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# RD Sharma solutions for Class 12 Mathematics chapter 6 - Determinants

## Chapter 6: Determinants

Ex. 6.1Ex. 6.2Ex. 6.3Ex. 6.4Ex. 6.5Others

#### Chapter 6: Determinants Exercise 6.1 solutions [Pages 10 - 11]

Ex. 6.1 | Q 1.1 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}5 & 20 \\ 0 & - 1\end{bmatrix}$

Ex. 6.1 | Q 1.2 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}- 1 & 4 \\ 2 & 3\end{bmatrix}$

Ex. 6.1 | Q 1.3 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{bmatrix}$

Ex. 6.1 | Q 1.4 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{bmatrix}$

Ex. 6.1 | Q 1.5 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1\end{bmatrix}$

Ex. 6.1 | Q 1.6 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}a & h & g \\ h & b & f \\ g & f & c\end{bmatrix}$

Ex. 6.1 | Q 1.7 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}2 & - 1 & 0 & 1 \\ - 3 & 0 & 1 & - 2 \\ 1 & 1 & - 1 & 1 \\ 2 & - 1 & 5 & 0\end{bmatrix}$

Ex. 6.1 | Q 2.1 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}$

Ex. 6.1 | Q 2.2 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}$

Ex. 6.1 | Q 2.3 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}$

Ex. 6.1 | Q 2.4 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}$

Ex. 6.1 | Q 3 | Page 10

Evaluate

$\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 .$

Ex. 6.1 | Q 4 | Page 10

Show that

$\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1$

Ex. 6.1 | Q 5 | Page 10

Evaluate

$\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}$ by two methods.

Ex. 6.1 | Q 6 | Page 10

Evaluate
$∆ = \begin{vmatrix}0 & \sin \alpha & - \cos \alpha \\ - \sin \alpha & 0 & \sin \beta \\ \cos \alpha & - \sin \beta & 0\end{vmatrix}$

Ex. 6.1 | Q 7 | Page 10

$∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}$

Ex. 6.1 | Q 8 | Page 10

If $A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}$ , verify that |AB| = |A| |B|.

Ex. 6.1 | Q 9 | Page 10

If A $\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}$ , then show that |3 A| = 27 |A|.

Ex. 6.1 | Q 10.1 | Page 10

Find the value of x, if
$\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}$

Ex. 6.1 | Q 10.2 | Page 10

Find the value of x, if

$\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = \begin{vmatrix}x & 3 \\ 2x & 5\end{vmatrix}$

Ex. 6.1 | Q 10.3 | Page 10

Find the value of x, if

$\begin{vmatrix}3 & x \\ x & 1\end{vmatrix} = \begin{vmatrix}3 & 2 \\ 4 & 1\end{vmatrix}$

Ex. 6.1 | Q 10.4 | Page 10

Find the value of x, if

$\begin{vmatrix}3x & 7 \\ 2 & 4\end{vmatrix} = 10$ , find the value of x.

Ex. 6.1 | Q 10.5 | Page 10

Find the value of x, if

$\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}$

Ex. 6.1 | Q 10.6 | Page 10

Find the value of x, if

$\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}$

Ex. 6.1 | Q 11 | Page 11

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

Ex. 6.1 | Q 12.1 | Page 11

For what value of x the matrix A is singular?
$A = \begin{bmatrix}1 + x & 7 \\ 3 - x & 8\end{bmatrix}$

Ex. 6.1 | Q 12.2 | Page 11

For what value of x the matrix A is singular?

$A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}$

#### Chapter 6: Determinants Exercise 6.2 solutions [Pages 57 - 62]

Ex. 6.2 | Q 1.1 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}$

Ex. 6.2 | Q 1.2 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{vmatrix}$

Ex. 6.2 | Q 1.3 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}$

Ex. 6.2 | Q 1.4 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{vmatrix}$

Ex. 6.2 | Q 1.5 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25\end{vmatrix}$

Ex. 6.2 | Q 1.6 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}$

Ex. 6.2 | Q 1.7 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}$

Ex. 6.2 | Q 2.01 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}$

Ex. 6.2 | Q 2.02 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}$

Ex. 6.2 | Q 2.03 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}$

Ex. 6.2 | Q 2.04 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}$

Ex. 6.2 | Q 2.05 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}a + b & 2a + b & 3a + b \\ 2a + b & 3a + b & 4a + b \\ 4a + b & 5a + b & 6a + b\end{vmatrix}$

