Matrices And Determinants - Properties and Evaluation of Determinants (Lecture-02)
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) $\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|=-(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)=-\left(a^3+b^3+c^3-3 a b c\right)$
SOLVED EXAMPLES
1. If $\Delta _{\mathrm{r}}=\left|\begin{array}{ccc}2 \mathrm{r}-1 & { }^{\mathrm{m}} \mathrm{C} _{\mathrm{r}} & 1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2} \mathrm{~m} & \sin ^{2}(\mathrm{~m}+1)\end{array}\right|$, then the value of $\sum\limits _{\mathrm{r}=0}^{\mathrm{m}} \Delta _{\mathrm{r}}$
(a) 0
(b) 1
(c) $\mathrm{m}^{2}-1$
(d) $2^{\mathrm{m}}$
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Solution:
$\sum\limits _{r=0}^{m} \Delta _{r}=\left|\begin{array}{ccc} \sum\limits _{r=0}^{m}(2 r-1) & \sum\limits _{r=0}^{m}{ }^{m} C _{r} & \sum\limits _{r=0}^{m} 1 \\ m^{2}-1 & 2^{m} & m+1 \\ \sin ^{2}\left(m^{2}\right) & \sin ^{2} m & \sin ^{2}(m+1) \end{array}\right|$
Adding the first row we get
$\begin{aligned} & \sum\limits _{\mathrm{r}=0}^{\mathrm{m}} \Delta _{\mathrm{r}}=\left|\begin{array}{ccc} -1+(1+3+5+\ldots . .+2 \mathrm{~m}-1) & { }^{\mathrm{m}} \mathrm{C} _{0}+{ }^{\mathrm{m}} \mathrm{C} _{1}+\ldots .+{ }^{\mathrm{m}} \mathrm{C} _{\mathrm{m}} & 1+1+1 \ldots \mathrm{m}+1 \text { times } \\ \mathrm{m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2} \mathrm{~m} & \sin ^{2}(\mathrm{~m}+1) \end{array}\right| \\ & =\left|\begin{array}{ccc} \mathrm{m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2} \mathrm{~m} & \sin ^{2}(\mathrm{~m}+1) \end{array}\right|=0 \end{aligned}$
Answer: (a)
2. If $f(\mathrm{x}), \mathrm{g}(\mathrm{x})$ and $\mathrm{h}(\mathrm{x})$ are polynomials of degree 2, then $\phi(\mathrm{x})=\left|\begin{array}{ccc}f(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ f^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\ f^{\prime \prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x})\end{array}\right|$ is a polynomial of degree
(a) 2
(b) 3
(c) 4
(d) None of these
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Solution
$\phi^{\prime}(\mathrm{x})=\left|\begin{array}{ccc}f^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\ f^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\ f^{\prime \prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x})\end{array}\right|+\left|\begin{array}{ccc}f(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ f^{\prime \prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x}) \\ f^{\prime \prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x})\end{array}\right|+\left|\begin{array}{ccc}f(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ f^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\ f^{\prime \prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x})\end{array}\right|=0+0+0$
$\left(\because f(\mathrm{x}), \mathrm{g}(\mathrm{x}), \mathrm{h}(\mathrm{x})\right.$ are polynomials of degree $3, f^{\prime \prime \prime}(\mathrm{x}), \mathrm{g}^{\prime \prime \prime}(\mathrm{x})$ and $\mathrm{h}^{\prime \prime \prime}(\mathrm{x})=0$ )
$\Rightarrow \phi^{1}(\mathrm{x})=0$
$\therefore \phi(\mathrm{x})$ is a constant.
Answer: (d)
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) $a^{3}-3 b$
(c) $\mathrm{a}^{3}$
(d) None of these
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Solution:
$\alpha+\beta+\gamma=-a, \alpha \beta+\beta \gamma+\gamma \alpha=0, \alpha \beta \gamma=-b$.
