DETERMINANTS

Minors & cofactors of elements of a determinant.

It we delete the row and column passing through the element ${\mathrm{a}}_{\mathrm{i}}$, the determinant thus obtained is called the minor of $a_{ij}$ denoted by $M_{ij}$ and cofactor of $a_{ij}$ is $(-1{)}^{i+j}{M}_{ij}$ and is denoted by $A_{ij}$ or $C_{ij}$.

Note :

If all the elements in a row (or column), except one element, are zeros the determinant reduces to a determinant of an order less by one.

$\mathrm{Eg}:\left|\begin{array}{ccc}5& 0& 0\\ 7& 3& -2\\ 1& 5& 4\end{array}\right|=5\left|\begin{array}{cc}3& -2\\ 5& 4\end{array}\right|$

Also a determinant can be replaced by a determinant of higher order by one.

Eg: $\left|\begin{array}{cc}3& 2\\ 8& -5\end{array}\right|=\left|\begin{array}{ccc}1& 0& 0\\ 0& 3& 2\\ 0& 8& -5\end{array}\right|$

Singular or non singular matrix :

A square matrix A is said to be non-singular if $|\mathrm{A}|\ne 0$, and is said to be singular if $|\mathrm{A}|=0$

Properties of determinants

For a square matrix A,

(i) If a row (column) is a zero vector, then $|\mathrm{A}|=0$

(ii) If any two rows (columns) are proportional, then $|\mathrm{A}|=0$

(iii) If the rows & columns are interchanged, then $|\mathrm{A}|$ remains the same i.e. $|\mathrm{A}|=\left|{\mathrm{A}}^{\mathrm{T}}\right|$

(iv) If any two rows (columns) are interchanged, value of $|\mathrm{A}|$ differ by a negative sign.

(v) $\left|\begin{array}{ccc}\mathrm{ka}& \mathrm{kb}& \mathrm{kc}\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|=\mathrm{k}\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|$

(vi) If A is a square matrix of order $n$, then $|\mathrm{kA}|={\mathrm{k}}^{\mathrm{n}}|\mathrm{A}|$

(vii) $\left|\begin{array}{ccc}{\mathrm{a}}_1+{\mathrm{a}}_2& {\mathrm{b}}_1+{\mathrm{b}}_2& {\mathrm{c}}_1+{\mathrm{c}}_2\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|+\left|\begin{array}{ccc}{\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|$

(viii) If a scalar multiple of any row (column) is added to another row (column) then $|\mathrm{A}|$ is unchanged.

i.e. $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|=\left|\begin{array}{ccc}\mathrm{a}+\lambda \mathrm{p}& \mathrm{b}+\lambda \mathrm{q}& \mathrm{c}+\lambda \mathrm{r}\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|$

(ix) The sum of the products of elements of any row (column) of a determinant with the cofactors of the corresponding elements of any other row (column) is zero

If, A = $\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\text{, then }{a}_{11}{C}_{21}+a_{12}{C}_{22}+a_{13}{C}_{23}=0$

(x) The sum of the products of elements of any row (or column) of a determinant with the cofactors of the corresponding elements of same row is $|\mathrm{A}|$

i.e. ${\mathrm{a}}_{11}{\mathrm{C}}_{11}+{\mathrm{a}}_{12}{\mathrm{C}}_{12}+{\mathrm{a}}_{13}{\mathrm{C}}_{13}=|\mathrm{A}|$.

(xi) If r rows (columns) become identical when ’ $a$ ’ is substituted for $\mathrm{x}$, then $(\mathrm{x}-\mathrm{a}{)}^{r-1}$ is a factor of the given determinant.

