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#### Question 1:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:
(i) $A=\left[\begin{array}{cc}5& 20\\ 0& -1\end{array}\right]$

(ii) $A=\left[\begin{array}{cc}-1& 4\\ 2& 3\end{array}\right]$

(iii) $A=\left[\begin{array}{ccc}1& -3& 2\\ 4& -1& 2\\ 3& 5& 2\end{array}\right]$

(iv) $A=\left[\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right]$

(v) $A=\left[\begin{array}{ccc}0& 2& 6\\ 1& 5& 0\\ 3& 7& 1\end{array}\right]$

(vi) $A=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$

(vii) $A=\left[\begin{array}{cccc}2& -1& 0& 1\\ -3& 0& 1& -2\\ 1& 1& -1& 1\\ 2& -1& 5& 0\end{array}\right]$

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

#### Question 2:

Evaluate the following determinants:

(i) $\left|\begin{array}{cc}x& -7\\ x& 5x+1\end{array}\right|$

(ii)

(iii)

(iv)  $\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|$

(i)

(ii)

(iii)

(iv)

#### Question 3:

Evaluate ${\left|\begin{array}{ccc}2& 3& 7\\ 13& 17& 5\\ 15& 20& 12\end{array}\right|}^{2}.$

Show that

#### Question 5:

Evaluate $\left|\begin{array}{ccc}2& 3& -5\\ 7& 1& -2\\ -3& 4& 1\end{array}\right|$ by two methods.

Let

First method

Second method is the Sarus Method, where we adjoin the first two columns to the right to get

Evaluate

Let

#### Question 7:

Given: $∆=\left|\begin{array}{ccc}\mathrm{cos}\alpha \mathrm{cos}\beta & \mathrm{cos}\alpha \mathrm{sin}\beta & -\mathrm{sin}\alpha \\ -\mathrm{sin}\beta & \mathrm{cos}\beta & 0\\ \mathrm{sin}\alpha \mathrm{cos}\beta & \mathrm{sin}\alpha \mathrm{sin}\beta & \mathrm{cos}\alpha \end{array}\right|$

#### Question 8:

If , verify that |AB| = |A| |B|.

#### Question 9:

If A $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 0& 0& 4\end{array}\right]$, then show that |3 A| = 27 |A|.

#### Question 10:

Find the values of x, if

(i)

(ii) $\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right|$

(iii) $\left|\begin{array}{cc}3& x\\ x& 1\end{array}\right|=\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$

(iv) If $\left|\begin{array}{cc}3x& 7\\ 2& 4\end{array}\right|=10$, find the value of x.

(v) $\left|\begin{array}{cc}x+1& x-1\\ x-3& x+2\end{array}\right|=\left|\begin{array}{cc}4& -1\\ 1& 3\end{array}\right|$

(vi) $\left|\begin{array}{cc}2x& 5\\ 8& x\end{array}\right|=\left|\begin{array}{cc}6& 5\\ 8& 3\end{array}\right|$

(i)

(ii)

(iii)

(iv)

(v)

(vi)

#### Question 11:

Find the integral value of x, if $\left|\begin{array}{ccc}{x}^{2}& x& 1\\ 0& 2& 1\\ 3& 1& 4\end{array}\right|=28.$

Integral value of x is 2. Thus,  is not an integer.

#### Question 12:

For what value of x the matrix A is singular?

(i) Matrix A will be singular if

$\left|A\right|=\left|\begin{array}{cc}1+x& 7\\ 3-x& 8\end{array}\right|=0\phantom{\rule{0ex}{0ex}}⇒8+8x-21+7x=0\phantom{\rule{0ex}{0ex}}⇒15x-13=0\phantom{\rule{0ex}{0ex}}⇒15x=13\phantom{\rule{0ex}{0ex}}⇒x=\frac{13}{15}$

(ii) Matrix A will be singular if

#### Question 1:

Evaluate the following determinant:
(i) $\left|\begin{array}{ccc}1& 3& 5\\ 2& 6& 10\\ 31& 11& 38\end{array}\right|$

