Question 1 :
If $\displaystyle  \begin{vmatrix} x & y   \\ 1 & 6   \end{vmatrix} $ = $\displaystyle  \begin{vmatrix} 1 & 8   \\ 1 & 6   \end{vmatrix} $ then x+2y=
Question 2 :
If every row of a matrix $A$ contains $p$ elements and its column contains $q$ elements, then the order of $A$ is
Question 3 :
If $A = \begin{bmatrix}1\end{bmatrix}$, then the order of the matrix is
Question 4 :
If P=$\displaystyle  \begin{bmatrix} 4 & 3 &2   \end{bmatrix}  $ and Q=$\displaystyle  \begin{bmatrix} -1 & 2 &3   \end{bmatrix}  $ then P-Q=
Question 5 :
If$\displaystyle \begin{vmatrix} 2 & 3 \\ 4 & 4 \end{vmatrix} $+$\displaystyle \begin{vmatrix} x & 3 \\ y & 1 \end{vmatrix} $=$\displaystyle \begin{vmatrix} 10 & 6 \\ 8 & 5 \end{vmatrix} $,then (x,y)=
Question 6 :
The value of x satisfying the equation 2$\displaystyle \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} $+$\displaystyle \begin{vmatrix} x^{2} & 9 \\ -1 & 0 \end{vmatrix} $=$\displaystyle \begin{vmatrix} 5x & 6 \\ 0 & 1 \end{vmatrix} $+$\displaystyle \begin{vmatrix} 0 & 5 \\ 1 & 3 \end{vmatrix} $are
Question 7 :
The number of possible orders of a matrix containing $24$ elements are:
Question 8 :
<b>If $A$ is a square of order $3$, then</b> $\left| Adj\left( Adj{ A }^{ 2 } \right)\right| =$
Question 9 :
If $A = \displaystyle \left[ \begin{matrix} 1 &2 \\ 3& 4 \end{matrix} \right] $, then number of elements in $A$ are
Question 10 :
If $A+B = \begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$ and $A-2B = \begin{bmatrix}-1 & 1 \\ 0 & -1\end{bmatrix}$, then $A$ =
Question 11 :
If a matrix is of order $2 \times 3$, then the number of elements in the matrix is<br>
Question 12 :
If A is $3 \times 4$ matrix and B is matrix such that A'B and BA' are both defined, then B is of the type.<br>
Question 13 :
If $\triangle =\left| \begin{matrix} arg{ z }_{ 1 } & arg{ z }_{ 2 } & arg{ z }_{ 3 } \\ arg{ z }_{ 2 } & arg{ z }_{ 3 } & arg{ z }_{ 1 } \\ arg{ z }_{ 3 } & arg{ z }_{ 1 } & arg{ z }_{ 2 } \end{matrix} \right|$, the, $\triangle$ is divided by:
Question 14 :
Unit matrix is a diagonal matrix in which all the diagonal elements are unity. Unit matrix of order 'n' is denoted by $I_n(or \ I)$ i.e. $A = [a_{ij}]_n$ is a  unit matrix when $a_{ij} = 0$ for $i \neq j \ and \ a_{ij} = 1$
Question 15 : <table class="wysiwyg-table"><tbody><tr><td></td><td>Day 1</td><td>Day 2</td><td>Day 3</td></tr><tr><td>Model X</td><td>$20$</td><td>$18$</td><td>$3$</td></tr><tr><td>Model Y</td><td>$16$</td><td>$5$</td><td>$8$</td></tr><tr><td>Model Z</td><td>$19$</td><td>$11$</td><td>$10$</td></tr></tbody></table>The table above shows the number of TV sets that were sold during a three-day sale. The prices of models $X, Y$ and $Z$ are $ $99$, $ $199$, and $ $299$, respectively. Which of the following matrix representations gives the total income, in dollars, received from the sale of the TV sets for each of the three days?