$\mathrm{Eg}:\mathrm{Let}A(x)=\left|\begin{array}{ccc}x-2& x-3& x-4\\ x-3& x-5& 2x-8\\ x-4& x-6& 3x-10\end{array}\right|$

Put $x=1$

$\mathrm{\Delta }=\left|\begin{array}{lll}-1& -2& -3\\ -2& -4& -6\\ -3& -5& -7\end{array}\right|=0$

$({\mathrm{R}}_1\mathrm{\&}{\mathrm{R}}_2$ are proportional)

(xii) If $\mathrm{A}=\mathrm{diag}({\mathrm{a}}_{11},{\mathrm{a}}_{22},\dots \dots \dots \dots \dots \dots ..{\mathrm{a}}_{\mathrm{nn}})$ then

$|A|=a_{11}{a}_{22}\dots \dots \dots \dots a_{nn}$

(xiii) $|\mathrm{AB}|=|\mathrm{A}||\overrightarrow{\mathrm{B}}|=|\mathrm{BA}|=\left|{\mathrm{AB}}^{\mathrm{T}}\right|=\left|{\mathrm{A}}^{\mathrm{T}}\mathrm{B}\right|=\left|{\mathrm{A}}^{\mathrm{T}}{\mathrm{B}}^{\mathrm{T}}\right|$

(xiv) Let $\mathrm{\Delta }(x)=\left|\begin{array}{lll}a(x)& b(x)& c(x)\\ p(x)& q(x)& r(x)\\ u(x)& v(x)& w(x)\end{array}\right|$, then

${\displaystyle \frac{d}{dx}}\mathrm{\Delta }(x)=\left|\begin{array}{ccc}{a}^1(x)& b^1(x)& c^1(x)\\ p(x)& q(x)& r(x)\\ u(x)& v(x)& w(x)\end{array}\right|+\left|\begin{array}{ccc}a(x)& b(x)& c(x)\\ p^1(x)& q^1(x)& r^1(x)\\ u(x)& v(x)& w(x)\end{array}\right|+\left|\begin{array}{ccc}a(x)& b(x)& c(x)\\ p(x)& q(x)& r(x)\\ u^1(x)& v^1(x)& w^1(x)\end{array}\right|$

(xv) Product of determinants of same order

$\begin{array}{rl}& \left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\times \left|\begin{array}{lll}{\alpha }_1& {\beta }_1& {\gamma }_1\\ {\alpha }_2& {\beta }_2& {\gamma }_2\\ {\alpha }_3& {\beta }_3& {\gamma }_3\end{array}\right|\\ & =\left|\begin{array}{lll}{a}_1{\alpha }_1+b_1{\beta }_1+c_1{\gamma }_1& a_1{\alpha }_2+b_1{\beta }_2+c_1{\gamma }_2& a_1{\alpha }_3+b_1{\beta }_3+c_1{\gamma }_3\\ a_2{\alpha }_1+b_2{\beta }_1+c_2{\gamma }_1& a_2{\alpha }_2+b_2{\beta }_2+c_2{\gamma }_2& a_2{\alpha }_3+b_2{\beta }_3+c_2{\gamma }_3\\ a_3{\alpha }_1+b_3{\beta }_1+c_3{\gamma }_1& a_3{\alpha }_2+b_3{\beta }_2+c_3{\gamma }_2& a_3{\alpha }_3+b_3{\beta }_3+c_3{\gamma }_3\end{array}\right|\end{array}$

Multiplication can also be performed row by column ; column by row or column by column as required in the problem.

(xvi) $\left|{\mathrm{A}}^{\mathrm{n}}\right|=|\mathrm{A}{|}^{\mathrm{n}}$

(xvii) Determinant of a skew symmetric matrix of odd order is zero

Use of Determinants :

(i) Area of triangle whose vertices are $({\mathrm{x}}_1,{\mathrm{y}}_1)({\mathrm{x}}_2,{\mathrm{y}}_2)({\mathrm{x}}_3,{\mathrm{y}}_3)$ is given by

$\mathrm{\Delta }={\displaystyle \frac{1}{2}}\left|\begin{array}{lll}{\mathrm{x}}_1& {\mathrm{y}}_1& 1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& 1\\ {\mathrm{x}}_3& {\mathrm{x}}_3& 1\end{array}\right|$