(ii) $\left|\begin{array}{ccc}67& 19& 21\\ 39& 13& 14\\ 81& 24& 26\end{array}\right|$

(iii) $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$

(iv) $\left|\begin{array}{ccc}1& -3& 2\\ 4& -1& 2\\ 3& 5& 2\end{array}\right|$

(v) $\left|\begin{array}{ccc}1& 4& 9\\ 4& 9& 16\\ 9& 16& 25\end{array}\right|$

(vi) $\left|\begin{array}{ccc}6& -3& 2\\ 2& -1& 2\\ -10& 5& 2\end{array}\right|$

(vii) $\left|\begin{array}{cccc}1& 3& 9& 27\\ 3& 9& 27& 1\\ 9& 27& 1& 3\\ 27& 1& 3& 9\end{array}\right|$

(viii)

(viii)

#### Question 2:

Without expanding, show that the values of each of the following determinants are zero:
(i) $\left|\begin{array}{ccc}8& 2& 7\\ 12& 3& 5\\ 16& 4& 3\end{array}\right|$

(ii) $\left|\begin{array}{ccc}6& -3& 2\\ 2& -1& 2\\ -10& 5& 2\end{array}\right|$

(iii) $\left|\begin{array}{ccc}2& 3& 7\\ 13& 17& 5\\ 15& 20& 12\end{array}\right|$

(iv) $\left|\begin{array}{ccc}1/a& {a}^{2}& bc\\ 1/b& {b}^{2}& ac\\ 1/c& {c}^{2}& ab\end{array}\right|$

(v) $\left|\begin{array}{ccc}a+b& 2a+b& 3a+b\\ 2a+b& 3a+b& 4a+b\\ 4a+b& 5a+b& 6a+b\end{array}\right|$

(vi) $\left|\begin{array}{ccc}1& a& {a}^{2}-bc\\ 1& b& {b}^{2}-ac\\ 1& c& {c}^{2}-ab\end{array}\right|$

(vii) $\left|\begin{array}{ccc}49& 1& 6\\ 39& 7& 4\\ 26& 2& 3\end{array}\right|$

(viii) $\left|\begin{array}{ccc}0& x& y\\ -x& 0& z\\ -y& -z& 0\end{array}\right|$

(ix) $\left|\begin{array}{ccc}1& 43& 6\\ 7& 35& 4\\ 3& 17& 2\end{array}\right|$

(x) $\left|\begin{array}{cccc}{1}^{2}& {2}^{2}& {3}^{2}& {4}^{2}\\ {2}^{2}& {3}^{2}& {4}^{2}& {5}^{2}\\ {3}^{2}& {4}^{2}& {5}^{2}& {6}^{2}\\ {4}^{2}& {5}^{2}& {6}^{2}& {7}^{2}\end{array}\right|$

(xi) $\left|\begin{array}{ccc}a& b& c\\ a+2x& b+2y& c+2z\\ x& y& z\end{array}\right|$

(xii) $\left|\begin{array}{ccc}{\left({2}^{x}+{2}^{-x}\right)}^{2}& {\left({2}^{x}-{2}^{-x}\right)}^{2}& 1\\ {\left({3}^{x}+{3}^{-x}\right)}^{2}& {\left({3}^{x}-{3}^{-x}\right)}^{2}& 1\\ {\left({4}^{x}+{4}^{-x}\right)}^{2}& {\left({4}^{x}-{4}^{-x}\right)}^{2}& 1\end{array}\right|$

(xiii) $\left|\begin{array}{ccc}\mathrm{sin}\alpha & \mathrm{cos}\alpha & \mathrm{cos}\left(\alpha +\delta \right)\\ \mathrm{sin}\beta & \mathrm{cos}\beta & \mathrm{cos}\left(\beta +\delta \right)\\ \mathrm{sin}\gamma & \mathrm{cos}\gamma & \mathrm{cos}\left(\gamma +\delta \right)\end{array}\right|$

