QBManager

Matrix and Determinant

79041 If $z_{1} = 2 + 3 i, z_{2} = 3 + 2 i$ where $i = \sqrt{- 1}$ then $[\begin{array}{cc} z_{1} & z_{2} \\ {\overset{―}{z}}_{2} & {\overset{―}{Z}}_{1} \end{array}] [\begin{array}{cc} {\overset{―}{Z}}_{1} & - z_{2} \\ {\overset{―}{z}}_{2} & z_{1} \end{array}] =$

1

13 I

2 I

3

26 I

4 Zero matrix

Explanation:

(C) :
$[\begin{array}{cc} z_{1} & z_{2} \\ {\bar{z}}_{2} & {\bar{z}}_{1} \end{array}] [\begin{array}{cc} {\bar{z}}_{1} & - z_{2} \\ {\bar{z}}_{2} & z_{1} \end{array}] = [\begin{array}{cc} 2 + 3 i & 3 + 2 i \\ - 3 + 2 i & 2 - 3 i \end{array}] [\begin{array}{cc} 2 - 3 i & - 3 - 2 i \\ 3 - 2 i & 2 + 3 i \end{array}]$
$(2 + 3 i) (2 - 3 i) + (3 + 2 i) (3 - 2 i) (2 + 3 i) (- 3 - 2 i) + (3 + 2 i) (2 + 3 i)$
$(- 3 + 2 i) (2 - 3 i) + (2 - 3 i) (3 - 2 i) (- 3 + 2 i) (- 3 - 2 i) + (2 - 3 i) (2 + 3 i)$
$= | \begin{array}{cc} 13 - 13 i^{2} & 0 \\ 0 & 13 - 13 i^{2} \end{array} |$
$= | \begin{array}{cc} 26 & 0 \\ 0 & 26 \end{array} | = 26 | \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} | = 26 I$

Shift-II]

Matrix and Determinant

79042 What is the value of $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} | =$ ?

1

a^{3} + b^{3} + c^{3} + 3 abc

2

a^{3} + b^{3} + c^{3} - 3 a b c

3

a^{3} + b^{3} + c^{3} - 6 a b c

4

a^{3} + b^{3} + c^{3} + 6 a b c

Explanation:

(B) $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} |$
$R_{2} \to R_{2} - R_{1}$
$= | \begin{array}{ccc} a & b & c \\ a - b - a & b - c - b & c - a - c \\ b + c & c + a & a + b \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ b + c & c + a & a + b \end{array} |$
$R_{3} \to R_{3} + R_{2}$
$= | \begin{array}{lcc} a & b & c \\ - b & - c & - a \\ b + c - b & c + a - c & a + b - c \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ c & a & b \end{array} |$
$= a (- bc + a^{2}) - b (- b^{2} + ac) + c (- ab + c^{2})$
$= - abc + a^{3} + b^{3} - a b c - a b c + c^{3}$
$= a^{3} + b^{3} + c^{3} - 3 abc$

Shift-II]

Matrix and Determinant

79043 Let $a \in$ and $A = [\begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array}]$ be a matrix.
Then equation $\det A = 16$

1 a parabola

2 a circle

3 an ellipse

4 a rectangular hyperbola

Explanation:

(D): $| \begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array} | = 16$
$= a | \begin{array}{ccc} 1 & a & a - y \\ 1 & a + x & a \\ 1 & a & a \end{array} | = 16 \Rightarrow$
$= a^{3} | \begin{array}{ccc} 1 & 1 & 1 - y / a \\ 1 & 1 + x / a & 1 \\ 1 & 1 & 1 \end{array} | = 16$
$= a^{3} [(1 + x / a - 1) - 1 (1 - 1) + (1 - y / a) (1 - 1 - x / a)] = 16$
$= a^{3} [x / a - x / a + x y / a^{2}] = 16$
$= x y = \frac{16}{a}$
A rectangular hyperbola

Shift-II]

Matrix and Determinant

79046 The number of positive roots of the equation
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$ is :

1 1

2 2

3 3

4 0

Explanation:

(B) : The number of positive roots
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$
$x (x^{2} - 12) - 3 (2 x - 14) + 7 (12 - 7 x) = 0$
$x^{3} - 12 x - 6 x + 42 + 84 - 49 x = 0$
$x^{3} - 67 x + 126 = 0$
$(x - 2) (x^{2} + 2 x - 63) = 0$
$(x - 2) (x - 7) (x + 9) = 0$
$n = 2, 7, - 9$
There are two positive roots.