(ii) If $a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0\mathrm{\&}{a}_{3}x+b_{3}y+c_3=0$ are the sides of a triangle, the area of the triangle is given by

$\mathrm{\Delta }={\displaystyle \frac{1}{2{\mathrm{C}}_1{\mathrm{C}}_2{\mathrm{C}}_3}}{\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|}^2$ where ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3$ are the cofactor of ${\mathrm{c}}_1,{\mathrm{c}}_2\mathrm{\&}{\mathrm{c}}_3$ respectively in $\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$

(ii) ${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0$ represents a pair of straight lines, then

$abc+2fgh-a{f}^2-b^2-c^2=\left|\begin{array}{lll}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$

Cramer’s rule for solving simultaneous linear equations

Consider the system of equations

${\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1\mathrm{z}={\mathrm{d}}_1$

${\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2\mathrm{z}={\mathrm{d}}_2$

${\mathrm{a}}_3\mathrm{x}+{\mathrm{b}}_3\mathrm{y}+{\mathrm{c}}_3\mathrm{z}={\mathrm{d}}_3$

Here $\mathrm{\Delta }=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$,

${\mathrm{\Delta }}_1=\left|\begin{array}{lll}{\mathrm{d}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{d}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{d}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$

${\mathrm{\Delta }}_2=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{d}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{d}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{d}}_3& {\mathrm{c}}_3\end{array}\right|$

${\mathrm{\Delta }}_3=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{d}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{d}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{d}}_3\end{array}\right|$

By cramer’s rule,

$\mathrm{x}={\displaystyle \frac{{\mathrm{\Delta }}_1}{\mathrm{\Delta }}},\mathrm{y}={\displaystyle \frac{{\mathrm{\Delta }}_2}{\mathrm{\Delta }}},\mathrm{z}={\displaystyle \frac{{\mathrm{\Delta }}_3}{\mathrm{\Delta }}}$

In the above system, if ${\mathrm{d}}_1={\mathrm{d}}_2={\mathrm{d}}_3=0$, it is called a homogeneous system.

Solved Examples

1. The value of the determinant

$\left|\begin{array}{lll}\cos (\mathrm{A}-\mathrm{P})& \cos (\mathrm{A}-\mathrm{Q})& \cos (\mathrm{A}-\mathrm{R})\\ \cos (\mathrm{B}-\mathrm{P})& \cos (\mathrm{B}-\mathrm{Q})& \cos (\mathrm{B}-\mathrm{R})\\ \cos (\mathrm{C}-\mathrm{P})& \cos (\mathrm{C}-\mathrm{Q})& \cos (\mathrm{C}-\mathrm{R})\end{array}\right|\text{ is }$

(a) 0

(b) 1

(c) $\sin 2\mathrm{A}\sin 2\mathrm{B}\sin 2\mathrm{C}$

(d) None of these

2. If $\left|\begin{array}{lll}{(a_1-b_1)}^2& {(a_1-b_2)}^2& {(a_1-b_3)}^2\\ {(a_2-b_1)}^2& {(a_2-b_2)}^2& {(a_2-b_3)}^2\\ {(a_3-b_1)}^2& {(a_3-b_2)}^2& {(a_3-b_3)}^2\end{array}\right|=k(a_1-a_2)(a_2-a_3)(a_3-a_1)(b_1-b_2)(b_2-b_3)(b_3-b_1)\text{, then }k$

is equal to

(a) 1

(b) 2

(c) 4

(d) 8

3. If the value of determinant $\left|\begin{array}{lll}\mathrm{a}& 1& 1\\ 1& \mathrm{b}& 1\\ 1& 1& \mathrm{c}\end{array}\right|$ is positive, then

(a) abc $>1$

(b) abc $>-8$

(c) abc $<-8$

(d) $abc>-2$

4. If $\alpha ,\beta$ and $\gamma$ are such that $\alpha +\beta +\gamma =0$, then $\left|\begin{array}{ccc}1& \cos \gamma & \cos \beta \\ \cos \gamma & 1& \cos \alpha \\ \cos \beta & \cos \alpha & 1\end{array}\right|$ is equal to