(xiv) $\left|\begin{array}{ccc}{\mathrm{sin}}^{2}23°& {\mathrm{sin}}^{2}67°& \mathrm{cos}180°\\ -{\mathrm{sin}}^{2}67°& -{\mathrm{sin}}^{2}23°& {\mathrm{cos}}^{2}180°\\ \mathrm{cos}180°& {\mathrm{sin}}^{2}23°& {\mathrm{sin}}^{2}67°\end{array}\right|$

(xv) $\left|\begin{array}{ccc}\mathrm{cos}\left(x+y\right)& -\mathrm{sin}\left(x+y\right)& \mathrm{cos}2y\\ \mathrm{sin}x& \mathrm{cos}x& \mathrm{sin}y\\ -\mathrm{cos}x& \mathrm{sin}x& -\mathrm{cos}y\end{array}\right|$

(xvi) $\left|\begin{array}{ccc}\sqrt{23}+\sqrt{3}& \sqrt{5}& \sqrt{5}\\ \sqrt{15}+\sqrt{46}& 5& \sqrt{10}\\ 3+\sqrt{115}& \sqrt{15}& 5\end{array}\right|$

(xvii)

(xii)

(xiii)

(xiv)

(xv)

(xvi)
$\left|\begin{array}{ccc}\sqrt{23}+\sqrt{3}& \sqrt{5}& \sqrt{5}\\ \sqrt{15}+\sqrt{46}& 5& \sqrt{10}\\ 3+\sqrt{115}& \sqrt{15}& 5\end{array}\right|\phantom{\rule{0ex}{0ex}}=\left|\begin{array}{ccc}\sqrt{3}& \sqrt{5}& \sqrt{5}\\ \sqrt{15}& 5& \sqrt{10}\\ 3& \sqrt{15}& 5\end{array}\right|+\left|\begin{array}{ccc}\sqrt{23}& \sqrt{5}& \sqrt{5}\\ \sqrt{46}& 5& \sqrt{10}\\ \sqrt{115}& \sqrt{15}& 5\end{array}\right|\phantom{\rule{0ex}{0ex}}=\sqrt{3}\left|\begin{array}{ccc}1& \sqrt{5}& \sqrt{5}\\ \sqrt{5}& 5& \sqrt{10}\\ \sqrt{3}& \sqrt{15}& 5\end{array}\right|+\sqrt{23}\left|\begin{array}{ccc}1& \sqrt{5}& \sqrt{5}\\ \sqrt{2}& 5& \sqrt{10}\\ \sqrt{5}& \sqrt{15}& 5\end{array}\right|\phantom{\rule{0ex}{0ex}}=\sqrt{3}×\sqrt{5}\left|\begin{array}{ccc}1& 1& \sqrt{5}\\ \sqrt{5}& \sqrt{5}& \sqrt{10}\\ \sqrt{3}& \sqrt{3}& 5\end{array}\right|+\sqrt{23}×\sqrt{5}\left|\begin{array}{ccc}1& \sqrt{5}& 1\\ \sqrt{2}& 5& \sqrt{2}\\ \sqrt{5}& \sqrt{15}& \sqrt{5}\end{array}\right|\phantom{\rule{0ex}{0ex}}=0+0\phantom{\rule{0ex}{0ex}}=0$

(xvii)

#### Question 3:

Evaluate :

$\left|\begin{array}{ccc}a& b+c& {a}^{2}\\ b& c+a& {b}^{2}\\ c& a+b& {c}^{2}\end{array}\right|$

#### Question 4:

Evaluate :

$\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|$

#### Question 5:

Evaluate :

$\left|\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|$

#### Question 6:

Evaluate :

$\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|$

#### Question 7:

Evaluate the following:

$\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right|$

Let $∆=\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right|$.