Question 16 :
If $\begin{bmatrix} i&0 \\3 &-i \end{bmatrix}+X=\begin{bmatrix} i&2 \\3 &4+i\end{bmatrix} - X$, then $X$ is equal to<br/>
Question 17 :
$\begin{vmatrix} x+5 & x \\ x+9 & x-2 \end{vmatrix}=0$ then x=
Question 18 : <p>The rank of the matrix</p>$\left[ {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> \lambda &2&4 \\ <br> 2&{ - 3}&1 <br>\end{array}} \right]$ is 3 if<br>
Question 19 :
If $A$ and $B$ are square matrices such that $B = -A^{-1} BA, \,$ then $\, (A + B)^2$ is equal to 
Question 20 :
A matrix has $16$ elements Which of the following can be the order of the matrix?
Question 21 :
If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix}$ = $(A+Bx)(x-A)^2$,<br>then the ordered pair $(A , B)$ is equal to:
Question 22 :
$\displaystyle \begin{vmatrix} 1 & a & {a}^{2}-bc \\ 1 & b & {b}^{2}-ca \\ 1 & c & {c}^{2}-ab \end{vmatrix}$=?
Question 23 :
If $A$ is $2\times 3$ matrix and $AB$ is a $2\times 5$ matrix, then $B$ must be a
Question 24 :
If $A = \bigl(\begin{smallmatrix}1 & -2\\ -3 & 4\end{smallmatrix}\bigr)$ and $A + B = O$, then B is<br>
Question 25 :
$\left[ \begin{matrix} x \\ 3 \end{matrix}\begin{matrix} 6 \\ 2x \end{matrix} \right]$ is a singular matrix, then $x$ is equal to
Question 26 :
Out of the following matrices, choose that matrix which is a scalar matrix.
Question 27 :
If $A = \dfrac {1}{\pi} \begin{bmatrix}\sin^{-1}(\pi x) & \tan^{-1} \left (\dfrac {\pi}{\pi}\right )\\ \sin^{-1} \left (\dfrac {x}{\pi}\right ) &\cot^{-1} (\pi x)\end{bmatrix}, B =\dfrac {1}{\pi} \begin{bmatrix}-\cos^{-1}(\pi x) &\tan^{-1} \left (\dfrac {x}{\pi}\right ) \\ \sin^{-1} \left (\dfrac {x}{\pi}\right ) & -\tan^{-1} (\pi x)\end{bmatrix}$, then $A - B$ is equal to<br/>
Question 28 :
A is of order $m \times n$ and B is of order $p \times q$, addition of A and B is possible only if<br>
Question 29 :
If the number of elements in a matrix is $60$ then how many different order of matrix are possible 
Question 30 :
If A =$\begin{bmatrix}1 & 2 \\ 3 & 4 \end{bmatrix}$, B =$\begin{bmatrix}2 & 3 \\ 4 & 5 \end{bmatrix}$, and 4A - 3B + C = 0, then C =
Question 31 :
Let $L$ denote the set of all straight lines in a plane, Let a relation $R$ be defined by $lRm$, iff $l$ is perpendicular to $m$ for all $l \in L$. Then, $R$ is
Question 32 :
Let N denote the set of all natural numbers and R a relation on $N\times N$. Which of the following is an equivalence relation?
Question 33 :
Let $f: N\rightarrow R$ such that $f(x)=\dfrac{2x-1}{2}$ and $g: Q\rightarrow R$such that $g(x)=x+2$ be two function. Then $(gof)\left(\dfrac{3}{2}\right)$ is equal to
Question 34 :
The relation $R$ in $N\times N$ such that $(a,b)R(c,d)\Leftrightarrow a+d=b+c$ is
Question 35 :
Let $f(x,y)=xy^{2}$ if $x$ and $y$ satisfy $x^{2}+y^{2}=9$ then the minimum value of $f(x,y)$ is
Question 37 :
If $f: A \rightarrow B$is a bijective function and if n(A) = 5, then n(B) is equal to
Question 39 :
If $R$ is the largest equivalence relation on a set $A$ and $S$ is any relation on $A$, then
Question 40 :
Let $A=\left\{ 2,3,4,5,....,17,18 \right\} $. Let $\simeq $ be the equivalence relation on $A\times A$, cartesian product of $A$ with itself, defined by $(a,b)\simeq (c,d)$, iff $ad=bc$. The the number of ordered pairs of the equivalence class of $(3,2)$ is
Question 42 :
Let S, T, U be three non-void sets and $f:S\rightarrow T, g:T\rightarrow U$ be so that g o f$:S\rightarrow$ U is surjective. Then?