Ex. 6.2 | Q 2.06 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}$

Ex. 6.2 | Q 2.07 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}$

Ex. 6.2 | Q 2.08 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}$

Ex. 6.2 | Q 2.09 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}$

Ex. 6.2 | Q 2.1 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}$

Ex. 6.2 | Q 2.11 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}$

Ex. 6.2 | Q 2.12 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}$

Ex. 6.2 | Q 2.13 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sin\alpha & \cos\alpha & \cos(\alpha + \delta) \\ \sin\beta & \cos\beta & \cos(\beta + \delta) \\ \sin\gamma & \cos\gamma & \cos(\gamma + \delta)\end{vmatrix}$

Ex. 6.2 | Q 2.14 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sin^2 23^\circ & \sin^2 67^\circ & \cos180^\circ \\ - \sin^2 67^\circ & - \sin^2 23^\circ & \cos^2 180^\circ \\ \cos180^\circ & \sin^2 23^\circ & \sin^2 67^\circ\end{vmatrix}$

Ex. 6.2 | Q 2.15 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}$

Ex. 6.2 | Q 2.16 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}$

Ex. 6.2 | Q 2.17 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .$

Ex. 6.2 | Q 3 | Page 58

Evaluate :

$\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}$

Ex. 6.2 | Q 4 | Page 58

Evaluate :

$\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix}$

Ex. 6.2 | Q 5 | Page 58

Evaluate :

$\begin{vmatrix}x + \lambda & x & x \\ x & x + \lambda & x \\ x & x & x + \lambda\end{vmatrix}$

Ex. 6.2 | Q 6 | Page 58

Evaluate :

$\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}$

Ex. 6.2 | Q 7 | Page 58

Evaluate the following:

$\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}$

Ex. 6.2 | Q 8 | Page 58

Evaluate the following:

$\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}$

Ex. 6.2 | Q 9 | Page 58

Evaluate the following:

$\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}$

Ex. 6.2 | Q 10 | Page 58

$If ∆ = \begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}, ∆_1 = \begin{vmatrix}1 & 1 & 1 \\ yz & zx & xy \\ x & y & z\end{vmatrix},\text{ then prove that }∆ + ∆_1 = 0 .$

Ex. 6.2 | Q 11 | Page 58

Prove that :

$\begin{vmatrix}a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b\end{vmatrix} = a^3 + b^3 + c^3 - 3abc$

Ex. 6.2 | Q 12 | Page 58

Prove that :

$\begin{vmatrix}b + c & a - b & a \\ c + a & b - c & b \\ a + b & c - a & c\end{vmatrix} = 3abc - a^3 - b - c^3$

Ex. 6.2 | Q 13 | Page 58

Prove that :

$\begin{vmatrix}a + b & b + c & c + a \\ b + c & c + a & a + b \\ c + a & a + b & b + c\end{vmatrix} = 2\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$

Ex. 6.2 | Q 14 | Page 58

Prove that :

$\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2 \left( a + b + c \right)^3$

Ex. 6.2 | Q 15 | Page 59

Prove that :

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

Ex. 6.2 | Q 16 | Page 59

Prove that :

$\begin{vmatrix}1 & b + c & b^2 + c^2 \\ 1 & c + a & c^2 + a^2 \\ 1 & a + b & a^2 + b^2\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right)$

Ex. 6.2 | Q 17 | Page 59

Prove that :

$\begin{vmatrix}a & a + b & a + 2b \\ a + 2b & a & a + b \\ a + b & a + 2b & a\end{vmatrix} = 9 \left( a + b \right) b^2$

Ex. 6.2 | Q 18 | Page 59

Prove that :

$\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix} = \begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix}$

Ex. 6.2 | Q 19 | Page 59

Prove that :

$\begin{vmatrix}z & x & y \\ z^2 & x^2 & y^2 \\ z^4 & x^4 & y^4\end{vmatrix} = \begin{vmatrix}x & y & z \\ x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4\end{vmatrix} = \begin{vmatrix}x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4 \\ x & y & z\end{vmatrix} = xyz \left( x - y \right) \left( y - z \right) \left( z - x \right) \left( x + y + z \right) .$

Ex. 6.2 | Q 20 | Page 59

Prove that :