#### Question 8:

Evaluate the following:

$\left|\begin{array}{ccc}0& x{y}^{2}& x{z}^{2}\\ {x}^{2}y& 0& y{z}^{2}\\ {x}^{2}z& z{y}^{2}& 0\end{array}\right|$

Let $∆=\left|\begin{array}{ccc}0& x{y}^{2}& x{z}^{2}\\ {x}^{2}y& 0& y{z}^{2}\\ {x}^{2}z& z{y}^{2}& 0\end{array}\right|$.

#### Question 9:

Evaluate the following:

$\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$

Let $∆=\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$.

#### Question 11:

Prove that :

$\left|\begin{array}{ccc}a& b& c\\ a-b& b-c& c-a\\ b+c& c+a& a+b\end{array}\right|={a}^{3}+{b}^{3}+{c}^{3}-3abc$

#### Question 12:

Prove that :

$\left|\begin{array}{ccc}b+c& a-b& a\\ c+a& b-c& b\\ a+b& c-a& c\end{array}\right|=3abc-{a}^{3}-b-{c}^{3}$

#### Question 13:

Prove that :

$\left|\begin{array}{ccc}a+b& b+c& c+a\\ b+c& c+a& a+b\\ c+a& a+b& b+c\end{array}\right|=2\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

Prove that :

Prove that :

Prove that :

Prove that :

#### Question 18:

Prove that :

$\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|=\left|\begin{array}{ccc}1& a& {a}^{2}\\ 1& b& {b}^{2}\\ 1& c& {c}^{2}\end{array}\right|$

Prove that :

Prove that :

Prove that :

Hence proved.

Prove that :

= RHS

Hence proved.

Prove that :

Hence proved.

#### Question 24:

Prove that :

$\left|\begin{array}{ccc}{a}^{2}& bc& ac+{c}^{2}\\ {a}^{2}+ab& {b}^{2}& ac\\ ab& {b}^{2}+bc& {c}^{2}\end{array}\right|=4{a}^{2}{b}^{2}{c}^{2}$

Prove that :

#### Question 26:

Prove that :

$\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|=1$

Prove that :

Prove that

Hence proved.

#### Question 29:

Prove that $\left|\begin{array}{ccc}{a}^{2}+1& ab& ac\\ ab& {b}^{2}+1& bc\\ ca& cb& {c}^{2}+1\end{array}\right|=1+{a}^{2}+{b}^{2}+{c}^{2}$

Hence proved.

Hence proved.

#### Question 32:

$\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|=4abc$

Hence proved.

#### Question 33:

$\left|\begin{array}{ccc}{b}^{2}+{c}^{2}& ab& ac\\ ba& {c}^{2}+{a}^{2}& bc\\ ca& cb& {a}^{2}+{b}^{2}\end{array}\right|=4{a}^{2}{b}^{2}{c}^{2}$

$∆=\left|\begin{array}{ccc}{b}^{2}+{c}^{2}& ab& ac\\ ba& {c}^{2}+{a}^{2}& bc\\ ca& cb& {a}^{2}+{b}^{2}\end{array}\right|$

#### Question 34:

$\left|\begin{array}{ccc}0& {b}^{2}a& {c}^{2}a\\ {a}^{2}b& 0& {c}^{2}b\\ {a}^{2}c& {b}^{2}c& 0\end{array}\right|=2{a}^{3}{b}^{3}{c}^{3}$

#### Question 35:

Prove that $\left|\begin{array}{ccc}\frac{{a}^{2}+{b}^{2}}{c}& c& c\\ a& \frac{{b}^{2}+{c}^{2}}{a}& a\\ b& b& \frac{{c}^{2}+{a}^{2}}{b}\end{array}\right|=4abc$

Hence proved.

#### Question 36:

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

Hence proved.

#### Question 37:

Prove the following identities:
$\left|\begin{array}{ccc}x+\lambda & 2x& 2x\\ 2x& x+\lambda & 2x\\ 2x& 2x& x+\lambda \end{array}\right|=\left(5x+\lambda \right){\left(\lambda -x\right)}^{2}$

#### Question 38:

Using properties of determinants prove that

Hence proved.

#### Question 39:

Prove the following identities:

$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$

Hence proved.