Question 43 :
The function $f: R\rightarrow R$ given by $f(x) = x^{3} - 1$ is
Question 44 :
Let $R$ be a reflexive relation in a finite set having $n$ elements and let there be $m$ ordered pairs in $R$. Then,
Question 45 :
Let N denote the set of all natural numbers. Define two binary relations on N as $R_1=\{(x, y)\epsilon N\times N : 2x+y=10\}$ and $R_2=\{(x, y)\epsilon N\times N:x+2y=10\}$. Then.
Question 46 : Match the following lists :<br/><table class="wysiwyg-table"><tbody><tr><td>1) $f:R\rightarrow R$ defind by $f(x)=ax+b$ is  $(a\neq 0)$</td><td><b>a)</b> injection but not surjection</td></tr><tr><td>2) $f:R\rightarrow R$ defind by $f(x)=[x]$ is</td><td><b>b)</b> surjection but not injection</td></tr><tr><td>3) $f:R\rightarrow [0,\infty )$ defind by $f(x)=\left | x \right |$ is </td><td><b>c)</b> bijection</td></tr><tr><td>4)  $f:N\rightarrow N$ defind by $f(x)=x^{3}$ is</td><td><b>d)</b> neither injection nor surjection</td></tr></tbody></table>
Question 47 :
Let $A=\left \{ 1, 2, 3 \right \}$. Which of the following is not an equivalence relation on A?
Question 48 :
The number of bijection from the set $A$ to itselfwhen $A$ contains $106$ elements is
Question 49 :
If A ={1, 3, 5, 7} and B = {1, 2, 3, 4, 5, 6, 7, 8}, then the number of one-to-one functions from A into B is
Question 50 :
Let Z be the set of all integers and let R be a relation on Z defined by $a$ R $b\Leftrightarrow (a-b)$ is divisible by $3$. Then, R is?
Question 51 :
Which of the following functions from $Z$ to itself are bijections?
Question 52 :
Let $\displaystyle f\left ( x \right )=\frac{ax^{2}+2x+1}{2x^{2}-2x+1}$, the value of $a$ for which $\displaystyle f:R\rightarrow \left [ -1,2 \right ]$ is onto , is<br>
Question 54 :
If $f:R\rightarrow S$ defined by<br/>$f(x)=4\sin { x } -3\cos { x } +1$ is onto, then $S$ is equal to
Question 55 :
The function $f:\left[ -\dfrac {1}{2},\dfrac {1}{2} \right] \rightarrow \left[ -\dfrac {\pi }{2},\dfrac {\pi }{2} \right] $ defined by $f(x)=\sin ^{ -1 }{ \left( 3x-4{ x }^{ 3 } \right)  } $ is
Question 57 :
Let $f:N\rightarrow N$ ($N$ being the set of positive integers) be a function defined by $f(x)=$ the biggest positive integer obtained by reshuffling the digits of $x$. For example, $f(296)=962$<br>$f$ is
Question 58 :
Let $R$ a relation on the set $N$ be defined by $\left\{ \left( x,y \right) |x,y\in N,2x+y=41 \right\}$. Then $R$ is
Question 60 :
If $f:R\rightarrow \left [\dfrac {\pi}{6}, \dfrac {\pi}{2}\right ), f(x)=\sin^{-1}\left (\dfrac {x^2-a}{x^2+1}\right )$ is a onto function, then set of values of $a$ is
Question 61 :
Consider the following statements:<br/>1. $\tan^{-1} 1+ \tan^{-1} (0.5) = \dfrac {\pi}2$<br/>2. $\sin^{-1}{\cfrac{1}{3} }+ \cos^{-1}{\cfrac{1}{3}} =\cfrac{\pi}{2}$<br/>Which of the above statements is/are correct ? 
