Determinants

Q38 If $A$ is a matrix of order $3 \times 3$, then |3A| is equal to ………….
FillBlank Q39 If $A$ is invertible matrix of order $3 \times 3$, then $\left| {{A^{ - 1}}} \right|$ is equal to …………..

FillBlank Q40 If $x,y,z \in R$, then the value of $\left| {\begin{array}{llllllllllllllllllll}{{{\left( {{2^x} + {2^{ - x}}} \right)}^2}}&{{{\left( {{2^x} - {2^{ - x}}} \right)}^2}}&1\\{{{\left( {{3^x} + {3^{ - x}}} \right)}^2}}&{{{\left( {{3^x} - {3^{ - x}}} \right)}^2}}&1\\{{{\left( {{4^x} + {4^{ - x}}} \right)}^2}}&{{{\left( {{4^x} - {4^{ - x}}} \right)}^2}}&1\end{array}} \right|$ is……..

FillBlank Q41 If $\cos 2\theta = 0$, then $\left| {\begin{array}{cccccccccccccccccccc}0&{\cos \theta }&{\sin \theta }\\{\cos \theta }&{\sin \theta }&0\\{\sin \theta }&0&{\cos \theta }\end{array}} \right|$ is equal to……………. FillBlank Q42 If $A$ is a matrix of order $3 \times 3$, then ${\left( {{A^2}} \right)^{ - 1}}$ is equal to……….. FillBlank Q43 If $A$ is a matrix of order $3 \times 3$, then the number of minors in determinant of $A$ are…………. FillBlank Q44 The sum of products of elements of any row with the cofactors of corresponding elements is equal to………………. FillBlank Q45 If $x = - 9$ is a root of $\left| {\begin{array}{llllllllllllllllllll}x&3&7\\2&x&2\\7&6&x\end{array}} \right| = 0$, then other two roots are…………….
FillBlank Q46 $\left| {\begin{array}{cccccccccccccccccccc}0&{xyz}&{x - z}\\{y - x}&0&{y - z}\\{z - x}&{z - y}&0\end{array}} \right|$ is equal to…………….
FillBlank Q47 . If $f(x) = \left| {\begin{array}{llllllllllllllllllll}{{{(1 + x)}^{17}}}&{{{(1 + x)}^{19}}}&{{{(1 + x)}^{23}}}\\{{{(1 + x)}^{23}}}&{{{(1 + x)}^{29}}}&{{{(1 + x)}^{34}}}\\{{{(1 + x)}^{41}}}&{{{(1 + x)}^{43}}}&{{{(1 + x)}^{47}}}\end{array}} \right|$
$= A + Bx + C{x^2} +$……….., then $A$ is equal to…………… FillBlank Q18 If $A = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2&0\\{ - 2}&{ - 1}&{ - 2}\\0&{ - 1}&1\end{array}} \right|$, then find the value of ${A^{ - 1}}$.
Using ${A^{ - 1}}$, solve the system of linear equations $x - 2y = 10$, $2x - y - z = 8$ and $- 2y + z = 7$. LA Q19 Using matrix method, solve the system of equations $3x + 2y - 2z = 3$, $x + 2y + 3z = 6$ and $2x - y + z = 2$. LA Q20 If $A = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&2&{ - 4}\\{ - 4}&2&{ - 4}\\2&{ - 1}&5\end{array}} \right|$ and $B = \left| {\begin{array}{cccccccccccccccccccc}1&{ - 1}&0\\2&3&4\\0&1&2\end{array}} \right|$, then find BA and use this to solve the system of equations $y + 2z = 7,x - y = 3$ and $2x + 3y + 4z = 17$.
LA Q21 If $a + b + c \ne 0$ and $\left| {\begin{array}{llllllllllllllllllll}a&b&c\\b&c&a\\c&a&b\end{array}} \right| = 0$, then prove that $a = b = c$.
LA Q22 Prove that $\left| {\begin{array}{llllllllllllllllllll}{bc - {a^2}}&{ca - {b^2}}&{ab - {c^2}}\\{ca - {b^2}}&{ab - {c^2}}&{bc - {a^2}}\\{ab - {c^2}}&{bc - {a^2}}&{ca - {b^2}}\end{array}} \right|$ is divisible by $(a + b + c)$ and find the quotient.
LA Q23 If $x + y + z = 0$, then prove that $\left| {\begin{array}{llllllllllllllllllll}{xa}&{yb}&{zc}\\{yc}&{za}&{xb}\\{zb}&{xc}&{ya}\end{array}} \right| = xyz\left| {\begin{array}{cccccccccccccccccccc}a&b&c\\c&a&b\\b&c&a\end{array}} \right|$. LA Q24 If $\left| {\begin{array}{cccccccccccccccccccc}{2x}&5\\8&x\end{array}} \right| = \left| {\begin{array}{cccccccccccccccccccc}6&{ - 2}\\7&3\end{array}} \right|$, then the value of $x$ is MCQ Q25 The value of $\left| {\begin{array}{llllllllllllllllllll}{a - b}&{b + c}&a\\{b - a}&{c + a}&b\\{c - a}&{a + b}&c\end{array}} \right|$ is MCQ Q26 If the area of a triangle with vertices $( - 3,0),(3,0)$ and $(0,k)$ is 9 sq units. Then, the value of $k$ will be MCQ Q27 The determinant $\left| {\begin{array}{llllllllllllllllllll}{{b^2} - ab}&{b - c}&{bc - ac}\\{ab - {a^2}}&{a - b}&{{b^2} - ab}\\{bc - ac}&{c - a}&{ab - {a^2}}\end{array}} \right|$ equals to MCQ Q28 The number of distinct real roots of $\left| {\begin{array}{llllllllllllllllllll}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}} \right| = 0$ in the
interval $- \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is