#### Question 41:

$\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+a& a\\ 1& 1& 1+a\end{array}\right|={a}^{3}+3{a}^{2}$

#### Question 42:

Prove the following identities:

$\left|\begin{array}{ccc}2y& y-z-x& 2y\\ 2z& 2z& z-x-y\\ x-y-z& 2x& 2x\end{array}\right|={\left(x+y+z\right)}^{3}$

Show that

#### Question 44:

Prove the following identities:

$\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|={a}^{2}\left(a+x+y+z\right)$

#### Question 45:

Prove the following identities:

$\left|\begin{array}{ccc}{a}^{3}& 2& a\\ {b}^{3}& 2& b\\ {c}^{3}& 2& c\end{array}\right|=2\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)\begin{array}{}\\ \\ \end{array}$

#### Question 46:

Without expanding, prove that

Hence proved.

#### Question 47:

Show that

Given: a, b, c are in A.P.

$2b=a+c$

#### Question 48:

Show that

Given:α, β, γ areinA.P.

Now,
$2\beta =\alpha +\gamma$

#### Question 49:

If a, b, c are real numbers such that $\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|=0$, then show that either .

#### Question 50:

Let $∆=\left|\begin{array}{ccc}p& b& c\\ a& q& c\\ a& b& r\end{array}\right|$.
Now,

#### Question 51:

Show that x = 2 is a root of the equation

$\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0$ and solve it completely.

#### Question 52:

​Solve the following determinant equations:

(i) $\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|=0$

(ii)

(iii) $\left|\begin{array}{ccc}3x-8& 3& 3\\ 3& 3x-8& 3\\ 3& 3& 3x-8\end{array}\right|=0$

(iv)

(v) $\left|\begin{array}{ccc}x+1& 3& 5\\ 2& x+2& 5\\ 2& 3& x+4\end{array}\right|=0$

(vi)

(vii) $\left|\begin{array}{ccc}15-2x& 11-3x& 7-x\\ 11& 17& 14\\ 10& 16& 13\end{array}\right|=0$

(viii) $\left|\begin{array}{ccc}1& 1& x\\ p+1& p+1& p+x\\ 3& x+1& x+2\end{array}\right|=0$

(ix) $\left|\begin{array}{ccc}3& -2& \mathrm{sin}\left(3\theta \right)\\ -7& 8& \mathrm{cos}\left(2\theta \right)\\ -11& 14& 2\end{array}\right|=0$

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

#### Question 53:

If $a,b$ and $c$ are all non-zero and $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=$0, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+$1$=$0.

We have,

$\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=$0

#### Question 54:

If $\left|\begin{array}{ccc}a& b-y& c-z\\ a-x& b& c-z\\ a-x& b-y& c\end{array}\right|=$0, then using properties of determinants, find the value of $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}$, where $x,y,z\ne$0.

$\left|\begin{array}{ccc}a& b-y& c-z\\ a-x& b& c-z\\ a-x& b-y& c\end{array}\right|=$0

#### Question 1:

Find the area of the triangle with vertices at the points:
(i) (3, 8), (−4, 2) and (5, −1)
(ii) (2, 7), (1, 1) and (10, 8)
(iii) (−1, −8), (−2, −3) and (3, 2)
(iv) (0, 0), (6, 0) and (4, 3).

(i)

(ii)

(iii)

(iv)

#### Question 2:

Using determinants show that the following points are collinear:
(i) (5, 5), (−5, 1) and (10, 7)
(ii) (1, −1), (2, 1) and (4, 5)
(iii) (3, −2), (8, 8) and (5, 2)
(iv) (2, 3), (−1, −2) and (5, 8)

(i) If the points  (5, 5), (−5, 1) and (10, 7) are collinear, then

Thus, these points are colinear.

(ii) If the points (1, −1), (2, 1) and (4, 5) are collinear, then

Thus, these points are collinear.

(iii) If the points (3, −2), (8, 8) and (5, 2)  are collinear, then

Thus the points are colinear.