Matrix and Determinant

79041 If $z_{1} = 2 + 3 i, z_{2} = 3 + 2 i$ where $i = \sqrt{- 1}$ then $[\begin{array}{cc} z_{1} & z_{2} \\ {\overset{―}{z}}_{2} & {\overset{―}{Z}}_{1} \end{array}] [\begin{array}{cc} {\overset{―}{Z}}_{1} & - z_{2} \\ {\overset{―}{z}}_{2} & z_{1} \end{array}] =$

1

13 I

2 I

3

26 I

4 Zero matrix

Explanation:

(C) :
$[\begin{array}{cc} z_{1} & z_{2} \\ {\bar{z}}_{2} & {\bar{z}}_{1} \end{array}] [\begin{array}{cc} {\bar{z}}_{1} & - z_{2} \\ {\bar{z}}_{2} & z_{1} \end{array}] = [\begin{array}{cc} 2 + 3 i & 3 + 2 i \\ - 3 + 2 i & 2 - 3 i \end{array}] [\begin{array}{cc} 2 - 3 i & - 3 - 2 i \\ 3 - 2 i & 2 + 3 i \end{array}]$
$(2 + 3 i) (2 - 3 i) + (3 + 2 i) (3 - 2 i) (2 + 3 i) (- 3 - 2 i) + (3 + 2 i) (2 + 3 i)$
$(- 3 + 2 i) (2 - 3 i) + (2 - 3 i) (3 - 2 i) (- 3 + 2 i) (- 3 - 2 i) + (2 - 3 i) (2 + 3 i)$
$= | \begin{array}{cc} 13 - 13 i^{2} & 0 \\ 0 & 13 - 13 i^{2} \end{array} |$
$= | \begin{array}{cc} 26 & 0 \\ 0 & 26 \end{array} | = 26 | \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} | = 26 I$

Shift-II]

Matrix and Determinant

79042 What is the value of $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} | =$ ?

1

a^{3} + b^{3} + c^{3} + 3 abc

2

a^{3} + b^{3} + c^{3} - 3 a b c

3

a^{3} + b^{3} + c^{3} - 6 a b c

4

a^{3} + b^{3} + c^{3} + 6 a b c

Explanation:

(B) $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} |$
$R_{2} \to R_{2} - R_{1}$
$= | \begin{array}{ccc} a & b & c \\ a - b - a & b - c - b & c - a - c \\ b + c & c + a & a + b \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ b + c & c + a & a + b \end{array} |$
$R_{3} \to R_{3} + R_{2}$
$= | \begin{array}{lcc} a & b & c \\ - b & - c & - a \\ b + c - b & c + a - c & a + b - c \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ c & a & b \end{array} |$
$= a (- bc + a^{2}) - b (- b^{2} + ac) + c (- ab + c^{2})$
$= - abc + a^{3} + b^{3} - a b c - a b c + c^{3}$
$= a^{3} + b^{3} + c^{3} - 3 abc$

Shift-II]

Matrix and Determinant

79043 Let $a \in$ and $A = [\begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array}]$ be a matrix.
Then equation $\det A = 16$

1 a parabola

2 a circle

3 an ellipse

4 a rectangular hyperbola

Explanation:

(D): $| \begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array} | = 16$
$= a | \begin{array}{ccc} 1 & a & a - y \\ 1 & a + x & a \\ 1 & a & a \end{array} | = 16 \Rightarrow$
$= a^{3} | \begin{array}{ccc} 1 & 1 & 1 - y / a \\ 1 & 1 + x / a & 1 \\ 1 & 1 & 1 \end{array} | = 16$
$= a^{3} [(1 + x / a - 1) - 1 (1 - 1) + (1 - y / a) (1 - 1 - x / a)] = 16$
$= a^{3} [x / a - x / a + x y / a^{2}] = 16$
$= x y = \frac{16}{a}$
A rectangular hyperbola

Shift-II]

Matrix and Determinant

79046 The number of positive roots of the equation
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$ is :

1 1

2 2

3 3

4 0

Explanation:

(B) : The number of positive roots
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$
$x (x^{2} - 12) - 3 (2 x - 14) + 7 (12 - 7 x) = 0$
$x^{3} - 12 x - 6 x + 42 + 84 - 49 x = 0$
$x^{3} - 67 x + 126 = 0$
$(x - 2) (x^{2} + 2 x - 63) = 0$
$(x - 2) (x - 7) (x + 9) = 0$
$n = 2, 7, - 9$
There are two positive roots.