It can be shown that
$\left|\begin{array}{lll} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|=-(\alpha+\beta+\gamma)\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\alpha \beta-\beta \gamma-\gamma \alpha\right)$
$\begin{aligned} & =-(\alpha+\beta+\gamma)\left((\alpha+\beta+\gamma)^{2}-3(\alpha \beta+\beta \gamma+\gamma \alpha)\right) \\ & =-(-a)\left((-a)^{2}-3 \times 0\right)=a^{3} \end{aligned}$
Answer: (c)
4. If $f(x)=\left|\begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & x(x-1)(x-2) & x(x+1)(x-1)\end{array}\right|$, then $f(100)$ is equal to
(a) 0
(b) 1
(c) 100
(d) -100
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Solution:
$\mathrm{C} _{3} \rightarrow \mathrm{C} _{3}-\mathrm{C} _{2} \operatorname{gives} \mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc}1 & \mathrm{x} & 1 \\ 2 \mathrm{x} & \mathrm{x}(\mathrm{x}-1) & 2 \mathrm{x} \\ 3 \mathrm{x}(\mathrm{x}-1) & \mathrm{x}(\mathrm{x}-1)(\mathrm{x}-2) & 3 \mathrm{x}(\mathrm{x}-1)\end{array}\right|$
$\therefore f(\mathrm{x})=0\left(\because \mathrm{C} _{1} \& \mathrm{C} _{3}\right.$ are identical $)$
$\Rightarrow f(100)=0$
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) $\dfrac{1}{2}$
(b) 0
(c) 2
(d) 1
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Solution:
$\left|\begin{array}{lll} \mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 \\ \mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2 \\ \mathrm{a}_3 & \mathrm{~b}_3 & \mathrm{c}_3 \end{array}\right|^2=\left|\begin{array}{lll} \mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 \\ \mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2 \\ \mathrm{a}_3 & \mathrm{~b}_3 & \mathrm{c}_3 \end{array}\right|\left|\begin{array}{lll} \mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 \\ \mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2 \\ \mathrm{a}_3 & \mathrm{~b}_3 & \mathrm{c}_3 \end{array}\right|$
$=\left|\begin{array}{ccc}\sum \mathrm{a} _{1}{ }^{2} & \sum \mathrm{a} _{1} \mathrm{a} _{2} & \sum \mathrm{a} _{1} \mathrm{a} _{3} \\ \sum \mathrm{a} _{1} \mathrm{a} _{2} & \sum \mathrm{a} _{2}{ }^{2} & \sum \mathrm{a} _{2} \mathrm{a} _{3} \\ \sum \mathrm{a} _{1} \mathrm{a} _{3} & \sum \mathrm{a} _{2} \mathrm{a} _{3} & \sum \mathrm{a} _{3}{ }^{2}\end{array}\right|=\left|\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|=1$
Answer: (d)
6. If $0 \leq[\mathrm{x}]<2,-1 \leq[\mathrm{y}]<1,1 \leq[\mathrm{z}]<3$ ([.] denotes the greatest integer function), then the maximum value of
$\Delta=\left|\begin{array}{ccc} {[x]+1} & {[y]} & {[z]} \\ {[x]} & {[y]+1} & {[z]} \\ {[x]} & {[y]} & {[z]+1} \end{array}\right| \text { is }$
(a) 2
(b) 6
(c) 4
(d) None of these
Show Answer
Solution:
Solving the determinant we get
$\Delta=1+[\mathrm{x}]+[\mathrm{y}]+[\mathrm{z}]$
$=1+1+0+2=4(\because$ maximum values of $[\mathrm{x}],[\mathrm{y}]$ and $[\mathrm{z}]$ are 1,0 and 2 respectively $)$
Answer: (c)
7. Let $\left|\begin{array}{ccc}x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6\end{array}\right|=a x^{4}+b x^{3}+c x^{2}+d x+e$ then, $5 a+4 b+3 c+2 d+e$ is equal to
(a) 0
(b) -16
(c) 16
(d) None of these
Show Answer
Solution:
$\mathrm{R} _{3} \rightarrow \mathrm{R} _{3}-\mathrm{R} _{2}$ gives
$\Delta=\left|\begin{array}{ccc}\mathrm{x} & 2 & \mathrm{x} \\ \mathrm{x}^{2} & \mathrm{x} & 6 \\ \mathrm{x}-\mathrm{x}^{2} & 0 & 0\end{array}\right|=12-\mathrm{x}^{3}-12 \mathrm{x}^{2}+\mathrm{x}^{4}$
$\therefore \mathrm{a}=1, \mathrm{~b}=-1, \mathrm{c}=-12, \mathrm{~d}=12, \mathrm{e}=0$
Put the values to get $5 a+4 b+3 c+2 d+e=-11$
Answer: (d)
EXERCISE
1. If $\Delta(x)=\left|\begin{array}{ccc}\mathrm{e}^{\mathrm{x}} & \sin 2 \mathrm{x} & \tan \left(\mathrm{x}^{2}\right) \\ \log _{\mathrm{e}}(1+\mathrm{x}) & \cos \mathrm{x} & \sin \mathrm{x} \\ \cos \left(\mathrm{x}^{2}\right) & \mathrm{e}^{\mathrm{x}}-1 & \sin \left(\mathrm{x}^{2}\right)\end{array}\right|=\mathrm{A}+\mathrm{Bx}+\mathrm{Cx}^{2}+\ldots \ldots \ldots \ldots \ldots$. then $\mathrm{B}=$