Question 62 :
Solve $\cos { \left[ \tan ^{ -1 }{ \left[ \sin { \left( \cot ^{ -1 }{ x }  \right)  }  \right]  }  \right]  } $
Question 63 :
$\quad \sin ^{ -1 }{ x } +\sin ^{ -1 }{ \cfrac { 1 }{ x } } +\cos ^{ -1 }{ x } +\cos ^{ -1 }{ \cfrac { 1 }{ x } = } $
Question 65 :
The value of $\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$ is 
Question 67 :
Consider $x = 4\tan^{-1}\left (\dfrac {1}{5}\right ), y = \tan^{-1} \left (\dfrac {1}{70}\right )$ and $z = \tan^{-1}\left (\dfrac {1}{99}\right )$.What is $x$ equal to?
Question 68 :
Solve ${\cos ^{ - 1}}\left( {\frac{4}{5}} \right) + {\cos ^{ - 1}}\left( {\frac{{63}}{{65}}} \right) = $
Question 70 :
Calculate the value of $\displaystyle \sin^{-1} \cos \left ( \sin^{-1} x\right ) + \cos^{-1} \sin \left ( \cos^{-1} x \right ) $. where $\displaystyle\left | x \right | \leq 1$
Question 71 :
What is $\sin { \left[ \sin ^{ -1 }{ \left( \cfrac { 3 }{ 5 } \right) } +\sin ^{ -1 }{ \left( \cfrac { 4 }{ 5 } \right) } \right] } $ equal to?
Question 73 :
If two angles of a triangle are $\tan ^{ -1 }{ (2) } $ and $\tan ^{ -1 }{ (3) } $, then the third angle is
Question 74 :
Consider the following :<br>1. ${\sin}^{-1}\dfrac{4}{5}+{\sin}^{-1}\dfrac{3}{5}=\dfrac{\pi}{2}$<br>2. ${\tan}^{-1}\sqrt{3}+{\tan}^{-1}1=-{\tan}^{-1}(2+\sqrt{3})$<br>Which of the above is/are correct?
Question 75 :
If $\sin ^{ -1 }{ \left( \cfrac { x }{ 13 } \right) } +co\sec ^{ -1 }{ \left( \cfrac { 13 }{ 12 } \right) } =\cfrac { \pi }{ 2 } $, then the value if $x$ is
Question 76 :
If $\dfrac {(x + 1)^{2}}{x^{3} + x} = \dfrac {A}{x} + \dfrac {Bx + C}{x^{2} + 1}$, then $\csc^{-1}\left (\dfrac {1}{A}\right ) + \cot^{-1}\left (\dfrac {1}{B}\right ) + \sec^{-1}C =$ ____
Question 77 :
The value of $\tan { \left[ \dfrac { 1 }{ 2 } \cos ^{ -1 }{ \left( \dfrac { 2 }{ 3 } \right) } \right] } $ is
Question 78 :
The number of solutions for the equation $2\sin ^{ -1 }{ \sqrt { { x }^{ 2 }-x+1 }  } +\cos ^{ -1 }{ \sqrt { { x }^{ 2 }-x }  } =\dfrac { 3\pi }{ 2 } $ is
Question 79 :
The value of $ \cos \left( \sin^{-1} \left( \dfrac {2}{3} \right) \right) $ is equal to :
Question 80 :
$ \sin \left( 2 \sin^{-1} \sqrt{\dfrac{63}{65}} \right) $<br/>is equal to :
Question 81 :
$\tan { ^{ -1 }\left( 3/5 \right) } +\tan { ^{ -1 }\left( 1/4 \right) } =$
Question 82 :
$\displaystyle \sum _{ k=1 }^{ k=n }{ \tan ^{ -1 }{ \frac { 2k }{ 2+{ k }^{ 2 }+{ k }^{ 4 } }  }  } =\tan ^{ -1 }{ \left( \dfrac { 6 }{ 7 }  \right)  } $, then the value of '$n$' is equal to
Question 83 :
The value of $a$ for which $\displaystyle ax^{2}+sin^{-1}(x^{2}-2x+2)+cos^{-1}(x^{2}-2x+2)=0$ has areal solution is
Question 84 :
Let $a, b, c$ be a positive real numbers $\theta = \tan^{-1} \sqrt{\dfrac{a(a + b +c)}{bc}} + \tan^{-1} \sqrt{\dfrac{b(a + b+ c)}{ca}} + \tan^{-1} \sqrt{\dfrac{c(a + b + c)}{ab}}$, then $\tan \theta$<br>
Question 85 :
If $3\cos ^{ -1 }{ x } +\sin ^{ -1 }{ x } =\pi $, then $x=.....$
Question 86 :
If $sin^{-1} \left( tan \dfrac {17\pi} {{4}}  \right )- sin^{-1} \left ( \sqrt{ \dfrac {3}{x}} \right ) - \left (\dfrac {\pi}{6} \right ) = 0$, then x is a root of the equation
Question 89 :
Domain of $f(x)=\cot ^{ -1 }{ x } +\cos ^{ -1 }{ x } +co\sec ^{ -1 }{ x } $ is
Question 90 :
The number of solution of the equation $ 1+x^{2}+2x\:\sin \left ( \cos^{-1}y \right )= 0 $ is :
Question 91 :
$\begin{vmatrix} 2^3 & 3^3 & 3.2^2+3.2+1\\ 3^3 & 4^3 & 3.3^2+3.3+1\\ 4^3 & 5^3 & 3.4^2+3.4+1\end{vmatrix}$ is equal to?
Question 92 :
Let $\omega\neq{1}$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $ \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ { \omega }^{ 2 } & \omega & 1 \end{bmatrix}$Where each of $a,\ b$ and $c$ is either $\omega$ or ${\omega}^{2}$. Then the number of distinct matrices in the set $S$ is
Question 93 :
If the value of the determinant $\begin{vmatrix}m & 2\\ -5 & 7\end{vmatrix}$ is $31$, find $m$.
Question 94 :
$x = \left| \begin{gathered}   - 1\,\,\,\,\,\, - 2\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\, - 2\,\,\,\,\,\,\,\,\,\,1 \hfill \\ \end{gathered}  \right|$, then $x=$
Question 95 :
If $A$ is a $3\times 3$ matrix and $\text{det}  (3A)=k(\text{det}  A)$, then $k=$
Question 96 :
If $\omega$ is a non-real cube root of unity and n is not a multiple of 3, then $\displaystyle \Delta =\left | \begin{matrix}<br>1 & \omega^{n} &\omega^{2n} \\ <br>\omega^{2n}&1 &\omega^{n} \\ <br>\omega^{n}&\omega^{2n} &1 <br>\end{matrix} \right |$ is equal to<br>
Question 97 :
If abc $\neq $0 and if $\begin{vmatrix}<br/>a & b & c\\ <br/>b & c & a\\ <br/>c & a & b<br/>\end{vmatrix}$ = 0 then $\dfrac{a^{3}+b^{3}+c^{3}}{abc}$ 
Question 98 :
$\mathrm{If}$ $\left|\begin{array}{lll}<br>1 & 0 & 0\\<br>2 & 3 & 4\\<br>5 & -6 & x<br>\end{array}\right|$ $= 45$ $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$ $\mathrm{x}=$<br><br>
Question 99 :
Find the value of the following determinant:<br/>$\begin{vmatrix}3\sqrt{6} & -4\sqrt{2}\\ 5\sqrt{3} & 2\end{vmatrix}$
Question 100 :
If $A$ is any skew-symmetric matrix of odd order then $\left| A \right| $ equals
Question 101 :
If $x, y, z$ are positive numbers, then value of the determinant $\begin{vmatrix}1 & log_xy & log_xz \\ log_yx & 1 & log_yz\\ log_zx & log_zy & 1\end{vmatrix}$ is equal to<br/>
Question 102 :
If $m = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$ and $n = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$, then what is the value of the determinant of $m \cos \theta - n \sin \theta$?