MCQ Q29 If $A,B$and $C$ are angles of a triangle, then the determinant $\left| {\begin{array}{cccccccccccccccccccc}{ - 1}&{\cos C}&{\cos B}\\{\cos C}&{ - 1}&{\cos A}\\{\cos B}&{\cos A}&{ - 1}\end{array}} \right|$ is equal to MCQ Q30 If $f(t) = \left[ {\begin{array}{cccccccccccccccccccc}{\cos t}&t&1\\{2\sin t}&t&{2t}\\{\sin t}&t&t\end{array}} \right]$, then $\mathop {\lim }\limits_{t \to 0} \frac{{f(t)}}{{{t^2}}}$ is equal to MCQ Q31 The maximum value of $\Delta = \left| {\begin{array}{cccccccccccccccccccc}1&1&1\\1&{1 + \sin \theta }&1\\{1 + \cos \theta }&1&1\end{array}} \right|$ is (where, $\theta$ is real number) MCQ Q32 If $f(x) = \left| {\begin{array}{cccccccccccccccccccc}0&{x - a}&{x - b}\\{x + a}&0&{x - c}\\{x + b}&{x + c}&0\end{array}} \right|$, then MCQ Q33 If $A = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&\lambda &{ - 3}\\0&2&5\\1&1&3\end{array}} \right|$, then ${A^{ - 1}}$ exists, if

MCQ Q34 If $A$ and $B$ are invertible matrices, then which of the following is not correct? MCQ Q35 If x, y and z are all different from zero and $\left| {\begin{array}{cccccccccccccccccccc}{1 + x}&1&1\\1&{1 + y}&1\\1&1&{1 + z}\end{array}} \right| = 0$, then the value of ${x^{ - 1}} + {y^{ - 1}} + {z^{ - 1}}$ is MCQ Q36 The value of $\left| {\begin{array}{cccccccccccccccccccc}x&{x + y}&{x + 2y}\\{x + 2y}&x&{x + y}\\{x + y}&{x + 2y}&x\end{array}} \right|$ is MCQ Q37 If there are two values of a which makes determinant, $\Delta = \left| {\begin{array}{cccccccccccccccccccc}1&{ - 2}&5\\2&a&{ - 1}\\0&4&{2a}\end{array}} \right| = 86$, then the sum of these number is MCQ Q1 $\left| {\begin{array}{cccccccccccccccccccc}{{x^2} - x + 1}&{x - 1}\\{x + 1}&{x + 1}\end{array}} \right|$

