If A and B are symmetric matrices of the same order, then which one of the following is true?

If the matrix \(M=\left[\begin{array}{ccc}0 & -1 & 3 \alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0\end{array}\right]\) is skew-symmetric, then

If \(f(x)=\left\{\begin{array}{ll}\frac{1-\tan x}{4 x-\pi} & x \neq \frac{\pi}{4} \\ k & x=\frac{\pi}{4}\end{array}\right.\) is continuous at \(x=\frac{\pi}{4}\), then the value of \(k\) is

The domain of the function f(x) = \(\frac{x^2+1}{x^2-3 x+2}\) is

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

The area of the smaller region bounded by the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\) and the straight line \(3 x+4 y=12\) is :

If \(\vec{a}+\vec{b}+\vec{c}=0,|\vec{a}|=3,|\vec{b}|=25,|\vec{c}|=7\), then the angle between \(\vec{a}\) and \(\vec{b}\) is:

Read the passage carefully and answer the questions
Alcohols and phenols are formed when a hydrogen atom in a hydrocarbon, aliphatic and aromatic respectively is replaced by -OH group. These are further classified on the basis of number of hydroxy groups and according to the hybridization of the carbon atom \(sp ^2\) or \(sp ^3\) to which the -OH group is attached. They find applications in industry as well as in day-to-day life. One of the most commercially obtained alcohol also known as wood spirit is obtained by destructive distillation of wood. Both alcohols and phenols are acidic in nature. The acidity of alcohols and phenols vary according to the nature of substituent(s) attached to them. Alcohols and phenols react with carboxylic acids, acid chlorides and acid anhydrides to form esters. Introduction of acetyl group in alcohols and phenols is called acetylation. Phenol undergo electrophilic aromatic substitution reaction because the -OH activates the benzene ring. It undergoes nitration, halogenation and sulphonation reactions.
When phenol is treated with concentrated sulphuric acid, then it gets converted into phenol-2,4-disulphonic acid \(( X )\). On further treatment of X with conc. nitric acid, product Y is formed. Identify Y .

Which of the following is correct for equipotential surfaces ?
(A) Electric field lines are always perpendicular to equipotential surfaces.
(B) Equipotential surfaces are always spherical in shape.
(C) Work done in moving a charge on equipotential surface is always negative.
(D) Two equipotential surfaces never intersect each other.
(E) Workdone in moving a charge on equipotential surface is always zero.

Nichrome element is used in household appliances because of the following characteristics
1. Its melting point is high
2. Its resistivity is large
3. It is tensile
4. It does not oxidize easily by the oxygen of the air when heated
Choose the correct answer from the options given below:

The correct sequence A, B, C, D, E with reference to atomic spectral series is :
(A) Lyman series \(\frac{1}{\lambda}=R\left(\frac{1}{A^2}-\frac{1}{n^2}\right)\)
(B) Balmer series \(\frac{1}{\lambda}=R\left(\frac{1}{B^2}-\frac{1}{n^2}\right)\)
(C) Pfund series \(\frac{1}{\lambda}=R\left(\frac{1}{C^2}-\frac{1}{n^2}\right)\)
(D) Brackett series \(\frac{1}{\lambda}=R\left(\frac{1}{D^2}-\frac{1}{n^2}\right)\)
(E) Paschen series \(\frac{1}{\lambda}=R\left(\frac{1}{E^2}-\frac{1}{n^2}\right)\)
Choose the correct answer from the options given below :

Arrange the following in decreasing order of their basic strength in aqueous solution
(A)C₆H₅NH₂
(B)C₂H₅NH₂
(C) (C₂H₅)₂NH
(D)NH₃

A straight wire of mass 500 g and length 7 m carries a current of 1.4 A. It is suspended in mid air by a uniform horizontal magnetic field B perpendicular to the length of the wire. The magnitude of magnetic field is: