Some useful results

(i) $\left|\begin{array}{lll}1& \mathrm{a}& {\mathrm{a}}^2\\ 1& \mathrm{b}& {\mathrm{b}}^2\\ 1& \mathrm{c}& {\mathrm{c}}^2\end{array}\right|=(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})$

(ii) $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ {\mathrm{a}}^2& {\mathrm{b}}^2& {\mathrm{c}}^2\\ \mathrm{bc}& \mathrm{ca}& \mathrm{ab}\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ {\mathrm{a}}^2& {\mathrm{b}}^2& {\mathrm{c}}^2\\ {\mathrm{a}}^3& {\mathrm{b}}^3& {\mathrm{c}}^3\end{array}\right|=(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})(\mathrm{ab}+\mathrm{bc}+\mathrm{ca})$

(iii) $\left|\begin{array}{lll}\mathrm{a}& \mathrm{bc}& \mathrm{abc}\\ \mathrm{b}& \mathrm{ca}& \mathrm{abc}\\ \mathrm{c}& \mathrm{ab}& \mathrm{abc}\end{array}\right|=\left|\begin{array}{lll}\mathrm{a}& {\mathrm{a}}^2& {\mathrm{a}}^3\\ \mathrm{b}& {\mathrm{b}}^2& {\mathrm{b}}^3\\ \mathrm{c}& {\mathrm{c}}^2& {\mathrm{c}}^3\end{array}\right|=\mathrm{abc}(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})$

(iv) $\left|\begin{array}{ccc}1& 1& 1\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\\ {\mathrm{a}}^3& {\mathrm{b}}^3& {\mathrm{c}}^3\end{array}\right|=(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})(\mathrm{a}+\mathrm{b}+\mathrm{c})$

(v) $\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=-(a^3+b^3+c^3-3abc)$

SOLVED EXAMPLES

1. If ${\mathrm{\Delta }}_{\mathrm{r}}=\left|\begin{array}{ccc}2\mathrm{r}-1& {}^{\mathrm{m}}{\mathrm{C}}_{\mathrm{r}}& 1\\ {\mathrm{m}}^2-1& 2^{\mathrm{m}}& \mathrm{m}+1\\ {\sin }^2\left({\mathrm{m}}^2\right)& {\sin }^2\mathrm{m}& {\sin }^2(\mathrm{m}+1)\end{array}\right|$, then the value of $\sum _{\mathrm{r}=0}^{\mathrm{m}}{\mathrm{\Delta }}_{\mathrm{r}}$

(a) 0

(b) 1

(c) ${\mathrm{m}}^2-1$

(d) $2^{\mathrm{m}}$

2. If $f(\mathrm{x}),\mathrm{g}(\mathrm{x})$ and $\mathrm{h}(\mathrm{x})$ are polynomials of degree 2, then $\varphi (\mathrm{x})=\left|\begin{array}{ccc}f(\mathrm{x})& \mathrm{g}(\mathrm{x})& \mathrm{h}(\mathrm{x})\\ f^{\prime }(\mathrm{x})& {\mathrm{g}}^{\prime }(\mathrm{x})& {\mathrm{h}}^{\prime }(\mathrm{x})\\ f^{\prime \prime }(\mathrm{x})& {\mathrm{g}}^{\prime \prime }(\mathrm{x})& {\mathrm{h}}^{\prime \prime }(\mathrm{x})\end{array}\right|$ is a polynomial of degree

(a) 2

(b) 3

(c) 4

(d) None of these

3. If $\alpha ,\beta ,\gamma$ are roots of $x^3+a^2+b=0$, then the value of $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$ is

(a) $-{\mathrm{a}}^3$

(b) $a^3-3b$

(c) ${\mathrm{a}}^3$

(d) None of these

4. If $f(x)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& x(x+1)\\ 3x(x-1)& x(x-1)(x-2)& x(x+1)(x-1)\end{array}\right|$, then $f(100)$ is equal to

(a) 0

(b) 1

(c) 100

(d) -100

5. If ${\mathrm{a}}_{\mathrm{i}}{}^2+{\mathrm{b}}_{\mathrm{i}}{}^2+{\mathrm{c}}_{\mathrm{i}}{}^2=1;\mathrm{i}=1,2,3$ and ${\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{\mathrm{j}}+{\mathrm{b}}_{\mathrm{i}}{\mathrm{b}}_{\mathrm{j}}+{\mathrm{c}}_{\mathrm{i}}{\mathrm{c}}_{\mathrm{j}}=0$, then the value of determinant