SA Q2 $\left| {\begin{array}{cccccccccccccccccccc}{a + x}&y&z\\x&{a + y}&z\\x&y&{a + z}\end{array}} \right|$
SA Q3 $\left| {\begin{array}{cccccccccccccccccccc}0&{x{y^2}}&{x{z^2}}\\{{x^2}y}&0&{y{z^2}}\\{{x^2}z}&{z{y^2}}&0\end{array}} \right|$
SA Q4 $\left| {\begin{array}{cccccccccccccccccccc}{3x}&{ - x + y}&{ - x + z}\\{x - y}&{3y}&{z - y}\\{x - z}&{y - z}&{3z}\end{array}} \right|$
SA Q5 $\left| {\begin{array}{cccccccccccccccccccc}{x + 4}&x&x\\x&{x + 4}&x\\x&x&{x + 4}\end{array}} \right|$
SA Q6 $\left| {\begin{array}{cccccccccccccccccccc}{a - b - c}&{2a}&{2a}\\{2b}&{b - c - a}&{2b}\\{2c}&{2c}&{c - a - b}\end{array}} \right|$
SA Q7 $\left| {\begin{array}{cccccccccccccccccccc}{{y^2}{z^2}}&{yz}&{y + z}\\{{z^2}{x^2}}&{zx}&{z + x}\\{{x^2}{y^2}}&{xy}&{x + y}\end{array}} \right| = 0$
SA Q8 $\left| {\begin{array}{cccccccccccccccccccc}{y + z}&z&y\\z&{z + x}&x\\y&x&{x + y}\end{array}} \right| = 4xyz$ SA Q9 $\left| {\begin{array}{cccccccccccccccccccc}{{a^2} + 2a}&{2a + 1}&1\\{2a + 1}&{a + 2}&1\\3&3&1\end{array}} \right| = {(a - 1)^3}$
SA Q10 If $A + B + C = 0$, then prove that $\left| {\begin{array}{cccccccccccccccccccc}1&{\cos C}&{\cos B}\\{\cos C}&1&{\cos A}\\{\cos B}&{\cos A}&1\end{array}} \right| = 0$.
SA Q11 If the coordinates of the vertices of an equilateral triangle with sides of length '$a$' are $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$ and $\left( {{x_3},{y_3}} \right)$, then
${\left| {\begin{array}{llllllllllllllllllll}{{x_1}}&{{y_1}}&1\\{{x_2}}&{{y_2}}&1\\{{x_3}}&{{y_3}}&1\end{array}} \right|^2} = \frac{{3{a^4}}}{4}$
SA Q12 Find the value of $\theta$ satisfying $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&{\sin 3\theta }\\{ - 4}&3&{\cos 2\theta }\\7&{ - 7}&{ - 2}\end{array}} \right] = 0$
SA Q13 If $\left[ {\begin{array}{llllllllllllllllllll}{4 - x}&{4 + x}&{4 + x}\\{4 + x}&{4 - x}&{4 + x}\\{4 + x}&{4 + x}&{4 - x}\end{array}} \right] = 0$, then find the value of $x$. SA Q14 If ${a_1},{a_2},{a_3}, \ldots ,{a_r}$ are in GP, then prove that the determinant
$\left| {\begin{array}{llllllllllllllllllll}{{a_{r + 1}}}&{{a_{r + 5}}}&{{a_{r + 9}}}\\{{a_{r + 7}}}&{{a_{r + 11}}}&{{a_{r + 15}}}\\{{a_{r + 11}}}&{{a_{r + 17}}}&{{a_{r + 21}}}\end{array}} \right|$ is independent of $r$. SA Q15 Show that the points $(a + 5,a - 4),(a - 2,a + 3)$ and $(a,a)$ do not lie on a straight line for any value of $a$. SA Q16 Show that $\Delta ABC$ is an isosceles triangle, if the determinant
$\Delta = \left| {\begin{array}{cccccccccccccccccccc}1&1&1\\{1 + \cos A}&{1 + \cos B}&{1 + \cos C}\\{{{\cos }^2}A + \cos A}&{{{\cos }^2}B + \cos B}&{{{\cos }^2}C + \cos C}\end{array}} \right| = 0$
SA Q17 Find ${A^{ - 1}}$, if $A = \left| {\begin{array}{llllllllllllllllllll}0&1&1\\1&0&1\\1&1&0\end{array}} \right|$ and show that ${A^{ - 1}} = \frac{{{A^2} - 3I}}{2}$. SA Q38 If $\Delta = \left| {\begin{array}{llllllllllllllllllll}a&p&x\\b&q&y\\c&r&z\end{array}} \right| = 16$, then ${\Delta _1} = \left| {\begin{array}{llllllllllllllllllll}{p + x}&{a + x}&{a + p}\\{q + y}&{b + y}&{b + q}\\{r + z}&{c + z}&{c + r}\end{array}} \right| = 32$.