Matrix and Determinant

79041 If $z_{1} = 2 + 3 i, z_{2} = 3 + 2 i$ where $i = \sqrt{- 1}$ then $[\begin{array}{cc} z_{1} & z_{2} \\ {\overset{―}{z}}_{2} & {\overset{―}{Z}}_{1} \end{array}] [\begin{array}{cc} {\overset{―}{Z}}_{1} & - z_{2} \\ {\overset{―}{z}}_{2} & z_{1} \end{array}] =$

1

13 I

2 I

3

26 I

4 Zero matrix

Explanation:

(C) :
$[\begin{array}{cc} z_{1} & z_{2} \\ {\bar{z}}_{2} & {\bar{z}}_{1} \end{array}] [\begin{array}{cc} {\bar{z}}_{1} & - z_{2} \\ {\bar{z}}_{2} & z_{1} \end{array}] = [\begin{array}{cc} 2 + 3 i & 3 + 2 i \\ - 3 + 2 i & 2 - 3 i \end{array}] [\begin{array}{cc} 2 - 3 i & - 3 - 2 i \\ 3 - 2 i & 2 + 3 i \end{array}]$
$(2 + 3 i) (2 - 3 i) + (3 + 2 i) (3 - 2 i) (2 + 3 i) (- 3 - 2 i) + (3 + 2 i) (2 + 3 i)$
$(- 3 + 2 i) (2 - 3 i) + (2 - 3 i) (3 - 2 i) (- 3 + 2 i) (- 3 - 2 i) + (2 - 3 i) (2 + 3 i)$
$= | \begin{array}{cc} 13 - 13 i^{2} & 0 \\ 0 & 13 - 13 i^{2} \end{array} |$
$= | \begin{array}{cc} 26 & 0 \\ 0 & 26 \end{array} | = 26 | \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} | = 26 I$

Shift-II]

Matrix and Determinant

79042 What is the value of $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} | =$ ?

1

a^{3} + b^{3} + c^{3} + 3 abc

2

a^{3} + b^{3} + c^{3} - 3 a b c

3

a^{3} + b^{3} + c^{3} - 6 a b c

4

a^{3} + b^{3} + c^{3} + 6 a b c

Explanation:

(B) $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} |$
$R_{2} \to R_{2} - R_{1}$
$= | \begin{array}{ccc} a & b & c \\ a - b - a & b - c - b & c - a - c \\ b + c & c + a & a + b \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ b + c & c + a & a + b \end{array} |$
$R_{3} \to R_{3} + R_{2}$
$= | \begin{array}{lcc} a & b & c \\ - b & - c & - a \\ b + c - b & c + a - c & a + b - c \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ c & a & b \end{array} |$
$= a (- bc + a^{2}) - b (- b^{2} + ac) + c (- ab + c^{2})$
$= - abc + a^{3} + b^{3} - a b c - a b c + c^{3}$
$= a^{3} + b^{3} + c^{3} - 3 abc$

Shift-II]

Matrix and Determinant

79043 Let $a \in$ and $A = [\begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array}]$ be a matrix.
Then equation $\det A = 16$

1 a parabola

2 a circle

3 an ellipse

4 a rectangular hyperbola

Explanation:

(D): $| \begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array} | = 16$
$= a | \begin{array}{ccc} 1 & a & a - y \\ 1 & a + x & a \\ 1 & a & a \end{array} | = 16 \Rightarrow$
$= a^{3} | \begin{array}{ccc} 1 & 1 & 1 - y / a \\ 1 & 1 + x / a & 1 \\ 1 & 1 & 1 \end{array} | = 16$
$= a^{3} [(1 + x / a - 1) - 1 (1 - 1) + (1 - y / a) (1 - 1 - x / a)] = 16$
$= a^{3} [x / a - x / a + x y / a^{2}] = 16$
$= x y = \frac{16}{a}$
A rectangular hyperbola

Shift-II]

Matrix and Determinant

79046 The number of positive roots of the equation
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$ is :

1 1

2 2

3 3

4 0

Explanation:

(B) : The number of positive roots
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$
$x (x^{2} - 12) - 3 (2 x - 14) + 7 (12 - 7 x) = 0$
$x^{3} - 12 x - 6 x + 42 + 84 - 49 x = 0$
$x^{3} - 67 x + 126 = 0$
$(x - 2) (x^{2} + 2 x - 63) = 0$
$(x - 2) (x - 7) (x + 9) = 0$
$n = 2, 7, - 9$
There are two positive roots.