(a) 0
(b) 1
(c) 2
(d) 4
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Answer: a2. Let $f(\mathrm{x})=\left|\begin{array}{ccc}\mathrm{x}+\mathrm{a} & \mathrm{x}+\mathrm{b} & \mathrm{x}+\mathrm{a}-\mathrm{c} \\ \mathrm{x}+\mathrm{b} & \mathrm{x}+\mathrm{c} & \mathrm{x}-1 \\ \mathrm{x}+\mathrm{c} & \mathrm{x}+\mathrm{d} & \mathrm{x}-\mathrm{b}+\mathrm{d}\end{array}\right| \& \int _{0}^{2} f(\mathrm{x}) \mathrm{dx}=-16$
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) $\pm 4$
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Answer: b3. 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
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Answer: a4. Match the following:
| Column I | Column II | ||
|---|---|---|---|
| (a) | (a) If $\left|\begin{array}{ccc}x & x+y & x+y+z \\ 2 x & 3 x+2 y & 4 x+3 y+2 z \\ 3 x & 6 x+3 y & 10 x+6 y+3 z\end{array}\right|=343$ then $x=$ | (p) | 2 |
| (b) | If $\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}=-2$ and | (q) | 4 |
| $f(x)=\left|\begin{array}{lll} \left(1+a^2 x\right) & \left(1+b^2\right) x & \left(1+c^2\right) x \\ \left(1+a^2\right) x & \left(1+b^2 x\right) & \left(1+c^2\right) x \\ \left(1+a^2\right) x & \left(1+b^2\right) x & \left(1+c^2 x\right) \end{array}\right|$ | |||
| then $f(\mathrm{x})$ is a polynomial of degree | |||
| (c) | If $\left|\begin{array}{ccc}\dfrac{a^{2}+b^{2}}{c} & c & c \\ a & \dfrac{b^{2}+c^{2}}{a} & a \\ b & b & \dfrac{c^{2}+a^{2}}{b}\end{array}\right|=k a b c$ then $\mathrm{k}=$ | (r) | 0 |
| (d) | If A, B, C are the angles of triangle then $\left|\begin{array}{ccc}\sin 2 A & \sin C & \sin B \\ \sin C & \sin 2 B & \sin A \\ \sin B & \sin A & \sin 2 C\end{array}\right|=$ | (s) | 7 |
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Answer: $\mathrm{a} \rightarrow \mathrm{s} ; \mathrm{b} \rightarrow \mathrm{p} ; \mathrm{c} \rightarrow \mathrm{q} ; \mathrm{d} \rightarrow \mathrm{r}$5. Let $\mathrm{k}$ be a positive real number and let
$A=\left|\begin{array}{ccc}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1\end{array}\right| \quad$ and $\quad B=\left|\begin{array}{ccc}0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0\end{array}\right|$. If
$\operatorname{det}(\operatorname{adj} \mathrm{A})+\operatorname{det}(\operatorname{adj} B)=10^{6}$, then $[\mathrm{k}]=$
(a) 4
(b) 5
(c) 6
(d) None of these
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Answer: a6. 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
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Answer: b7. If $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are complex numbers, then $\Delta=\left|\begin{array}{ccc}0 & -\mathrm{y} & -\mathrm{z} \\ \overline{\mathrm{y}} & 0 & -\mathrm{x} \\ \overline{\mathrm{z}} & \overline{\mathrm{x}} & 0\end{array}\right|$ is
(a) purely real
(b) purely imaginary
(c) complex
(d) 0
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Answer: b8. 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 $\mathrm{a} \neq 0, \mathrm{~b} \neq 0, \mathrm{c} \neq 0$, then $\mathrm{a}^{-1}+\mathrm{b}^{-1}+\mathrm{c}^{-1}=$
(a) 4
(b) -3
(c) -2
(d) -1
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Answer: b9. 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 $\left(1, a, a^{2}\right),\left(1, b, b^{2}\right),\left(1, c, c^{2}\right)$ are noncoplanar, then $a b c=$
(a) 2
(b) -1
(c) 1
(d) 0
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Answer: b10. Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be real numbers with $\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}=1$. Then the equation
$\left|\begin{array}{ccc} a x-b y-c & b x+a y & c x+a \\ b x+a y & -a x+b y-c & c y+b \\ c x+a & c y+b & -a x-b y+c \end{array}\right|=0 \text { represents }$
(a) a parabola
(b) pair oflines
(c) a straight line
(d) circle.