Correct Answer True TrueFalse Q48 ${\left( {{A^3}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^3}$, where $A$ is a square matrix and $|A| \ne 0$

Correct Answer True TrueFalse Q49 ${(aA)^{ - 1}} = \frac{1}{a}{A^{ - 1}}$, where $a$ is any real number and $A$ is a square matrix.

Correct Answer False TrueFalse Q50 $\left| {{A^{ - 1}}} \right| \ne |A{|^{ - 1}}$, where $A$ is a non-singular matrix.

Correct Answer False TrueFalse Q51 If $A$ and $B$ are matrices of order 3 and $|A| = 5,|B| = 3$, then $|3AB| = 27 \times 5 \times 3 = 405$

Correct Answer True TrueFalse Q52 If the value of a third order determinant is 12 , then the value of the determinant formed by replacing each element by its cofactor will be 144.

Correct Answer True TrueFalse Q53 . $\left| {\begin{array}{llllllllllllllllllll}{x + 1}&{x + 2}&{x + a}\\{x + 2}&{x + 3}&{x + b}\\{x + 3}&{x + 4}&{x + c}\end{array}} \right| = 0$, where a, b and c are in AP.

Correct Answer True TrueFalse Q54 $|{\mathop{\rm adj}\nolimits} A| = |A{|^2}$, where $A$ is a square matrix of order two.

Correct Answer False TrueFalse Q55 The determinant$\left| {\begin{array}{llllllllllllllllllll}{\sin A}&{\cos A}&{\sin A + \cos B}\\{\sin B}&{\cos A}&{\sin B + \cos B}\\{\sin C}&{\cos A}&{\sin C + \cos B}\end{array}} \right|$ is equal to zero.

Correct Answer True TrueFalse Q56 If the determinant $\left| {\begin{array}{cccccccccccccccccccc}{x + a}&{p + u}&{l + f}\\{y + b}&{q + v}&{m + g}\\{z + c}&{r + w}&{n + h}\end{array}} \right|$ splits into exactly $k$ determinants of order 3, each element of which contains only one term, then the value of $k$ is 8.

Correct Answer True TrueFalse Q58 The maximum value of $\left| {\begin{array}{cccccccccccccccccccc}1&1&1\\1&{1 + \sin \theta }&1\\1&1&{1 + \cos \theta }\end{array}} \right|$ is $\frac{1}{2}$.

Correct Answer True TrueFalse

Q1 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&4\\{ - 5}&{ - 1}\end{array}} \right|$ SA Q2 (i) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right|$ (ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{x^2} - x + 1}&{x - 1}\\{x + 1}&{x + 1}\end{array}} \right|$ SA Q3 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2\\4&2\end{array}} \right]$, then show that $|2A| = 4|A|$ SA Q4 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&1\\0&1&2\\0&0&4\end{array}} \right],$ then show that $|3A| = 27|A|.$ SA Q5 Evaluate the determinants

• $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&{ - 1}&{ - 2}\\0&0&{ - 1}\\3&{ - 5}&0\end{array}} \right|$

• $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&{ - 4}&5\\1&1&{ - 2}\\2&3&1\end{array}} \right|$

• $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&1&2\\{ - 1}&0&{ - 3}\\{ - 2}&3&0\end{array}} \right|$

• $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 1}&{ - 2}\\0&2&{ - 1}\\3&{ - 5}&0\end{array}} \right|$

SA Q6 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&{ - 2}\\2&1&{ - 3}\\5&4&{ - 9}\end{array}} \right]$, find $|A|.$ SA Q7 Find the values of x, if

(i) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&4\\5&1\end{array}} \right| = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{2x}&4\\6&x\end{array}} \right|$

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

Q1 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&a&{x + a}\\y&b&{y + b}\\z&c&{z + c}\end{array}} \right| = 0$ SA Q2 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{a - b}&{b - c}&{c - a}\\{b - c}&{c - a}&{a - b}\\{c - a}&{a - b}&{b - c}\end{array}} \right| = 0$ SA Q3 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&7&{65}\\3&8&{75}\\5&9&{86}\end{array}} \right| = 0$ SA Q4 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{bc}&{a(b + c)}\\1&{ca}&{b(c + a)}\\1&{ab}&{c(a + b)}\end{array}} \right| = 0$ SA Q5 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{b + c}&{q + r}&{y + z}\\{c + a}&{r + p}&{z + x}\\{a + b}&{p + q}&{x + y}\end{array}} \right| = 2\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}a&p&x\\b&q&y\\c&r&z\end{array}} \right|$ SA Q6 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&a&{ - b}\\{ - a}&0&{ - c}\\b&c&0\end{array}} \right| = 0$ SA Q7 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - {a^2}}&{ab}&{ac}\\{ba}&{ - {b^2}}&{bc}\\{ca}&{cb}&{ - {c^2}}\end{array}} \right| = 4{a^2}{b^2}{c^2}$ SA Q8 (i)$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&a&{{a^2}}\\1&b&{{b^2}}\\1&c&{{c^2}}\end{array}} \right| = (a - b)(b - c)(c - a).$
(ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}} \right| = (a - b)(b - c)(c - a)(a + b + c)$ SA Q9 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&{{x^2}}&{yz}\\y&{{y^2}}&{zx}\\z&{{z^2}}&{xy}\end{array}} \right| = (x - y)(y - z)(z - x)(xy + yz + zx)$ SA Q10 (i) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{x + 4}&{2x}&{2x}\\{2x}&{x + 4}&{2x}\\{2x}&{2x}&{x + 4}\end{array}} \right| = (5x + 4){(4 - x)^2}$
(ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{y + k}&y&y\\y&{y + k}&y\\y&y&{y + k}\end{array}} \right| = {k^2}(3y + k)$ SA Q11 (i)$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{a - b - c}&{2a}&{2a}\\{2b}&{b - c - a}&{2b}\\{2c}&{2c}&{c - a - b}\end{array}} \right| = {(a + b + c)^3}$
(ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{x + y + 2z}&x&y\\z&{y + z + 2x}&y\\z&x&{z + x + 2y}\end{array}} \right| = 2{(x + y + z)^3}$ SA Q12 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x&{{x^2}}\\{{x^2}}&1&x\\x&{{x^2}}&1\end{array}} \right| = {(1 - {x^3})^2}.$ SA Q13 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{1 + {a^2} - {b^2}}&{2ab}&{ - 2b}\\{2ab}&{1 - {a^2} + {b^2}}&{2a}\\{2b}&{ - 2a}&{1 - {a^2} - {b^2}}\end{array}} \right| = {(1 + {a^2} + {b^2})^2}$ SA Q14 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{a^2} + 1}&{ab}&{ac}\\{ab}&{{b^2} + 1}&{bc}\\{ca}&{cb}&{{c^2} + 1}\end{array}} \right|$

$= 1 + {a^2} + {b^2} + {c^2}.$ SA Q15 Let A be a square matrix of order 3 × 3, then $|kA|$ equal to
(A) $k|A|$

(B) ${k^2}|A|$

(C) ${k^3}|A|$

(D) $3k|A|$ SA Q16 Which of the following is correct ?
(A) Determinant is a square matrix.

(B) Determinant is a number associated to a matrix.

(C) Determinant is a number associated to a square matrix.

(D) None of these SA

Q1 Find area of the triangle with vertices at the point given in each of the following:
(i)$\left( {1,0} \right),\left( {6,0} \right),\left( {4,3} \right)$

(ii)$\left( {2,7} \right),\left( {1,1} \right),\left( {10,8} \right)$

(iii)$\left( { - 2, - 3} \right),\left( {3,2} \right),\left( { - 1, - 8} \right)$ SA Q2 Show that points $A(a,b + c),B(b,c + a),C(c,a + b)$ are collinear. SA Q2 (i) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\0&1&0\\0&0&1\end{array}} \right|$
(ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&4\\3&5&{ - 1}\\0&1&2\end{array}} \right|$ SA Q3 Find values of k if area of triangle is 4 sq. units and vertices are
(i) $(k,0),(4,0),(0,2)$
(ii) $( - 2,0),(0,4),(0,k)$ SA Q3 Using Cofactors of elements of second row, evaluate
$\Delta = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}5&3&8\\2&0&1\\1&2&3\end{array}} \right|$ . SA Q4 Using Cofactors of elements of third column, evaluate
$\Delta = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x&{yz}\\1&y&{zx}\\1&z&{xy}\end{array}} \right|$ SA Q4 (i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants. SA Q5 If area of triangle is 35 sq. units with vertices (2,– 6), (5, 4) and (k, 4), then k is

(A) 12

(B) – 2

(C) – 12, – 2

(D) 12, – 2 SA

Q1 (i) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 4}\\0&3\end{array}} \right|$
(ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}a&c\\b&d\end{array}} \right|$ SA

Q1 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2\\3&4\end{array}} \right]$ SA Q2 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\2&3&5\\{ - 2}&0&1\end{array}} \right]$ SA Q3 . $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&3\\{ - 4}&{ - 6}\end{array}} \right]$ SA Q4 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\3&0&{ - 2}\\1&0&3\end{array}} \right]$ SA Q5 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 2}\\4&3\end{array}} \right]$ SA Q6 . $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 1}&5\\{ - 3}&2\end{array}} \right]$ SA Q7 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2&3\\0&2&4\\0&0&5\end{array}} \right]$ SA Q8 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\3&3&0\\5&2&{ - 1}\end{array}} \right]$ SA Q9 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&1&3\\4&{ - 1}&0\\{ - 7}&2&1\end{array}} \right]$ SA Q10 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\0&2&{ - 3}\\3&{ - 2}&4\end{array}} \right]$ SA Q11 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\0&{\cos \alpha }&{\sin \alpha }\\0&{\sin \alpha }&{ - \cos \alpha }\end{array}} \right]$ SA Q12 Let $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&7\\2&5\end{array}} \right]$ and $B = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}6&8\\7&9\end{array}} \right]$ . Verify that ${(AB)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}$ . SA Q13 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&1\\{ - 1}&2\end{array}} \right]$ , show that ${A^2} - 5A + 7I = O.$ Hence, find ${A^{ - 1}}.$ SA Q14 . For the matrix $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&2\\1&1\end{array}} \right]$, find the numbers a and b such that ${A^2} + aA + bI = 0.$ SA Q15 For the matrix $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&1\\1&2&{ - 3}\\2&{ - 1}&3\end{array}} \right]$ , show that ${A^3} - 6{A^2} + 5A + 11I = O.$ Hence, find ${A^{ - 1}}$ . SA Q16 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 1}&1\\{ - 1}&2&{ - 1}\\1&{ - 1}&2\end{array}} \right]$ , verify that ${A^3} - 6{A^2} + 9A - 4I = O$ and hence, find ${A^{ - 1}}.$ SA Q17 Let A be a non-singular square matrix of order 3$\times$ 3. Then, $|adj\;A|$ is equal to