Matrix and Determinant

79041 If $z_{1} = 2 + 3 i, z_{2} = 3 + 2 i$ where $i = \sqrt{- 1}$ then $[\begin{array}{cc} z_{1} & z_{2} \\ {\overset{―}{z}}_{2} & {\overset{―}{Z}}_{1} \end{array}] [\begin{array}{cc} {\overset{―}{Z}}_{1} & - z_{2} \\ {\overset{―}{z}}_{2} & z_{1} \end{array}] =$

1

13 I

2 I

3

26 I

4 Zero matrix

Explanation:

(C) :
$[\begin{array}{cc} z_{1} & z_{2} \\ {\bar{z}}_{2} & {\bar{z}}_{1} \end{array}] [\begin{array}{cc} {\bar{z}}_{1} & - z_{2} \\ {\bar{z}}_{2} & z_{1} \end{array}] = [\begin{array}{cc} 2 + 3 i & 3 + 2 i \\ - 3 + 2 i & 2 - 3 i \end{array}] [\begin{array}{cc} 2 - 3 i & - 3 - 2 i \\ 3 - 2 i & 2 + 3 i \end{array}]$
$(2 + 3 i) (2 - 3 i) + (3 + 2 i) (3 - 2 i) (2 + 3 i) (- 3 - 2 i) + (3 + 2 i) (2 + 3 i)$
$(- 3 + 2 i) (2 - 3 i) + (2 - 3 i) (3 - 2 i) (- 3 + 2 i) (- 3 - 2 i) + (2 - 3 i) (2 + 3 i)$
$= | \begin{array}{cc} 13 - 13 i^{2} & 0 \\ 0 & 13 - 13 i^{2} \end{array} |$
$= | \begin{array}{cc} 26 & 0 \\ 0 & 26 \end{array} | = 26 | \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} | = 26 I$

Shift-II]

Matrix and Determinant

79042 What is the value of $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} | =$ ?

1

a^{3} + b^{3} + c^{3} + 3 abc

2

a^{3} + b^{3} + c^{3} - 3 a b c

3

a^{3} + b^{3} + c^{3} - 6 a b c

4

a^{3} + b^{3} + c^{3} + 6 a b c

Explanation:

(B) $| \begin{array}{ccc} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{array} |$
$R_{2} \to R_{2} - R_{1}$
$= | \begin{array}{ccc} a & b & c \\ a - b - a & b - c - b & c - a - c \\ b + c & c + a & a + b \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ b + c & c + a & a + b \end{array} |$
$R_{3} \to R_{3} + R_{2}$
$= | \begin{array}{lcc} a & b & c \\ - b & - c & - a \\ b + c - b & c + a - c & a + b - c \end{array} | = | \begin{array}{ccc} a & b & c \\ - b & - c & - a \\ c & a & b \end{array} |$
$= a (- bc + a^{2}) - b (- b^{2} + ac) + c (- ab + c^{2})$
$= - abc + a^{3} + b^{3} - a b c - a b c + c^{3}$
$= a^{3} + b^{3} + c^{3} - 3 abc$

Shift-II]

Matrix and Determinant

79043 Let $a \in$ and $A = [\begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array}]$ be a matrix.
Then equation $\det A = 16$

1 a parabola

2 a circle

3 an ellipse

4 a rectangular hyperbola

Explanation:

(D): $| \begin{array}{ccc} a & a & a - y \\ a & a + x & a \\ a & a & a \end{array} | = 16$
$= a | \begin{array}{ccc} 1 & a & a - y \\ 1 & a + x & a \\ 1 & a & a \end{array} | = 16 \Rightarrow$
$= a^{3} | \begin{array}{ccc} 1 & 1 & 1 - y / a \\ 1 & 1 + x / a & 1 \\ 1 & 1 & 1 \end{array} | = 16$
$= a^{3} [(1 + x / a - 1) - 1 (1 - 1) + (1 - y / a) (1 - 1 - x / a)] = 16$
$= a^{3} [x / a - x / a + x y / a^{2}] = 16$
$= x y = \frac{16}{a}$
A rectangular hyperbola

Shift-II]

Matrix and Determinant

79046 The number of positive roots of the equation
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$ is :

1 1

2 2

3 3

4 0

Explanation:

(B) : The number of positive roots
$| \begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} | = 0$
$x (x^{2} - 12) - 3 (2 x - 14) + 7 (12 - 7 x) = 0$
$x^{3} - 12 x - 6 x + 42 + 84 - 49 x = 0$
$x^{3} - 67 x + 126 = 0$
$(x - 2) (x^{2} + 2 x - 63) = 0$
$(x - 2) (x - 7) (x + 9) = 0$
$n = 2, 7, - 9$
There are two positive roots.

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