Tag: mathematics and statistics

Questions Related to mathematics and statistics

Let $\displaystyle f(x)=\left ( 2-\dfrac{x}{a} \right )^{\tan \left ( \dfrac{\pi :x }{2:a} \right )}, x\neq a$. The value which should be assigned to $f$ at $x=a$ so that it is continuous everywhere is

mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

  1. $\dfrac{2}{\pi} $

  2. $e^{-2/\pi }$

  3. $2$

  4. $e^{2/\pi }$

Show answer
Correct Option: D
Explanation:

For $f$ to be continuous, we must have
$\displaystyle f(a)=\lim _{x\rightarrow a}\left ( 2-\dfrac{x}{a} \right )^{\tan \left ( \dfrac{\pi :x }{2:a} \right )}$
$\Rightarrow log  f(x)=\tan \left ( \dfrac{\pi :x }{2:a} \right )log(2-\dfrac{x}{a})$
$=\displaystyle \dfrac{log(2-\dfrac{x}{a})}{\cot \left ( \dfrac{\pi }{2}\dfrac{x}{a} \right )}\left ( \dfrac{0}{0}form \right )$
$\displaystyle \lim _{x\rightarrow a} log f(x)=\lim _{x\rightarrow a}\dfrac{\dfrac{-1/a}{2-\dfrac{x}{a}}}{-\dfrac{\pi }{2a}\csc ^{2}\left ( \dfrac{\pi }{2}\dfrac{x}{a} \right )}$
(L'Hospital's Rule)
$=\displaystyle \dfrac{2}{\pi }\lim _{x\rightarrow a}\dfrac{1}{2-\dfrac{x}{a}}\times \lim _{x\rightarrow a}\sin ^{2}\left ( \dfrac{\pi }{2}\dfrac{x}{a} \right )$
$=\displaystyle \dfrac{2}{\pi }$
Therefore, $\displaystyle \lim _{x\rightarrow a}f(x)=e^{2/\pi }$
Hence, option 'D' is correct.

If $f(x)=\left{\begin{matrix} |x|-3, & x < 1\ |x-2|+a, & x\geq 1\end{matrix}\right.$ and $g(x)=\left{\begin{matrix} 2-|x|, & x < 2 \ sgn(x)-b, & x\geq 2\end{matrix}\right.$ and $h(x)=f(x)+g(x)$ is discontinuous at exactly one point, then which of the following values of a and b are possible.

mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

  1. $a=-3, b=0$

  2. $a=2, b=1$

  3. $a=2, b=0$

  4. $a=-3, b=1$

Show answer
Correct Option: A,B

If $\Delta =\begin{vmatrix} { a } _{ 11 } & { a } _{ 12 } & { a } _{ 13 } \ { a } _{ 21 } & { a } _{ 22 } & { a } _{ 23 } \ { a } _{ 31 } & { a } _{ 32 } & { a } _{ 33 } \end{vmatrix}$ and ${ A } _{ ij }$ is cofactors of ${ a } _{ ij }$, then the value of $\Delta $ is given by

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. ${ a } _{ 11 }{ A } _{ 31 }+{ a } _{ 12 }{ A } _{ 32 }+{ a } _{ 13 }{ A } _{ 33 }$

  2. ${ a } _{ 11 }{ A } _{ 11 }+{ a } _{ 12 }{ A } _{ 21 }+{ a } _{ 13 }{ A } _{ 31 }$

  3. ${ a } _{ 21 }{ A } _{ 11 }+{ a } _{ 22 }{ A } _{ 12 }+{ a } _{ 23 }{ A } _{ 13 }$

  4. ${ a } _{ 11 }{ A } _{ 11 }+{ a } _{ 21 }{ A } _{ 21 }+{ a } _{ 31 }{ A } _{ 31 }$

Show answer
Correct Option: D
Explanation:

$\Delta =\begin{vmatrix} { a } _{ 11 } & { a } _{ 12 } & { a } _{ 13 } \ { a } _{ 21 } & { a } _{ 22 } & { a } _{ 23 } \ { a } _{ 31 } & { a } _{ 32 } & { a } _{ 33 } \end{vmatrix}$


Also given ${ A } _{ ij }$ is cofactor of ${ a } _{ ij }$
$\Delta =a _{11} A _{11}+a _{21} A _{21}+a _{31} A _{31}$

$A=\left{\begin{array}{ll}
8 & 9\
10 & 11
\end{array}\right}$, then cofactor of $\mathrm{a} _{12}$ is:

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. 11

  2. 10

  3. -11

  4. -10

Show answer
Correct Option: D
Explanation:

By property of cofactor (2),
$A=\begin{bmatrix}
8 & 9\
10 & 11
\end{bmatrix}$
minor of $a _{12}=M _{12}=10$
So, cofactor of $a _{12}=M _{12}(-1)^{1+2}=-M _{12}=-10$

If $\triangle =\begin{bmatrix} { a } _{ 1 } & { b } _{ 1 } & { c } _{ 1 } \ { a } _{ 2 } & { b } _{ 2 } & { c } _{ 2 } \ { a } _{ 3 } & { b } _{ 3 } & { c } _{ 3 } \end{bmatrix}$ and ${A} _{2},{B} _{2},{C} _{2}$ are respectively cofactors of ${a} _{2},{b} _{2},{c} _{2}$ then ${a} _{1}{A} _{2}+{b} _{1}{B} _{2}+{c} _{1}{C} _{2}$ is equal to ?