(A) $|A|$

(B)$|A{|^2}$

(C) $|A{|^3}$

(D) $3|A|$ SA Q18 If A is an invertible matrix of order 2, then det $({A^{ - 1}})$ is equal to

(A) det (A)

(B) $\cfrac{1}{{\det (A)}}$

(C) 1

(D) 0 SA

Q1 $x + 2y = 2,2x + 3y = 3.$ SA Q2 $2x - y = 5,x + y = 4.$ SA Q3 $x + 3y = 5,2x + 6y = 8.$ SA Q4 $x + y + z = 1,2x + 3y + 2z = 2,ax + ay + 2az = 4.$ SA Q5 $3x - y - 2z = 2,2y - z = - 1,3x - 5y = 3.$ SA Q6 $5x - y + 4z = 5,2x + 3y + 5z = 2,5x - 2y + 6z = - 1.$ SA Q7 $5x + 2y = 4,7x + 3y = 5.$ SA Q8 $2x - y = - 2,3x + 4y = 3.$ SA Q9 $4x - 3y = 3,3x - 5y = 7.$ SA Q10 . $5x + 2y = 3,3x + 2y = 5.$ SA Q11 $2x + y + z = 1,x - 2y - z = 3/2,3y - 5z = 9.$ SA Q12 . $x - y + z = 4,2x + y - 3z = 0,x + y + z = 2.$ SA Q13 $2x + 3y + 3z = 5,x - 2y + z = - 4,3x - y - 2z = 3.$ SA Q14 $x - y + 2z = 7,3x + 4y - 5z = - 5,2x - y + 3z = 12$ SA Q15 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 3}&5\\3&2&{ - 4}\\1&1&{ - 2}\end{array}} \right]$ , find ${A^{ - 1}}.$Using ${A^{ - 1}},$ solve the system of equations
$2x - 3y + 5z = 11,3x + 2y - 4z = - 5,x + y - 2z = - 3.$ SA Q16 The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs.60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find cost of the each item per kg by matrix method. SA

Q1 Prove that the determinant $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&{\sin \theta }&{\cos \theta }\\{ - \sin \theta }&{ - x}&1\\{\cos \theta }&1&x\end{array}} \right|$ is independent of $\theta$. SA Q2 Without expanding the determinant, prove that
$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}a&{{a^2}}&{bc}\\b&{{b^2}}&{ca}\\c&{{c^2}}&{ab}\end{array}} \right| = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{{a^2}}&{{a^3}}\\1&{{b^2}}&{{b^3}}\\1&{{c^2}}&{{c^3}}\end{array}} \right|$ SA Q3 Evaluate $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{\cos \alpha \cos \beta }&{\cos \alpha \sin \beta }&{ - \sin \alpha }\\{ - \sin \beta }&{\cos \beta }&0\\{\sin \alpha \cos \beta }&{\sin \alpha \sin \beta }&{\cos \alpha }\end{array}} \right|$. SA Q4 If a, b and c are real numbers, and $\Delta = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{b + c}&{c + a}&{a + b}\\{c + a}&{a + b}&{b + c}\\{a + b}&{b + c}&{c + a}\end{array}} \right| = 0$, show that either $a + b + c = 0$or $a = b = c.$ SA Q5 Solve the equation$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{x + a}&x&x\\x&{x + a}&x\\x&x&{x + a}\end{array}} \right| = 0,a \ne 0$ SA Q6 Prove that $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{a^2}}&{bc}&{ac + {c^2}}\\{{a^2} + bc}&{{b^2}}&{ac}\\{ab}&{{b^2} + bc}&{{c^2}}\end{array}} \right| = 4{a^2}{b^2}{c^2}.$ SA Q7 If ${A^{ - 1}} = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&{ - 1}&1\\{ - 15}&6&{ - 5}\\5&{ - 2}&2\end{array}} \right]$ and $B = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2&{ - 2}\\{ - 1}&3&0\\0&{ - 2}&1\end{array}} \right]$ find ${(AB)^{ - 1}}.$ SA Q8 Let $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 2}&1\\{ - 2}&3&1\\1&1&5\end{array}} \right]$ . Verify that,