$\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ {\mathrm{b}}_1& {\mathrm{b}}_2& {\mathrm{b}}_3\\ {\mathrm{c}}_1& {\mathrm{c}}_2& {\mathrm{c}}_3\end{array}\right|$ is

(a) ${\displaystyle \frac{1}{2}}$

(b) 0

(c) 2

(d) 1

6. If $0\le [\mathrm{x}]<2,-1\le [\mathrm{y}]<1,1\le [\mathrm{z}]<3$ ([.] denotes the greatest integer function), then the maximum value of

$\mathrm{\Delta }=\left|\begin{array}{ccc}[x]+1& [y]& [z]\\ [x]& [y]+1& [z]\\ [x]& [y]& [z]+1\end{array}\right|\text{ is }$

(a) 2

(b) 6

(c) 4

(d) None of these

7. Let $\left|\begin{array}{ccc}x& 2& x\\ x^2& x& 6\\ x& x& 6\end{array}\right|=a{x}^4+b{x}^3+c{x}^2+dx+e$ then, $5a+4b+3c+2d+e$ is equal to

(a) 0

(b) -16

(c) 16

(d) None of these

EXERCISE

1. If $\mathrm{\Delta }(x)=\left|\begin{array}{ccc}{\mathrm{e}}^{\mathrm{x}}& \sin 2\mathrm{x}& \tan \left({\mathrm{x}}^2\right)\\ {\log }_{\mathrm{e}}(1+\mathrm{x})& \cos \mathrm{x}& \sin \mathrm{x}\\ \cos \left({\mathrm{x}}^2\right)& {\mathrm{e}}^{\mathrm{x}}-1& \sin \left({\mathrm{x}}^2\right)\end{array}\right|=\mathrm{A}+\mathrm{Bx}+{\mathrm{Cx}}^2+\dots \dots \dots \dots \dots$. then $\mathrm{B}=$

(a) 0

(b) 1

(c) 2

(d) 4

2. Let $f(\mathrm{x})=\left|\begin{array}{ccc}\mathrm{x}+\mathrm{a}& \mathrm{x}+\mathrm{b}& \mathrm{x}+\mathrm{a}-\mathrm{c}\\ \mathrm{x}+\mathrm{b}& \mathrm{x}+\mathrm{c}& \mathrm{x}-1\\ \mathrm{x}+\mathrm{c}& \mathrm{x}+\mathrm{d}& \mathrm{x}-\mathrm{b}+\mathrm{d}\end{array}\right|\mathrm{\&}{\int }_0^{2}f(\mathrm{x})\mathrm{dx}=-16$

Where $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ are in A.P., then the common difference of the A.P is

(a) $\pm 1$

(b) $\pm 2$

(c) $\pm 3$

(d) $\pm 4$

3. If $\left|\begin{array}{ccc}1& 1& 1\\ {}^m{C}_1& {}^{\mathrm{m}+3}{\mathrm{C}}_1& {}^{\mathrm{m}+6}{\mathrm{C}}_1\\ {}^{\mathrm{m}}{\mathrm{C}}_2& {}^{\mathrm{m}+3}{\mathrm{C}}_2& {}^{\mathrm{m}+6}{\mathrm{C}}_2\end{array}\right|=2^{\alpha }{3}^{\beta }{5}^{\gamma }$ then $\alpha +\beta +\gamma$ is equal to

(a) 3

(b) 5

(c) 7

(d) 0

4. Match the following:

5. Let $\mathrm{k}$ be a positive real number and let

$A=\left|\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right|$ and $B=\left|\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right|$. If

$\det (\mathrm{adj}\mathrm{A})+\det (\mathrm{adj}B)={10}^6$, then $[\mathrm{k}]=$

(a) 4

(b) 5

(c) 6

(d) None of these

6. If $\mathrm{A},\mathrm{B},\mathrm{C}$ be the angles of a triangle, then $\left|\begin{array}{ccc}-1+\cos \mathrm{B}& \cos \mathrm{C}+\cos \mathrm{B}& \cos \mathrm{B}\\ \cos \mathrm{C}+\cos \mathrm{A}& -1+\cos \mathrm{A}& \cos \mathrm{A}\\ -1+\cos \mathrm{B}& -1+\cos \mathrm{A}& -1\end{array}\right|=$