(iv) If the points (2, 3), (−1, −2) and (5, 8) are collinear, then

Thus the points are colinear.

#### Question 3:

If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.

If the points (a, 0), (0, b) and (1, 1) are collinear, then

#### Question 4:

Using determinants prove that the points (a, b), (a', b') and (aa', bb') are collinear if ab' = a'b.

If the points are collinear, then ∆ = 0. So,
ab' − a'b = 0

Thus, ab' = a'b

#### Question 5:

Find the value of $\lambda$ so that the points (1, −5), (−4, 5) and are collinear.

If the points (1, −5), (−4, 5) and  are collinear, then

#### Question 6:

Find the value of x if the area of ∆ is 35 square cms with vertices (x, 4), (2, −6) and (5, 4).

#### Question 7:

Using determinants, find the area of the triangle whose vertices are (1, 4), (2, 3) and (−5, −3). Are the given points collinear?

Therefore, (1, 4), (2, 3) and (−5, −3) are not collinear because, $\left|\begin{array}{ccc}1& 4& 1\\ 2& 3& 1\\ -5& -3& 1\end{array}\right|$ is not equal to 0.

#### Question 8:

Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).

Given:
Vertices of triangle: (− 3, 5), (3, − 6) and (7, 2)

#### Question 9:

Using determinants, find the value of k so that the points (k, 2 − 2 k), (−k + 1, 2k) and (−4 − k, 6 − 2k) may be collinear.

If the points (k, 2 − 2 k), (− k + 1, 2k) and (− 4 − k, 6 − 2k) are collinear, then

#### Question 10:

If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.

If the points (x, −2), (5, 2), (8, 8) are collinear, then

#### Question 11:

If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.

If the points (3, −2), (x, 2) and (8, 8) are collinear, then

#### Question 12:

Using determinants, find the equation of the line joining the points
(i) (1, 2) and (3, 6)
(ii) (3, 1) and (9, 3)

(i)
Given: A  =  (1, 2) and B  =  (3, 6)

Let the point P be (x, y).  So,
Area of triangle ABP = 0

(ii)
Given: A = (3, 1) and B = (9, 3)

Let the point P be (x, y). So,

Area of triangle ABP = 0

$⇒∆=\frac{1}{2}\left|\begin{array}{ccc}3& 1& 1\\ 9& 3& 1\\ x& y& 1\end{array}\right|=0\phantom{\rule{0ex}{0ex}}⇒3\left(3-y\right)-1\left(9-x\right)+1\left(9y-3x\right)=0\phantom{\rule{0ex}{0ex}}⇒9-3y-9+x+9y-3x=0\phantom{\rule{0ex}{0ex}}⇒-2x+6y=0\phantom{\rule{0ex}{0ex}}⇒x=3y\phantom{\rule{0ex}{0ex}}$

#### Question 13:

Find values of k, if area of triangle is 4 square units whose vertices are
(i) (k, 0), (4, 0), (0, 2)
(ii) (−2, 0), (0, 4), (0, k)

x − 2y = 4
−3x + 5y = −7

2xy = 1
7x − 2y = −7

2xy = 17
3x + 5y = 6

3x + y = 19
3xy = 23

2xy = − 2
3x + 4y = 3

#### Question 6:

3x + ay = 4
2x + ay = 2, a ≠ 0

2x + 3y = 10
x + 6y = 4

5x + 7y = − 2
4x + 6y = − 3

9x + 5y = 10
3y − 2x = 8

#### Question 10:

x + 2y = 1
3x + y = 4

Given: x + 2y = 1
3x + y = 4

#### Question 11:

3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11

Given: 3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11

#### Question 12:

x − 4yz = 11
2x − 5y + 2z = 39
− 3x + 2y + z = 1

Given: x − 4yz = 11
2x − 5y + 2z = 39
− 3x + 2y + z = 1

#### Question 13:

6x + y − 3z = 5
x + 3y − 2z = 5
2x + y + 4z = 8

Given: 6x + y − 3z = 5
x + 3y − 2z = 5
2x + y + 4z = 8

#### Question 14:

x+ y = 5
y + z = 3
x + z = 4

These equations can be written as
x + y + 0z = 5
0x + y + z = 3
x + 0y + z = 4

#### Question 15:

2y − 3z = 0
x + 3y = − 4
3x + 4y = 3

These equations can be written as
0x + 2y − 3z = 0
x + 3y + 0z = − 4
3x + 4y + 0z = 3

