documentclass[12pt,a4paper]{book}
usepackage{amsmath,amsthm}
usepackage{ifthen,amsfonts,amssymb,dsfont}
usepackage{etoolbox,multicol}
documentclass[12pt,a4paper]{book}
usepackage{amsmath,amsthm}
usepackage{ifthen,amsfonts,amssymb,dsfont}
usepackage{etoolbox,multicol}
usepackage[english]{babel}
usepackage{blindtext}
usepackage{geometry}
usepackage{lmodern}
usepackage[T1]{fontenc}
usepackage{enumitem,kantlipsum}
usepackage[dvips]{graphicx,color}
usepackage{verbatim, pstricks,ulem}
usepackage{syntonly}
newcommand{newmatrix}[2]{left(begin{array}{#1}   #2end{array}right)}

newboxallanswerkeys
setboxallanswerkeys=vbox{}
newenvironment{answerkey}{%
globalsetboxallanswerkeys=vboxbgroup
unvboxallanswerkeys
}%
{%
bigbreak
egroup
}

newcommand{showallanswerkeys}{parunvboxallanswerkeys}

newcommand*{getanswerkeys}[6]{%
item[theenumi.]begin{answerkey}(#5)end{answerkey}
}

newboxallanswers
setboxallanswers=vbox{}

newenvironment{answer}{%
globalsetboxallanswers=vboxbgroup
unvboxallanswers
}%
{%
bigbreak
egroup
}

newcommand{showallanswers}{parunvboxallanswers}

newcommand*{getanswer}[6]{%
ifthenelse{equal{#6}{}}{}{
item[theenumi.]begin{answer}(#5)~#6end{answer}
}}

setlengthparindent{0pt}

newcommand{fourch}[6]{
par
begin{tabular}{*{4}{@{}p{0.23textwidth}}}
(a)~#1 & (b)~#2 & (c)~#3 & (d)~#4
end{tabular}
getanswer{#1}{#2}{#3}{#4}{#5}{#6}
getanswerkeys{#1}{#2}{#3}{#4}{#5}{#6}
}

newcommand{twoch}[6]{
par
begin{tabular}{*{2}{@{}p{0.46textwidth}}}
(a)~#1 & (b)~#2
end{tabular}
par%vspace{4pt}
begin{tabular}{*{2}{@{}p{0.46textwidth}}}
(c)~#3 & (d)~#4
end{tabular}
getanswer{#1}{#2}{#3}{#4}{#5}{#6}
getanswerkeys{#1}{#2}{#3}{#4}{#5}{#6}
}

newcommand{onech}[6]{
%par
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(a) & #1
end{tabular}vspace{2pt}
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(b) & #2
end{tabular}vspace{2pt}
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(c) & #3
end{tabular}vspace{2pt}
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(d) & #4
end{tabular}

getanswer{#1}{#2}{#3}{#4}{#5}{#6}
getanswerkeys{#1}{#2}{#3}{#4}{#5}{#6}
}

newlengthwidthcha
newlengthwidthchb
newlengthwidthchc
newlengthwidthchd
newlengthwidthch
newlengthtabmaxwidth

setlengthtabmaxwidth{0.96textwidth}
newlengthfourthtabwidth
setlengthfourthtabwidth{0.25textwidth}
newlengthhalftabwidth
setlengthhalftabwidth{0.5textwidth}

newcommand{choice}[6]{%
settowidthwidthcha{AM.#1}setlength{widthch}{widthcha}%
settowidthwidthchb{BM.#2}%
ifdimwidthch<widthchbrelaxsetlength{widthch}{widthchb}fi%
settowidthwidthchb{CM.#3}%
ifdimwidthch<widthchbrelaxsetlength{widthch}  {widthchb}fi%
settowidthwidthchb{DM.#4}%
ifdimwidthch<widthchbrelaxsetlength{widthch} {widthchb}fi%

ifdimwidthch>halftabwidth
onech{#1}{#2}{#3}{#4}{#5}{#6}
elseifdimwidthch<halftabwidth
ifdimwidthch>fourthtabwidth
twoch{#1}{#2}{#3}{#4}{#5}{#6}
else
fourch{#1}{#2}{#3}{#4}{#5}{#6}
fififi}