(a) $\cos \alpha \cos \beta \cos \gamma$

(b) $\cos \alpha +\cos \beta +\cos \gamma$

(c) 1

(d) None of these

5. If $\mathrm{\Delta }=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1\\ \sin \alpha & \cos \alpha & 1\\ \cos (\alpha +\beta )& -\sin (\alpha +\beta )& 1\end{array}\right|$, then $\mathrm{\Delta }\in$

(a) $[1-\sqrt{2},1+\sqrt{2}]$

(b) $[-1,1]$

(c) $[-\sqrt{2},\sqrt{2}]$

(d) None of these

Exercise

1. If ${\mathrm{D}}_{\mathrm{k}}=\left|\begin{array}{ccc}1& \mathrm{n}& \mathrm{n}\\ 2\mathrm{k}& {\mathrm{n}}^2+\mathrm{n}+2& {\mathrm{n}}^2+\mathrm{n}\\ 2\mathrm{k}-1& {\mathrm{n}}^2& {\mathrm{n}}^2+\mathrm{n}+2\end{array}\right|$ and $\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{D}}_{\mathrm{k}}=48$, then $\mathrm{n}$ equals

(a) 4

(b) 6

(c) 8

(d) 10

2. If $f(x)=\left|\begin{array}{ccc}\sin x& \mathrm{cosec}x& \tan x\\ \sec x& x\sin x& x\tan x\\ x^2-1& \cos x& x^2+1\end{array}\right|$, then ${\int }_{-a}^a|f(x)|dx=$

(a) 1

(b) -1

(c) $2\mathrm{a}$

(d) 0

3. If $\mathrm{A},\mathrm{B},\mathrm{C}$ are the angles of $\mathrm{△}\mathrm{ABC}$, then $\left|\begin{array}{lll}{\sin }^2\mathrm{A}& \cot \mathrm{A}& 1\\ {\sin }^2\mathrm{B}& \cot \mathrm{B}& 1\\ {\sin }^2\mathrm{C}& \cot \mathrm{C}& 1\end{array}\right|=$

(a) ${\displaystyle \frac{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2}{4\mathrm{\Delta }}}$

(b) ${\displaystyle \frac{a^2+b^2+c^2}{4R^2\mathrm{\Delta }}}$

(c) ${\displaystyle \frac{a^2+b^2+c^2}{16R^2\mathrm{\Delta }}}$

(d) 0

4. A triangle has vertices $A_i(x_i,y_i)$ for $i=1,2,3$. Then the determinant

$\mathrm{\Delta }=\left|\begin{array}{lll}{\mathrm{x}}_2-{\mathrm{x}}_3& {\mathrm{y}}_2-{\mathrm{y}}_3& {\mathrm{y}}_1({\mathrm{y}}_2-{\mathrm{y}}_3)+{\mathrm{x}}_1({\mathrm{x}}_2-{\mathrm{x}}_3)\\ {\mathrm{x}}_3-{\mathrm{x}}_1& {\mathrm{y}}_3-{\mathrm{y}}_1& {\mathrm{y}}_2({\mathrm{y}}_3-{\mathrm{y}}_1)+{\mathrm{x}}_2({\mathrm{x}}_3-{\mathrm{x}}_1)\\ {\mathrm{x}}_1-{\mathrm{x}}_2& {\mathrm{y}}_2-{\mathrm{y}}_2& {\mathrm{y}}_3({\mathrm{y}}_1-{\mathrm{y}}_2)+{\mathrm{x}}_3({\mathrm{x}}_1-{\mathrm{x}}_2)\end{array}\right|=0$ means

(a) the medians for triangle ${\mathrm{A}}_1{\mathrm{A}}_2{\mathrm{A}}_3$ are concurrent