(i) ${[adj\;A]^{ - 1}} = adj({A^{ - 1}})$

(ii) ${({A^{ - 1}})^{ - 1}} = A$ SA Q9 Evaluate$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&y&{x + y}\\y&{x + y}&x\\{x + y}&x&y\end{array}} \right|$ SA Q10 Evaluate $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x&y\\1&{x + y}&y\\1&x&{x + y}\end{array}} \right|$ SA Q11 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}\alpha &{{\alpha ^2}}&{\beta + \gamma }\\\beta &{{\beta ^2}}&{\gamma + \alpha }\\\gamma &{{\gamma ^2}}&{\alpha + \beta }\end{array}} \right| = (\beta - \gamma )(\gamma - \alpha )(\alpha - \beta )(\alpha + \beta + \gamma )$ SA Q12 . $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&{{x^2}}&{1 + p{x^3}}\\y&{{y^2}}&{1 + p{y^3}}\\z&{{z^2}}&{1 + p{z^3}}\end{array}} \right| = (1 + pxyz)(x - y)(y - z)(z - x)$ SA Q13 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{3a}&{ - a + b}&{ - a + c}\\{ - b + a}&{3b}&{ - b + c}\\{ - c + a}&{ - c + b}&{3c}\end{array}} \right| = 3(a + b + c)(ab + bc + ca)$ SA Q14 $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{1 + p}&{1 + p + q}\\2&{3 + 2p}&{4 + 3p + 2q}\\3&{6 + 3p}&{10 + 6p + 3q}\end{array}} \right| = 1$ SA Q15 . $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{\sin \alpha }&{\cos \alpha }&{\cos (\alpha + \delta )}\\{\sin \beta }&{\cos \beta }&{\cos (\beta + \delta )}\\{\sin \gamma }&{\cos \gamma }&{\cos (\gamma + \delta )}\end{array}} \right| = 0$ SA Q16 Solve the system of the following equations
$\cfrac{2}{x} + \cfrac{3}{y} + \cfrac{{10}}{z} = 4,\cfrac{4}{x} - \cfrac{6}{y} + \cfrac{5}{z} = 1,\cfrac{6}{x} + \cfrac{9}{y} - \cfrac{{20}}{z} = 2$ SA Q17 If a, b, c, are in A.P, then the determinant $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{x + 2}&{x + 3}&{x + 2a}\\{x + 3}&{x + 4}&{x + 2b}\\{x + 4}&{x + 5}&{x + 2c}\end{array}} \right|$ is

(A) 0

(B) 1

(C) x

(D) 2x SA Q18 If$x,y,z$ are non-zero real numbers, then the inverse of matrix $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&0&0\\0&y&0\\0&0&z\end{array}} \right]$ is

(A)$\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{x^{ - 1}}}&0&0\\0&{{y^{ - 1}}}&0\\0&0&{{z^{ - 1}}}\end{array}} \right]$

(B) $xyz\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{x^{ - 1}}}&0&0\\0&{{y^{ - 1}}}&0\\0&0&{{z^{ - 1}}}\end{array}} \right]$

(C) $\cfrac{1}{{xyz}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&0&0\\0&y&0\\0&0&z\end{array}} \right]$

(D)$\cfrac{1}{{xyz}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\0&1&0\\0&0&1\end{array}} \right]$ SA Q19 Let $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{\sin \theta }&1\\{ - \sin \theta }&1&{\sin \theta }\\{ - 1}&{ - \sin \theta }&1\end{array}} \right],$ where $0 \le \theta \le 2\pi .$ Then,

(A) $Det(A) = 0$

(B) $Det(A) \in (2,\infty )$

(C) $Det(A) \in (2,4)$

(D) $Det(A) \in [2,4]$ SA