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Answer: c11. Let $\Delta \neq 0$ and $\Delta^{\mathrm{c}}$ denotes the determinants of cofactors, then $\Delta^{\mathrm{c}}=\Delta^{\mathrm{n}-1}$, where $\mathrm{n}(>0)$ is the order of $\Delta$. 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
$\left|\begin{array}{lll} b c-a^{2} & c a-b^{2} & a b-c^{2} \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right| \text { is }$
(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
$\left|\begin{array}{ccc} a^{2}+\lambda^{2} & a b+c \lambda & c a-b \lambda \\ a b-c \lambda & b^{2}+\lambda^{2} & b c+a \lambda \\ a c+b \lambda & b c-a \lambda & c^{2}+\lambda^{2} \end{array}\right| \times\left|\begin{array}{ccc} \lambda & c & -b \\ -c & \lambda & a \\ b & -a & \lambda \end{array}\right| \text { is }$
(a) $8 \lambda^{6}$
(b) $27 \lambda^{9}$
(c) $8 \lambda^{9}$
(d) $27 \lambda^{6}$
(v) Suppose $a, b, c \in R, a+b+c>0, A=b c-a^{2}, B=c a-b^{2} \& C=a b-c^{2}$ 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
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Answer: (i) c (ii) b (iii) d (iv) c (v) b12. 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) $2-\sqrt{3}$
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Answer: b13.* If $\left(\mathrm{x} _{1}-\mathrm{x} _{2}\right)^{2}+\left(\mathrm{y} _{1}-\mathrm{y} _{2}\right)^{2}=\mathrm{a}^{2}$
$\left(\mathrm{x} _{2}-\mathrm{x} _{3}\right)^{2}+\left(\mathrm{y} _{2}-\mathrm{y} _{3}\right)^{2}=\mathrm{b}^{2}$
$\left(\mathrm{x} _{3}-\mathrm{x} _{1}\right)^{2}+\left(\mathrm{y} _{3}-\mathrm{y} _{1}\right)^{2}=\mathrm{c}^{2}$ and
$4\left|\begin{array}{lll}x _{1} & y _{1} & 1 \\ x _{2} & y _{2} & 1 \\ x _{3} & y _{3} & 1\end{array}\right|=\lambda\left\{\lambda^{3}-\left(\lambda _{1}+\lambda _{2}+\lambda _{3}\right) \lambda^{2}+\left(\lambda _{1} \lambda _{2}+\lambda _{2} \lambda _{3}+\lambda _{3} \lambda _{1}\right) \lambda-\lambda _{1} \lambda _{2} \lambda _{3}\right\}$ then
(a) $\lambda \geq \dfrac{3}{2}\left(\lambda _{1} \lambda _{2} \lambda _{3}\right)^{1 / 3}$
(b) $\lambda _{1} \lambda _{2} \lambda _{3}=8 \mathrm{abc}$
(c) $\sum \lambda _{1} \lambda _{2}=4 \sum \mathrm{ab}$
(d) $2 \lambda=\lambda _{1}+\lambda _{2}+\lambda _{3}$
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Answer: a,b,c,d14. If $\quad A _{r}=\left|\begin{array}{cc}r & r-1 \\ r-1 & r\end{array}\right|$, where $r$ is a natural number, then the value of $\sqrt{\sum\limits _{r=1}^{2008} A _{r}}$ is………….
(a) 2008
(c) 2007
(b) 0
(d) None of these
A program to give wings to girl students
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Answer: a15.* 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