(b) the triangle ${\mathrm{A}}_1{\mathrm{A}}_2{\mathrm{A}}_3$ is right angled at ${\mathrm{A}}_3$

(b) the triangle ${\mathrm{A}}_1{\mathrm{A}}_2{\mathrm{A}}_3$ equilateral triangle

(b) altitudes of the triangle ${\mathrm{A}}_1{\mathrm{A}}_2{\mathrm{A}}_3$ are concurrent

5. Let $\{{\mathrm{D}}_1,{\mathrm{D}}_2,{\mathrm{D}}_3$……………..$D_n$}be the set of all third order determinants that can be formed with the distinct nonzero real numbers ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots \dots \dots .{\mathrm{a}}_9$, then

(a) $\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{D}}_{\mathrm{i}}=1$

(b) $\sum _{i=1}^n{D}_i=0$

(c) ${\mathrm{D}}_{\mathrm{i}}={\mathrm{D}}_{\mathrm{i}}\mathrm{\forall }\mathrm{i}\mathrm{\&}\mathrm{j}$

(d) None of these

6. If the value of $\left|\begin{array}{ccc}(-1{)}^{n}a& (-1{)}^{n+1}b& (-1{)}^{n+2}c\\ a+1& b-1& c+1\\ a-1& b+1& c-1\end{array}\right|+\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b-1& b+1\\ c& c+1& c-1\end{array}\right|$ is zero, then the value of $n$ is

(a) any even integer

(b) any odd integer

(c) any positive integer

(d) zero

7. ${\mathrm{\Delta }}_1=\left|\begin{array}{lll}{y}^5{z}^6(z^3-y^3)& x^4{z}^6(x^3-z^3)& x^4{y}^5(y^3-x^3)\\ y^2{z}^3(y^6-z^6)& x{z}^3(z^6-x^6)& x{y}^2(x^6-y^6)\\ y^2{z}^3(z^3-y^3)& x{z}^3(x^3-z^3)& x{y}^2(y^3-x^3)\end{array}\right|$ and

${\mathrm{\Delta }}_2=\left|\begin{array}{lll}x& y^2& z^3\\ x^4& y^5& z^6\\ x^7& y^8& z^9\end{array}\right|$ then ${\mathrm{\Delta }}_1{\mathrm{\Delta }}_2$ is equal to

(a) ${\mathrm{\Delta }}_2{}^3$

(b) ${\mathrm{\Delta }}_2{}^2$

(c) ${\mathrm{\Delta }}_2{}^4$

(d) ${\mathrm{\Delta }}_2{}^5$

8. Match the following :

9. If $\mathrm{p}+\mathrm{q}+\mathrm{r}=\mathrm{a}+\mathrm{b}+\mathrm{c}=0$ then the value of $\left|\begin{array}{lll}\mathrm{pa}& \mathrm{qb}& \mathrm{rc}\\ \mathrm{qc}& \mathrm{ra}& \mathrm{pb}\\ \mathrm{rb}& \mathrm{pc}& \mathrm{qa}\end{array}\right|$ is

(a) 0

(c) 1

(b) $ap+bq+cr$

(d) None of these

10. $\left|\begin{array}{ccc}\mathrm{bc}& \mathrm{ca}& \mathrm{ab}\\ \mathrm{p}& \mathrm{q}& \mathrm{r}\\ 1& 1& 1\end{array}\right|=$…………….where $\mathrm{a},\mathrm{b},\mathrm{c}\text{ are respectively the }{\mathrm{p}}^{\text{th }},{\mathrm{q}}^{\text{th }},{\mathrm{r}}^{\text{th }}$ terms of an H.P.