$\begin{vmatrix}\left( b + c \right)^2 & a^2 & bc \\ \left( c + a \right)^2 & b^2 & ca \\ \left( a + b \right)^2 & c^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)$

Ex. 6.2 | Q 21 | Page 59

Prove that :

$\begin{vmatrix}\left( a + 1 \right) \left( a + 2 \right) & a + 2 & 1 \\ \left( a + 2 \right) \left( a + 3 \right) & a + 3 & 1 \\ \left( a + 3 \right) \left( a + 4 \right) & a + 4 & 1\end{vmatrix} = - 2$

Ex. 6.2 | Q 22 | Page 59

Prove that :

$\begin{vmatrix}a^2 & a^2 - \left( b - c \right)^2 & bc \\ b^2 & b^2 - \left( c - a \right)^2 & ca \\ c^2 & c^2 - \left( a - b \right)^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)$

Ex. 6.2 | Q 23 | Page 59

Prove that :

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

Ex. 6.2 | Q 24 | Page 59

Prove that :

$\begin{vmatrix}a^2 & bc & ac + c^2 \\ a^2 + ab & b^2 & ac \\ ab & b^2 + bc & c^2\end{vmatrix} = 4 a^2 b^2 c^2$
Ex. 6.2 | Q 25 | Page 59

Prove that :

$\begin{vmatrix}x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4\end{vmatrix} = 16 \left( 3x + 4 \right)$
Ex. 6.2 | Q 26 | Page 59

Prove that :

$\begin{vmatrix}1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 4 + 3p + 2q \\ 3 & 6 + 3p & 10 + 6p + 3q\end{vmatrix} = 1$

Ex. 6.2 | Q 27 | Page 59

Prove that :

$\begin{vmatrix}a & b - c & c - b \\ a - c & b & c - a \\ a - b & b - a & c\end{vmatrix} = \left( a + b - c \right) \left( b + c - a \right) \left( c + a - b \right)$

Ex. 6.2 | Q 28 | Page 60

Prove that

$\begin{vmatrix}a^2 & 2ab & b^2 \\ b^2 & a^2 & 2ab \\ 2ab & b^2 & a^2\end{vmatrix} = \left( a^3 + b^3 \right)^2$
Ex. 6.2 | Q 29 | Page 60

Prove that

$\begin{vmatrix}a^2 + 1 & ab & ac \\ ab & b^2 + 1 & bc \\ ca & cb & c^2 + 1\end{vmatrix} = 1 + a^2 + b^2 + c^2$
Ex. 6.2 | Q 30 | Page 60
$\begin{vmatrix}1 & a & a^2 \\ a^2 & 1 & a \\ a & a^2 & 1\end{vmatrix} = \left( a^3 - 1 \right)^2$
Ex. 6.2 | Q 31 | Page 60
$\begin{vmatrix}a + b + c & - c & - b \\ - c & a + b + c & - a \\ - b & - a & a + b + c\end{vmatrix} = 2\left( a + b \right) \left( b + c \right) \left( c + a \right)$
Ex. 6.2 | Q 32 | Page 60

$\begin{vmatrix}b + c & a & a \\ b & c + a & b \\ c & c & a + b\end{vmatrix} = 4abc$

Ex. 6.2 | Q 33 | Page 60

$\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2$

Ex. 6.2 | Q 34 | Page 60

$\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3$

Ex. 6.2 | Q 35 | Page 60

Prove that

$\begin{vmatrix}\frac{a^2 + b^2}{c} & c & c \\ a & \frac{b^2 + c^2}{a} & a \\ b & b & \frac{c^2 + a^2}{b}\end{vmatrix} = 4abc$

Ex. 6.2 | Q 36 | Page 58

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

Ex. 6.2 | Q 37 | Page 60

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

Ex. 6.2 | Q 38 | Page 60

Using properties of determinants prove that

$\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2$

Ex. 6.2 | Q 39 | Page 60

Prove the following identities:

$\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz$

Ex. 6.2 | Q 40 | Page 61

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

Ex. 6.2 | Q 41 | Page 61

$\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2$

Ex. 6.2 | Q 42 | Page 61

Prove the following identity:

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

Ex. 6.2 | Q 43 | Page 61

Show that  $\begin{vmatrix}y + z & x & y \\ z + x & z & x \\ x + y & y & z\end{vmatrix} = \left( x + y + z \right) \left( x - z \right)^2$