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. $-\triangle$

  2. $0$

  3. $\triangle$

  4. $none\ of\ these$

Show answer
Correct Option: B
Explanation:

Co-factor of $ \displaystyle a _2 = (-1)^{i+j}\left|\begin{matrix} b _1 & c _1 \ b _3 & c _3 \end{matrix}\right| = - [b _1 \, c _3 - c _1 \, b _3] = A _2 $

i=2  j=1

of $ \displaystyle b _2 = (-1)^{2+2} \left|\begin{matrix} a _1 & c _1 \\ a _3 & c _3 \end{matrix}\right| = [a _1 \, c _3 - c _1 \, a _3] = B _2 $ 

of $ \displaystyle c _2 = (-1)^{2+3} \left|\begin{matrix} a _1 & b _1 \\ a _3 & b _3 \end{matrix}\right| = -[a _1 \, b _3 \, - b _1 \, a _3 ] = C _2 $

$ a _1 \, A _2 + b _1 \, B _2 +c _1 \, C _2 $

$ \displaystyle -a _1 \, b _1 \, c _3 + a _1 \, c _1 \, b _3 + b _1 \, a _1 \, c _3 - b _1 c _1 \, a _3 -c _1 \, a _1 \, b _3 + c _1 \, b _1 \, a _3 $

$ \displaystyle = 0 $

If $\Delta = \begin{vmatrix}a _1 & b _1 & c _1 \ a _2 & b _2 & c _2\ a _3 & b _3 & c _3\end{vmatrix}$ and $A _1, B _1, C _1$ denote the co-factors of $a _1, b _1, c _1$ respectively, then teh value os the determinant $\begin{vmatrix}A _1 & B _1 & C _1\ A _2 & B _2 & C _2\ A _3 & B _3 & C _3\end{vmatrix}$ is-

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. $\Delta$

  2. $\Delta^2$

  3. $\Delta^3$

  4. $0$

Show answer
Correct Option: B
Explanation:

Given,

$\Delta = \begin{vmatrix}a _1 & b _1 & c _1 \ a _2 & b _2 & c _2\ a _3 & b _3 & c _3\end{vmatrix}$ and $A _1, B _1, C _1$ denote the co-factors of $a _1, b _1, c _1$ respectively.
Now,
 $\begin{vmatrix}A _1 & B _1 & C _1\ A _2 & B _2 & C _2\ A _3 & B _3 & C _3\end{vmatrix}$
$=|adj \Delta|$
$=\Delta^{3-1}$
$=\Delta^2$.

If $\Delta  = \left| {\begin{array}{*{20}{c}}  {{a _1}}&{{b _1}}&{{c _1}} \   {{a _2}}&{{b _2}}&{{c _2}} \   {{a _3}}&{{b _3}}&{{c _3}} \end{array}} \right|$ and $A _2$, $B _2$, $C _2$ are respectively cofactors of $a _2,b _2,c _2$ then 


$a _1A _2+b _1B _2+c _1C _2$ is 

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. $ - \Delta $

  2. $0$

  3. $\Delta $

  4. none of these

Show answer
Correct Option: B
Explanation:
$\Delta =\begin{vmatrix} { a } _{ 1 } & { b } _{ 1 } & { c } _{ 1 } \\ { a } _{ 2 } & { b } _{ 2 } & { c } _{ 2 } \\ { a } _{ 3 } & { b } _{ 3 } & { c } _{ 3 } \end{vmatrix}$

$A _2=c _3\ b _1+b _1\ c _1$

$B _2=a _1\ c _3+a _3\ b _1$

$C _2=-a _1\ b _3+a _3\ b _1$

$\to \ a _1 A _2+b _1 B _2+c _1 C _2$

$a, b, c _3+a, c, b _3+b, a, c _3-b, a, c _3-b, c, a _3-c, a, b _3+b, c, a _3$

$=0$

$B$ is correct.

The value of a third order determinant is $11$, then the value of the square of the determinant formed by the cofactors will be?

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. $11$

  2. $121$

  3. $1331$

  4. $14641$

Show answer
Correct Option: D
Explanation:
third order determinant = determinant of $3\times 3$ matrix $A$
given $|A|=11$
det (cofactor matrix of $A$) =set (transpare of cofactor amtrix of $A$) (transpare does not change the det)
=det(adjacent of $A$)
$\left\{det\ (cofactor\ matrix\ of\ A) \right\}^2=\left\{det\ (adjacent\ of\ A)\right\}^2$
(Using for an $n\times n\ det\ (cofactor\ matrix\ of\ A)=det\ (A)^{n-1})$
we get
$det\ (cofactor\ matrix\ of\ A)^2=\left\{det (A)^{3-1}\right\}^2$
$=(11)^{2\times 2}=11^4$
$=146.41$

Consider the determinant, $\Delta=\begin{vmatrix} p & q & r \ x & y & z \ l & m & n \end{vmatrix}$ ${M} _{0}$ denotes the minor of an element in $i$th row and $j$th column and ${C} _{ij}$ denotes the cofactor of an element in $i$th row and $j$th column.
The value of $p.{C} _{21}+q.{C} _{22}+r.{C} _{23}$ is equal to

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. $0$

  2. $-\Delta$

  3. $\Delta$

  4. ${\Delta}^{2}$

Show answer
Correct Option: A
Explanation:

From the property of determinants, if any element is multiplied with cofactor of corresponding element of another row, and summed up for each element of original row, the sum comes out zero $\Rightarrow (A)$

The cofactor of the element $4$ in the determinant $\begin{vmatrix} 1 & 3 & 5 & 1\ 2 & 3 & 4 & 2\ 8 & 0 & 1 & 1\ 0 & 2 & 1 & 1\end{vmatrix}$ is?

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

  1. $4$

  2. $10$

  3. $-10$

  4. $-4$

Show answer
Correct Option: A