#### Question 16:

5x − 7y + z = 11
6x − 8yz = 15
3x + 2y − 6z = 7

Given: 5x − 7y + z = 11
6x − 8yz = 15
3x + 2y − 6z = 7

#### Question 17:

2x − 3y − 4z = 29
− 2x + 5yz = − 15
3xy + 5z = − 11

Given: 2x − 3y − 4z = 29
− 2x + 5yz = − 15
3xy + 5z = − 11

#### Question 18:

x + y = 1
x + z = − 6
xy − 2z = 3

These equations can be written as
x+ y + 0z = 1
x + 0y + z = − 6
xy − 2z = 3

#### Question 19:

x + y + z + 1 = 0
ax + by + cz + d = 0
a2x + b2y + x2z + d2 = 0

#### Question 20:

x + y + z + w = 2
x − 2y + 2z + 2w = − 6
2x + y − 2z + 2w = − 5
3xy + 3z − 3w = − 3

2x − 3z + w = 1
xy + 2w = 1
− 3y + z + w = 1
x + y + z = 1

#### Question 22:

2xy = 5
4x − 2y = 7

Given: 2xy = 5
4x − 2y = 7

Here, D1 and D2 are non-zero, but D is zero. Thus, the given system of linear equations is inconsistent.

#### Question 23:

3x + y = 5
− 6x − 2y = 9

Given: 3x + y = 5
− 6x − 2y = 9

Here, D1 and D2 are non-zero, but D is zero. Thus, the system of linear equations is inconsistent.

#### Question 24:

3xy + 2z = 3
2x + y + 3z = 5
x − 2yz = 1

Given: 3xy + 2z = 3
2x + y + 3z = 5
x − 2yz = 1

$D=\left|\begin{array}{ccc}3& -1& 2\\ 2& 1& 3\\ 1& -2& -1\end{array}\right|\phantom{\rule{0ex}{0ex}}=3\left(-1+6\right)+1\left(-2-3\right)+2\left(-4-1\right)\phantom{\rule{0ex}{0ex}}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{1=}\left|\begin{array}{ccc}3& -1& 2\\ 5& 1& 3\\ 1& -2& -1\end{array}\right|\phantom{\rule{0ex}{0ex}}=3\left(-1+6\right)+1\left(-5-3\right)+2\left(-10-1\right)\phantom{\rule{0ex}{0ex}}=-15\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{2}=\left|\begin{array}{ccc}3& 3& 2\\ 2& 5& 3\\ 1& 1& -1\end{array}\right|\phantom{\rule{0ex}{0ex}}=3\left(-5-3\right)-3\left(-2-3\right)+2\left(2-5\right)\phantom{\rule{0ex}{0ex}}=-15\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{D}_{3}=\left|\begin{array}{ccc}3& -1& 3\\ 2& 1& 5\\ 1& -2& 1\end{array}\right|\phantom{\rule{0ex}{0ex}}=3\left(1+10\right)+1\left(2-5\right)+3\left(-4-1\right)\phantom{\rule{0ex}{0ex}}=-15\phantom{\rule{0ex}{0ex}}$

Here, D is zero, but D1, D2 and D3​ are non-zero. Thus, the system of linear equations is inconsistent.

#### Question 25:

3xy + 2z = 6
2xy + z = 2
3x + 6y + 5z = 20.

Given: 3xy + 2z = 6
2xy + z = 2
3x + 6y + 5z = 20

Since D is non-zero, the system of linear equations is consistent and has a unique solution.