Question 103 :
The roots of the equation $\begin{vmatrix}x-1&1 &1\\1&x-1 &1\\1&1&x-1\end{vmatrix} = 0$ are
Question 104 :
If $\begin{vmatrix}p& q - y& r - z\\ p - x& q & r - z\\ p - x& q - y& r\end{vmatrix} = 0$, then the value of $\dfrac {p}{x} + \dfrac {q}{y} + \dfrac {r}{z}$ is<br>
Question 105 :
If $A=\begin{bmatrix}\alpha &2 \\2 &\alpha \end{bmatrix}$ and $|A^3|=125$, then $\alpha$ is equal to<br>
Question 106 :
The solution set of the equation$\begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5{ x }^{ 2 } \end{vmatrix}=0$, is<br>
Question 107 :
<br/>Let $\displaystyle f(\theta)=\begin{vmatrix}\cos\dfrac{\theta}{2}<br/> &1  &1 \\1 &\cos \dfrac{\theta}{2}  &-\cos\dfrac{\theta}{2} \\-\cos\dfrac{\theta}{2}  &1  &-1 \end{vmatrix}$ <br> $\displaystyle f(\pi) +f(-\pi) $ is equal to
Question 108 :
$|f(x)|={\begin{bmatrix}{}<br/>\mathrm{s}\mathrm{i}\mathrm{n}x & \mathrm{c}\mathrm{o}\mathrm{s}ecx & \mathrm{t}\mathrm{a}\mathrm{n}x\\<br/>\mathrm{s}\mathrm{e}\mathrm{c}x & x\mathrm{s}\mathrm{i}\mathrm{n}x & x\mathrm{t}\mathrm{a}\mathrm{n}x\\<br/>x^{2}-1 & \mathrm{c}\mathrm{o}\mathrm{s}x & x^{2}+1<br/>\end{bmatrix}}$ then . $.\displaystyle \int_{-a}^{a}|f(x)|d$ equals <br/><br/><br/>
Question 109 :
The determinant $\begin{vmatrix} { y }^{ 2 } & -xy & { x }^{ 2 } \\ a & b & c \\ a' & b' & c' \end{vmatrix}$ is equivalent to<br/>
Question 110 :
What is the value of a + b + c + d?
Question 111 :
If A is a square matrix of order 3, then $|(A - A^T)^{105}|$ is equal to
Question 113 :
$\det. \:[A_0 +  A_0^2B_0^2 + A_0^3 + A_0^4B_0^4 + \:.......\:10 \:terms]$ is equal to :<br/>
Question 114 :
The number of distinct real roots of the equation, $\begin{vmatrix} \cos x& \sin x & \sin x\\ \sin x & \cos x & \sin x\\ \sin x & \sin x & \cos x\end{vmatrix} = 0$ in the interval $\left [-\dfrac {\pi}{4}, \dfrac {\pi}{4}\right ]$ is/are :<br/>
Question 115 :
The determinant $\begin{vmatrix}<br>\sin \alpha & \cos \alpha & 1\\ <br>\sin \beta & \cos \beta & 1\\ <br>\sin \gamma & \cos \gamma & 1<br>\end{vmatrix}$ is equal to<br>
Question 117 :
Let k be a positive real number and let $A = \begin{bmatrix}2k-1 & 2\sqrt{k} & 2\sqrt{k}\\ 2\sqrt{k} & 1 & -2k\\ -2\sqrt{k} & 2k & -1\end{bmatrix}$ and $B=\begin{bmatrix}0 & 2k-1 & \sqrt{k}\\ 1-2k & 0 & 2\sqrt{k}\\ -\sqrt{k} & -2\sqrt{k} & 0\end{bmatrix}$ . If det $(adj A) + det (adj B) = 10^{6}, then [k]$ is equal to <br/><br/>[Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].
Question 118 :
If $f(x)=a+bx+{ cx }^{ 2 }$ and $\alpha ,\beta ,\gamma $ are the roots of the equation ${ x }^{ 3 }=1$, then $\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right| $ is equal to<br>
Question 119 :
The value of the determinant $\displaystyle \left| \begin{matrix} 1+a \\ 1 \\ 1 \end{matrix}\,\,\,\begin{matrix} 1 \\ 1+a \\ 1 \end{matrix}\,\,\,\begin{matrix} 1 \\ 1 \\ 1+a \end{matrix} \right| $ is 
Question 120 :
If $\displaystyle \begin{vmatrix}<br/>x^{k} & x^{k+2} &x^{k+3} \\ <br/>y^{k} & y^{k+2} & y^{k+3}\\ <br/>z^{k} & z^{k+2} & z^{k+3}<br/>\end{vmatrix}$<br/>$\displaystyle =\left ( x-y \right )\left ( y-z \right )\left ( z-x \right )\left ( \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} \right )$<br/>then<br/>