(a) -1

(b) 0

(c) 1

(d) 2

7. If $\mathrm{x},\mathrm{y},\mathrm{z}$ are complex numbers, then $\mathrm{\Delta }=\left|\begin{array}{ccc}0& -\mathrm{y}& -\mathrm{z}\\ \bar{\mathrm{y}}& 0& -\mathrm{x}\\ \bar{\mathrm{z}}& \bar{\mathrm{x}}& 0\end{array}\right|$ is

(a) purely real

(b) purely imaginary

(c) complex

(d) 0

8. If $\left|\begin{array}{ccc}1+\mathrm{a}& 1& 1\\ 1+\mathrm{b}& 1+2\mathrm{b}& 1\\ 1+\mathrm{c}& 1+\mathrm{c}& 1+3\mathrm{c}\end{array}\right|=0$ where $\mathrm{a}\ne 0,\mathrm{b}\ne 0,\mathrm{c}\ne 0$, then ${\mathrm{a}}^{-1}+{\mathrm{b}}^{-1}+{\mathrm{c}}^{-1}=$

(a) 4

(b) -3

(c) -2

(d) -1

9. If $\left|\begin{array}{lll}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$ and the vectors $(1,a,a^2),(1,b,b^2),(1,c,c^2)$ are noncoplanar, then $abc=$

(a) 2

(b) -1

(c) 1

(d) 0

10. Let $\mathrm{a},\mathrm{b},\mathrm{c}$ be real numbers with ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=1$. Then the equation

$\left|\begin{array}{ccc}ax-by-c& bx+ay& cx+a\\ bx+ay& -ax+by-c& cy+b\\ cx+a& cy+b& -ax-by+c\end{array}\right|=0\text{ represents }$

(a) a parabola

(b) pair oflines

(c) a straight line

(d) circle.

11. Let $\mathrm{\Delta }\ne 0$ and ${\mathrm{\Delta }}^{\mathrm{c}}$ denotes the determinants of cofactors, then ${\mathrm{\Delta }}^{\mathrm{c}}={\mathrm{\Delta }}^{\mathrm{n}-1}$, where $\mathrm{n}(>0)$ is the order of $\mathrm{\Delta }$. On the basis of above information, answer the following questions :-

(i) If $a,b,c$ are the roots of $x^3-p^2+r=0$ then

$\left|\begin{array}{lll}bc-a^2& ca-b^2& ab-c^2\\ ca-b^2& ab-c^2& bc-a^2\\ ab-c^2& bc-a^2& ca-b^2\end{array}\right|\text{ is }$

(a) ${\mathrm{p}}^2$

(b) ${\mathrm{p}}^4$

(c) ${\mathrm{p}}^6$

(d) ${\mathrm{p}}^9$

(ii) If ${\ell }_1,{\mathrm{m}}_1,{\mathrm{n}}_1;{\ell }_2,{\mathrm{m}}_2,{\mathrm{n}}_2;{\ell }_3,{\mathrm{m}}_3,{\mathrm{n}}_3$;are real quantities satisfying the six relations : ${\ell }_1{}^2+{\mathrm{m}}_1{}^2+{\mathrm{n}}_1{}^2$ $={\ell }_2{}^2+{\mathrm{m}}_2{}_2+{\mathrm{n}}_2{}_2={\ell }^3{}_2+{\mathrm{m}}_3{}^2+{\mathrm{n}}_3{}^2=1;{\ell }_1{\ell }_2+{\mathrm{m}}_1{\mathrm{m}}_2+{\mathrm{n}}_1{\mathrm{n}}_2={\ell }_2{\ell }_3+{\mathrm{m}}_2{\mathrm{m}}_3+{\mathrm{n}}_2{\mathrm{n}}_3=$ ${\ell }_3{\ell }_1+{\mathrm{m}}_3{\mathrm{m}}_1+{\mathrm{n}}_3{\mathrm{n}}_1=0$, then

$\left|\begin{array}{lll}{\ell }_1& {\mathrm{m}}_1& {\mathrm{n}}_1\\ {\ell }_2& {\mathrm{m}}_2& {\mathrm{n}}_2\\ {\ell }_3& {\mathrm{m}}_3& {\mathrm{n}}_3\end{array}\right|$ is

(a) 0

(b) $\pm 1$

(c) $\pm 2$

(d) $\pm 3$

(iii) If a, b, c are the roots of ${\mathrm{x}}^3-3{\mathrm{x}}^2+3\mathrm{x}+7=0$, then $\left|\begin{array}{ccc}2\mathrm{bc}-{\mathrm{a}}^2& {\mathrm{c}}^2& {\mathrm{b}}^2\\ {\mathrm{c}}^2& 2\mathrm{ac}-{\mathrm{b}}^2& {\mathrm{a}}^2\\ {\mathrm{b}}^2& {\mathrm{a}}^2& 2\mathrm{ab}-{\mathrm{c}}^2\end{array}\right|$ is