newcommand{mcq}[8]{
item #2
choice{#3}{#4}{#5}{#6}{#7}{#8}
}

defdatabaseone{
mcq{245}{Let $A$ be a symmetric matrix of order $n$ and $B$ be a skew-    symmetric matrix of order $n$. Then which one of the following is a skew-symmetric matrix?}{$A^{2}$}{$B^{2}$}{$AB+BA$}{$AB-BA$}{c}{}{}

mcq{248}{A square matrix is a diagonal matrix if and only if it is}{upper triangular}{lower triangular}{both upper triangular and lower triangular}{none of these}{c}{}{}

mcq{249}{Let $A$ be a non-singular matrix of order $3$. Then the rank of matrix $A^{2018}$ is}{$1$}{$2$}{$3$}{$9$}{c}{Note that $A^{2018}$ is a non-singular matrix of order $3$. Hence rank $(A^{2018})=3$.}{}

mcq{250}{If $A=newmatrix{ccc}{0 & 1 & 0\0 & 0 & 1\1 & 0 & 0}$, then rank of the matrix $A+A^{2}+A^{3}$ is}{$1$}{$2$}{$3$}{none of these}{a}{Note that $A+A^{2}+A^{3}=protectnewmatrix{ccc}{1 & 1 & 1\1 & 1 & 1\1 & 1 & 1}$. Hence rank $(A+A^{2}+A^{3})=1$}{}
}

begin{document}
chapter{One}
begin{enumerate}
databaseone
end{enumerate}

bigskip
Huge{Answer Key:}
bigskip

begin{center}
begin{multicols}{6}
showallanswerkeys
end{multicols}
end{center}

bigskip
Huge{Hint/Explanation of Selected Problems:}
bigskipbigskip
showallanswers

chapter{Two}
begin{enumerate}
databaseone
end{enumerate}

bigskip
Huge{Answer Key:}
bigskip

begin{center}
begin{multicols}{6}
showallanswerkeys
end{multicols}
end{center}

bigskip
Huge{Hint/Explanation of Selected Problems:}
bigskipbigskip
showallanswers
end{document}

documentclass[12pt,a4paper]{book}
usepackage{amsmath,amsthm}
usepackage{ifthen,amsfonts,amssymb,dsfont}
usepackage{etoolbox,multicol}


%%NO!!! two documentclass documentclass[12pt,a4paper]{book}
%% NO already loaded usepackage{amsmath,amsthm}
%% NO already loaded usepackage{ifthen,amsfonts,amssymb,dsfont}
%% NO already loaded usepackage{etoolbox,multicol}
usepackage[english]{babel}

% not in areal document usepackage{blindtext}
usepackage{geometry}
usepackage{lmodern}
usepackage[T1]{fontenc}
usepackage{enumitem,
% not in a real documentkantlipsum
}
usepackage[
% NO dvips
]{graphicx,color}
usepackage{verbatim, pstricks,ulem}
%NO! This disables almost all of latex if you use it usepackage{syntonly}

% No use pmatrix from amsmath newcommand{newmatrix}[2]{left(begin{array}{#1}   #2end{array}right)}

newboxallanswerkeys
setboxallanswerkeys=vbox{}
newenvironment{answerkey}{%
globalsetboxallanswerkeys=vboxbgroup
unvboxallanswerkeys
}%
{%
bigbreak
egroup
}

newcommand{showallanswerkeys}{parunvboxallanswerkeys}

newcommand*{getanswerkeys}[6]{%
item[theenumi.]begin{answerkey}(#5)end{answerkey}
}

newboxallanswers
setboxallanswers=vbox{}

newenvironment{answer}{%
% OK but this is not colour safe, don't use colour in this environment
globalsetboxallanswers=vboxbgroup
unvboxallanswers
}%
{%
bigbreak
egroup
}

newcommand{showallanswers}{parunvboxallanswers}

newcommand*{getanswer}[6]{%
ifthenelse{equal{#6}{}}{}{
item[theenumi.]begin{answer}(#5)~#6end{answer}
}}

setlengthparindent{0pt}

newcommand{fourch}[6]{
par
begin{tabular}{*{4}{@{}p{0.23textwidth}}}
(a)~#1 & (b)~#2 & (c)~#3 & (d)~#4
end{tabular}
getanswer{#1}{#2}{#3}{#4}{#5}{#6}
getanswerkeys{#1}{#2}{#3}{#4}{#5}{#6}
}