#### Question 26:

xy + z = 3
2x + yz = 2
x − 2y + 2z = 1

Here,
$D={D}_{1}={D}_{2}={D}_{3}=0$
Thus, the system of linear equations has infinitely many solutions.

#### Question 27:

x + 2y = 5
3x + 6y = 15

Hence, the system of linear equation has infinitely many solutions.

#### Question 28:

x + yz = 0
x − 2y + z = 0
3x + 6y − 5z = 0

Hence, the system of linear equations has infinitely many solutions.

#### Question 29:

2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2

Hence, the system of linear equations has infinitely many solutions.

#### Question 30:

xy + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10

Hence, the system of equations has infinitely many solutions.

#### Question 31:

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission

 Month Sale of units Total commission drawn (in Rs) A B C Jan 90 100 20 800 Feb 130 50 40 900 March 60 100 30 850

Find out the rates of commission on items A, B and C by using determinant method.

Let x, y and z be the rates of commission on items A, B and C respectively. Based on the given data, we get

Dividing all the equations by 10 on both sides, we get

Therefore, the rates of commission on items A, B and C are 2, 4 and 11, respectively.

#### Question 32:

An automobile company uses three types of steel S1, S2 and S3 for producing three types of cars C1, C2 and C3. Steel requirements (in tons) for each type of cars are given below :

 Cars C1 C2 C3 Steel S1 2 3 4 S2 1 1 2 S3 3 2 1

Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.

Therefore, 2 C1 cars, 3 C2 cars and 4 C​3 cars can be produced using the three types of steel.

#### Question 1:

Solve each of the following system of homogeneous linear equations.
x
+ y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0

Given: x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0

#### Question 2:

Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2xy + 3z = 0

#### Question 3:

Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0

Given: 3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0

#### Question 4:

Find the real values of $\lambda$ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions

#### Question 5:

If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.

The three equations can be expressed as

Expressing this as a determinant, we get

$∆=\left|\begin{array}{ccc}\left(a-1\right)& -1& -1\\ -1& \left(b-1\right)& -1\\ -1& -1& \left(c-1\right)\end{array}\right|$

If the matrix has a non-trivial solution, then

$\left|\begin{array}{ccc}\left(a-1\right)& -1& -1\\ -1& \left(b-1\right)& -1\\ -1& -1& \left(c-1\right)\end{array}\right|=0$

Hence proved.

#### Question 1:

If A is a singular matrix, then write the value of |A|.

Given: A is a singular matrix.

Thus, $\left|A\right|=0$

#### Question 2:

For what value of x, the following matrix is singular?

$\left[\begin{array}{cc}5-x& x+1\\ 2& 4\end{array}\right]$

If a matrix A is singular, then $\left|A\right|=0$

#### Question 3:

Write the value of the determinant $\left[\begin{array}{ccc}2& 3& 4\\ 2x& 3x& 4x\\ 5& 6& 8\end{array}\right].$

#### Question 4:

State whether the matrix $\left[\begin{array}{cc}2& 3\\ 6& 4\end{array}\right]$ is singular or non-singular.

#### Question 5:

Find the value of the determinant $\left[\begin{array}{cc}4200& 4201\\ 4205& 4203\end{array}\right]$.

#### Question 6:

Find the value of the determinant $\left[\begin{array}{ccc}101& 102& 103\\ 104& 105& 106\\ 107& 108& 109\end{array}\right]$

#### Question 7:

Write the value of the determinant $\left|\begin{array}{ccc}a& 1& b+c\\ b& 1& c+a\\ c& 1& a+b\end{array}\right|.$

#### Question 8:

If , find the value of |A| + |B|.

If , find |AB|.

#### Question 10:

Evaluate $\left|\begin{array}{cc}4785& 4787\\ 4789& 4791\end{array}\right|$

If w is an imaginary cube root of unity, find the value of $\left|\begin{array}{ccc}1& w& {w}^{2}\\ w& {w}^{2}& 1\\ {w}^{2}& 1& w\end{array}\right|$.