(a) 9

(b) 27

(c) 8

(d) 0

(iv) If ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2={\lambda }^2$ then the value of

$\left|\begin{array}{ccc}{a}^2+{\lambda }^2& ab+c\lambda & ca-b\lambda \\ ab-c\lambda & b^2+{\lambda }^2& bc+a\lambda \\ ac+b\lambda & bc-a\lambda & c^2+{\lambda }^2\end{array}\right|\times \left|\begin{array}{ccc}\lambda & c& -b\\ -c& \lambda & a\\ b& -a& \lambda \end{array}\right|\text{ is }$

(a) $8{\lambda }^6$

(b) $27{\lambda }^9$

(c) $8{\lambda }^9$

(d) $27{\lambda }^6$

(v) Suppose $a,b,c\in R,a+b+c>0,A=bc-a^2,B=ca-b^2\mathrm{\&}C=ab-c^2$ and

$\left|\begin{array}{lll}\mathrm{A}& \mathrm{B}& \mathrm{C}\\ \mathrm{B}& \mathrm{C}& \mathrm{A}\\ \mathrm{C}& \mathrm{A}& \mathrm{B}\end{array}\right|=49\text{, then }\left|\begin{array}{lll}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right|=$

(a) -7

(b) 7

(c) -2401

(d) 2401

12. The value of the determinant $\left|\begin{array}{ccc}1& {\mathrm{e}}^{\mathrm{i}\pi /3}& {\mathrm{e}}^{\mathrm{i}\pi /4}\\ {\mathrm{e}}^{-\mathrm{i}\pi /3}& 1& {\mathrm{e}}^{\mathrm{i}2\pi /3}\\ {\mathrm{e}}^{-i\pi /4}& {\mathrm{e}}^{-\mathrm{i}2\pi /3}& 1\end{array}\right|$ is

(a) $2+\sqrt{2}$

(b) $-(2+\sqrt{2})$

(c) $-2+\sqrt{3}$

(d) $2-\sqrt{3}$

13.* If ${({\mathrm{x}}_1-{\mathrm{x}}_2)}^2+{({\mathrm{y}}_1-{\mathrm{y}}_2)}^2={\mathrm{a}}^2$

${({\mathrm{x}}_2-{\mathrm{x}}_3)}^2+{({\mathrm{y}}_2-{\mathrm{y}}_3)}^2={\mathrm{b}}^2$

${({\mathrm{x}}_3-{\mathrm{x}}_1)}^2+{({\mathrm{y}}_3-{\mathrm{y}}_1)}^2={\mathrm{c}}^2$ and

$4\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\lambda \{{\lambda }^3-({\lambda }_1+{\lambda }_2+{\lambda }_3){\lambda }^2+({\lambda }_1{\lambda }_2+{\lambda }_2{\lambda }_3+{\lambda }_3{\lambda }_1)\lambda -{\lambda }_1{\lambda }_2{\lambda }_3\}$ then

(a) $\lambda \ge {\displaystyle \frac{3}{2}}{\left({\lambda }_1{\lambda }_2{\lambda }_3\right)}^{1/3}$

(b) ${\lambda }_1{\lambda }_2{\lambda }_3=8\mathrm{abc}$

(c) $\sum {\lambda }_1{\lambda }_2=4\sum \mathrm{ab}$

(d) $2\lambda ={\lambda }_1+{\lambda }_2+{\lambda }_3$

14. If $A_r=\left|\begin{array}{cc}r& r-1\\ r-1& r\end{array}\right|$, where $r$ is a natural number, then the value of $\sqrt{\sum _{r=1}^{2008}{A}_r}$ is………….

(a) 2008

(c) 2007

(b) 0

(d) None of these

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15.* If $f(\mathrm{x})=\left|\begin{array}{ccc}\mathrm{x}-3& 2{\mathrm{x}}^2-18& 3{\mathrm{x}}^3-81\\ \mathrm{x}-5& 2{\mathrm{x}}^2-50& 4{\mathrm{x}}^3-500\\ 1& 2& 3\end{array}\right|$, then $f(1)\cdot f(3)+f(3)\cdot f(5)+f(5)\cdot f(1)=$

(a) $f(3)$

(c) 2928

(b) 0

(d) None of these