Ex. 6.2 | Q 44 | Page 61

Prove the following identity:

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

Ex. 6.2 | Q 45 | Page 61

Prove the following identity:

|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)

Ex. 6.2 | Q 46 | Page 61

Without expanding, prove that

$\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}$

Ex. 6.2 | Q 47 | Page 61

Show that

$\begin{vmatrix}x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c\end{vmatrix} =\text{ 0 where a, b, c are in A . P .}$

Ex. 6.2 | Q 48 | Page 61
Show that
|(x-3,x-4,x-alpha),(x-2,x-3,x-beta),(x-1,x-2,x-gamma)|=0, where α, β, γ are in A.P.

Ex. 6.2 | Q 49 | Page 61

If a, b, c are real numbers such that
$\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0$ , then show that either
$a + b + c = 0 \text{ or, } a = b = c$

Ex. 6.2 | Q 50 | Page 61
$If \begin{vmatrix}p & b & c \\ a & q & c \\ a & b & r\end{vmatrix} = 0,\text{ find the value of }\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}, p \neq a, q \neq b, r \neq c .$

Ex. 6.2 | Q 51 | Page 61

Show that x = 2 is a root of the equation

$\begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 & 2x & x + 2\end{vmatrix} = 0$  and solve it completely.

Ex. 6.2 | Q 52.1 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}x + a & b & c \\ a & x + b & c \\ a & b & x + c\end{vmatrix} = 0$

Ex. 6.2 | Q 52.2 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}x + a & x & x \\ x & x + a & x \\ x & x & x + a\end{vmatrix} = 0, a \neq 0$

Ex. 6.2 | Q 52.3 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix} = 0$

Ex. 6.2 | Q 52.4 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}1 & x & x^2 \\ 1 & a & a^2 \\ 1 & b & b^2\end{vmatrix} = 0, a \neq b$

Ex. 6.2 | Q 52.5 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix} = 0$

Ex. 6.2 | Q 52.6 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}1 & x & x^3 \\ 1 & b & b^3 \\ 1 & c & c^3\end{vmatrix} = 0, b \neq c$

Ex. 6.2 | Q 52.7 | Page 61
​Solve the following determinant equation:
$\begin{vmatrix}15 - 2x & 11 - 3x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13\end{vmatrix} = 0$
Ex. 6.2 | Q 52.8 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2\end{vmatrix} = 0$
Ex. 6.2 | Q 52.9 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}3 & - 2 & \sin\left( 3\theta \right) \\ - 7 & 8 & \cos\left( 2\theta \right) \\ - 11 & 14 & 2\end{vmatrix} = 0$

Ex. 6.2 | Q 53 | Page 62

If $a, b$ and c  are all non-zero and

$\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c\end{vmatrix} =$ 0, then prove that
$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} +$1
= 0

Ex. 6.2 | Q 54 | Page 62

If $\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =$ 0, then using properties of determinants, find the value of  $\frac{a}{x} + \frac{b}{y} + \frac{c}{z}$  , where $x, y, z \neq$ 0

#### Chapter 6: Determinants Exercise 6.3 solutions [Pages 71 - 72]

Ex. 6.3 | Q 1.1 | Page 71

Find the area of the triangle with vertice at the point:

(3, 8), (−4, 2) and (5, −1)

Ex. 6.3 | Q 1.2 | Page 71

Find the area of the triangle with vertice at the point:

(2, 7), (1, 1) and (10, 8)

Ex. 6.3 | Q 1.3 | Page 71

Find the area of the triangle with vertice at the point:

(−1, −8), (−2, −3) and (3, 2)

Ex. 6.3 | Q 1.4 | Page 71

Find the area of the triangle with vertice at the point:

(0, 0), (6, 0) and (4, 3)

Ex. 6.3 | Q 2.1 | Page 71

Using determinants show that the following points are collinear:

(5, 5), (−5, 1) and (10, 7)

Ex. 6.3 | Q 2.2 | Page 71

Using determinants show that the following points are collinear:

(1, −1), (2, 1) and (4, 5)

Ex. 6.3 | Q 2.3 | Page 71

Using determinants show that the following points are collinear:

(3, −2), (8, 8) and (5, 2)

Ex. 6.3 | Q 2.4 | Page 71

Using determinants show that the following points are collinear:

(2, 3), (−1, −2) and (5, 8)

Ex. 6.3 | Q 3 | Page 71

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

Ex. 6.3 | Q 4 | Page 71

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

Ex. 6.3 | Q 5 | Page 71

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

Ex. 6.3 | Q 6 | Page 71

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

Ex. 6.3 | Q 7 | Page 71

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

Ex. 6.3 | Q 8 | Page 71

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

Ex. 6.3 | Q 9 | Page 71

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.