newcommand{twoch}[6]{
par
begin{tabular}{*{2}{@{}p{0.46textwidth}}}
(a)~#1 & (b)~#2
end{tabular}
par%vspace{4pt}
begin{tabular}{*{2}{@{}p{0.46textwidth}}}
(c)~#3 & (d)~#4
end{tabular}
getanswer{#1}{#2}{#3}{#4}{#5}{#6}
getanswerkeys{#1}{#2}{#3}{#4}{#5}{#6}
}

newcommand{onech}[6]{
%par
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(a) & #1
end{tabular}vspace{2pt}
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(b) & #2
end{tabular}vspace{2pt}
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(c) & #3
end{tabular}vspace{2pt}
begin{tabular}{@{}p{0.04textwidth}@{}p{0.91textwidth}}
(d) & #4
end{tabular}

getanswer{#1}{#2}{#3}{#4}{#5}{#6}
getanswerkeys{#1}{#2}{#3}{#4}{#5}{#6}
}

newlengthwidthcha
newlengthwidthchb
newlengthwidthchc
newlengthwidthchd
newlengthwidthch
newlengthtabmaxwidth

setlengthtabmaxwidth{0.96textwidth}
newlengthfourthtabwidth
setlengthfourthtabwidth{0.25textwidth}
newlengthhalftabwidth
setlengthhalftabwidth{0.5textwidth}

newcommand{choice}[6]{%
settowidthwidthcha{AM.#1}setlength{widthch}{widthcha}%
settowidthwidthchb{BM.#2}%
ifdimwidthch<widthchbrelaxsetlength{widthch}{widthchb}fi%
settowidthwidthchb{CM.#3}%
ifdimwidthch<widthchbrelaxsetlength{widthch}  {widthchb}fi%
settowidthwidthchb{DM.#4}%
ifdimwidthch<widthchbrelaxsetlength{widthch} {widthchb}fi%

ifdimwidthch>halftabwidth
onech{#1}{#2}{#3}{#4}{#5}{#6}
elseifdimwidthch<halftabwidth
ifdimwidthch>fourthtabwidth
twoch{#1}{#2}{#3}{#4}{#5}{#6}
else
fourch{#1}{#2}{#3}{#4}{#5}{#6}
fififi}

newcommand{mcq}[8]{
item #2
choice{#3}{#4}{#5}{#6}{#7}{#8}
}

defdatabaseone{
mcq{245}{Let $A$ be a symmetric matrix of order $n$ and $B$ be a
% no space in skew- symmetric
skew-symmetric matrix of order $n$. Then which one of the following is a skew-symmetric matrix?}{$A^{2}$}{$B^{2}$}{$AB+BA$}{$AB-BA$}{c}{}{}

mcq{248}{A square matrix is a diagonal matrix if and only if it is}{upper triangular}{lower triangular}{both upper triangular and lower triangular}{none of these}{c}{}{}

mcq{249}{Let $A$ be a non-singular matrix of order $3$. Then the rank of matrix $A^{2018}$ is}{$1$}{$2$}{$3$}{$9$}{c}{Note that $A^{2018}$ is a non-singular matrix of order $3$. Hence rank $(A^{2018})=3$.}{}

mcq{250}{If $A=begin{pmatrix}0 & 1 & 0\0 & 0 & 1\1 & 0 & 0end{pmatrix}$, then rank of the matrix $A+A^{2}+A^{3}$ is}{$1$}{$2$}{$3$}{none of these}{a}{Note that $A+A^{2}+A^{3}=protectbegin{pmatrix}1 & 1 & 1\1 & 1 & 1\1 & 1 & 1protectend{pmatrix}$. Hence rank $(A+A^{2}+A^{3})=1$}{}
}

begin{document}
chapter{One}
begin{enumerate}
databaseone
end{enumerate}

%bigskip
% NO (this makes entire document Huge Huge{Answer Key:}
section*{Answer Key:}
%bigskip

begin{center}
begin{multicols}{6}
showallanswerkeys
end{multicols}
end{center}

%bigskip
%Huge
section*{Hint/Explanation of Selected Problems:}
%bigskipbigskip
showallanswers

chapter{Two}
begin{enumerate}
databaseone
end{enumerate}

%bigskip
%Huge
section*{Answer Key:}
%bigskip

begin{center}
begin{multicols}{6}
showallanswerkeys
end{multicols}
end{center}

%bigskip
%Huge
section*{Hint/Explanation of Selected Problems:}
%bigskipbigskip
showallanswers
end{document}