\left(y-y_t\right)\), the sum of deviations of \(y\) from their corresponding trend value is equal to :

In a linear programming problem, the constraints on decision variables x and y are \(y-2 x \leq 0, y \geq 0,0 \leq x \leq 5\). The feasible region of the above problem :

Match List - I with List - II.

LIST I	LIST II
A. \(\left[1+\left(\frac{d y}{d x}\right)^2\right]=\left(\frac{d^2 y}{d x^2}\right)^2\)	I. order \(=2\), degree \(=2\)
B. \(\frac{d^3 y}{d x^3}-3 \frac{d^2 y}{d x^2}+2\left(\frac{d y}{d x}\right)^4=y^4\)	II. order \(=3\), degree \(=2\)
C. \(\left(1+\frac{d y}{d x}\right)^3=\left(\frac{d^3 y}{d x^3}\right)^2\)	III. order \(=2\), degree \(=1\)
D. \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^?=\frac{d^2 y}{d x^2}\)	IV. order \(=3\), degree \(=1\)

Choose the correct answer from the option given below:

The equation of the normal to the curve \(y=\sin x\) at \((0,0)\) is :

\(\int \frac{(x-3) e^x}{(x-1)^3} d x\) is equal to

The semi vertical angle of a right circular cone of maximum volume of a given slant height is

If matrices \(A=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\) and \(B=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\), then \(B A\) is equal to:

A thin infinite line charge has linear charge density of \( 2.0 \times 10^{-9} \text{ C/m} \). The electric field intensity at a distance of 3 m from the line charge is : (\( \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \))

If \(\int_0^a 3 x^2 d x=8\), then the value of \(a\) is:

A die is thrown 4 times and getting 3 is considered a success. The probability of 2 successes is :

Read the passage carefully and answer the questions
Ethers with the general formula \(C _n H _{2 n+2} O\) and functional group C-O-C may be divided as symmetrical and unsymmetrical.
Lower symmetrical ether may be obtained by the dehydration of alcohols at low temperatures in the presence of protic acids,
such as \(H _2 SO _4\) and \(H _3 PO _4\) whereas at high temperature intramolecular dehydration takes place.
Formation of ether from primary alcohols is also accompanied by the formation of small amount of alkenes.
Due to the steric hinderance, secondary and tertiary alcohols undergo elimination reaction rather than substitution reaction yielding alkene as major product. Ethers can also be obtained by heating alkyl halides with sodium or potassium alkoxides.
Substituted Ethers (secondary & tertiary) can also be prepared using this method.
If primary halides are replaced with secondary and tertiary halides, alkenes are the major products.
The major product in the following reaction is