(a) 0

(b) 1

(c) -1

(d) None of these

11. Suppose $f(\mathrm{x})$ is a function satisfying the following conditions :

(i) $f(0)=2,f(1)=1$

(ii) fhas minimum at $x=5/2$ and

(iii) for all $\mathrm{x},f^1(\mathrm{x})=\left|\begin{array}{ccc}2\mathrm{ax}& 2\mathrm{ax}-1& 2\mathrm{ax}+\mathrm{b}+1\\ \mathrm{b}& \mathrm{b}-1& -1\\ 2(\mathrm{ax}+\mathrm{b})& 2\mathrm{ax}+2\mathrm{b}+1& 2\mathrm{ax}+\mathrm{b}\end{array}\right|$ where $\mathrm{a},\mathrm{b}$ are constants, then $f(\mathrm{x})=$

(a) 0

(b) constant $\mathrm{\forall }x$

(c) ${\displaystyle \frac{1}{4}}(x^2-5x+2)$

(d) None of these

12. Let $\mathrm{\Delta }=\left|\begin{array}{ccc}-bc& b^2+bc& c^2+bc\\ a^2+ac& -ac& c^2+ac\\ a^2+ab& b^2+ab& -ab\end{array}\right|$ and the equation ${\mathrm{px}}^3+{\mathrm{qx}}^2+rx+s=0$ has roots $a,b,c$ where $a,b,c\in R^{+}$

(i) The value of $\mathrm{\Delta }$ is

(a) $\le {\displaystyle \frac{9{\mathrm{r}}^2}{{\mathrm{p}}^2}}$

(b) $\ge {\displaystyle \frac{27{\mathrm{s}}^2}{{\mathrm{p}}^2}}$

(c) $\le {\displaystyle \frac{27{\mathrm{s}}^3}{{\mathrm{p}}^3}}$

(d) None of these

(ii) The value of $\mathrm{\Delta }$ is

(a) ${\displaystyle \frac{{\mathrm{r}}^2}{{\mathrm{p}}^2}}$

(b) ${\displaystyle \frac{{\mathrm{r}}^3}{{\mathrm{p}}^3}}$

(c) ${\displaystyle \frac{-\mathrm{s}}{\mathrm{p}}}$

(d) None of these

(iii) If $\mathrm{\Delta }=27$ and $a^2+b^2+c^2=2$, then

(a) $3p+2q=0$

(c) $3\mathrm{p}+\mathrm{q}=0$

(b) $4p+3q=0$

(d) None of these

13. If $\mathrm{p}{\lambda }^4+\mathrm{q}{\lambda }^3+\mathrm{r}{\lambda }^2+\mathrm{s}\lambda +\mathrm{t}=\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ {\lambda }^2+1& 2-\lambda & \lambda -3\\ {\lambda }^2-3& \lambda +4& 3\lambda \end{array}\right|$, then $\mathrm{p}=$

(a) -5

(b) -4

(c) -3

(d) -2

14.* If $g(x)=\left|\begin{array}{lll}{a}^{-x}& e^{x{\log }_{c}a}& x^2\\ a^{-3x}& e^{3{\log }_{c}a}& x^4\\ a^{-5x}& e^{5^{x{\log }_{c}a}}& 1\end{array}\right|$, then

(a) graph of $\mathrm{g}(\mathrm{x})$ is symmetric about origin

(b) graph of $\mathrm{g}(\mathrm{x})$ is symmetric about $\mathrm{Y}$ axis

(c) ${\left({\displaystyle \frac{{\mathrm{d}}^4\mathrm{g}(\mathrm{x})}{{\mathrm{dx}}^4}}\right)}_{\mathrm{x}=0}=0$

(d) $f(\mathrm{x})=\mathrm{g}(\mathrm{x}){\log }_{\mathrm{e}}\left({\displaystyle \frac{\mathrm{a}-\mathrm{x}}{\mathrm{a}+\mathrm{x}}}\right)$ is an odd function

15.* If $f(\alpha ,\beta )=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1\\ \sin \alpha & \cos \alpha & 1\\ \cos (\alpha +\beta )& -\sin (\alpha +\beta )& 1\end{array}\right|$ then

(a) $f(300,200)=f(400,200)$

(c) $f(100,200)=f(200,200)$

(b) $f(200,400)=f(200,600)$

(d) None of these