Ex. 6.3 | Q 10 | Page 71

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

Ex. 6.3 | Q 11 | Page 72

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

Ex. 6.3 | Q 12.1 | Page 72

Using determinants, find the equation of the line joining the points

(1, 2) and (3, 6)

Ex. 6.3 | Q 12.2 | Page 72

Using determinants, find the equation of the line joining the points

(3, 1) and (9, 3)

Ex. 6.3 | Q 13.1 | Page 72

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

Ex. 6.3 | Q 13.2 | Page 72

Find values of k, if area of triangle is 4 square units whose vertices are

(−2, 0), (0, 4), (0, k)

#### Chapter 6: Determinants Exercise 6.4 solutions [Pages 84 - 85]

Ex. 6.4 | Q 1 | Page 84

x − 2y = 4
−3x + 5y = −7

Ex. 6.4 | Q 2 | Page 84

2x − y = 1
7x − 2y = −7

Ex. 6.4 | Q 3 | Page 84

2x − y = 17
3x + 5y = 6

Ex. 6.4 | Q 4 | Page 84

3x + y = 19
3x − y = 23

Ex. 6.4 | Q 5 | Page 84

2x − y = − 2
3x + 4y = 3

Ex. 6.4 | Q 6 | Page 84

3x + ay = 4
2x + ay = 2, a ≠ 0

Ex. 6.4 | Q 7 | Page 84

2x + 3y = 10
x + 6y = 4

Ex. 6.4 | Q 8 | Page 84

5x + 7y = − 2
4x + 6y = − 3

Ex. 6.4 | Q 9 | Page 84

9x + 5y = 10
3y − 2x = 8

Ex. 6.4 | Q 10 | Page 84

Given: x + 2y = 1
3x + y = 4

Ex. 6.4 | Q 11 | Page 84

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

Ex. 6.4 | Q 12 | Page 84

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

Ex. 6.4 | Q 13 | Page 84

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

Ex. 6.4 | Q 14 | Page 84

xy = 5
y + z = 3
x + z = 4

Ex. 6.4 | Q 15 | Page 84

2y − 3z = 0
x + 3y = − 4
3x + 4y = 3

Ex. 6.4 | Q 16 | Page 84

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

Ex. 6.4 | Q 17 | Page 84

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

Ex. 6.4 | Q 18 | Page 84

x + y = 1
x + z = − 6
x − y − 2z = 3

Ex. 6.4 | Q 19 | Page 84

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

Ex. 6.4 | Q 20 | Page 84

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

Ex. 6.4 | Q 21 | Page 84

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

Ex. 6.4 | Q 22 | Page 84

2x − y = 5
4x − 2y = 7

Ex. 6.4 | Q 23 | Page 84

3x + y = 5
− 6x − 2y = 9

Ex. 6.4 | Q 24 | Page 84

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

Ex. 6.4 | Q 25 | Page 84

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

Ex. 6.4 | Q 26 | Page 85

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

Ex. 6.4 | Q 27 | Page 85

x + 2y = 5
3x + 6y = 15

Ex. 6.4 | Q 28 | Page 85

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

Ex. 6.4 | Q 29 | Page 85

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

Ex. 6.4 | Q 30 | Page 85

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

Ex. 6.4 | Q 31 | Page 85

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 commissiondrawn (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.

Ex. 6.4 | Q 32 | Page 85

An automobile company uses three types of steel S1S2 and S3 for producing three types of cars C1C2and C3. Steel requirements (in tons) for each type of cars are given below :

 CarsC1 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.

#### Chapter 6: Determinants Exercise 6.5 solutions [Page 89]

Ex. 6.5 | Q 1 | Page 89

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

Ex. 6.5 | Q 2 | Page 89

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

Ex. 6.5 | Q 3 | Page 89

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

Ex. 6.5 | Q 4 | Page 89

Find the real values of λ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions
$2 \lambda x - 2y + 3z = 0$
$x + \lambda y + 2z = 0$
$2x + \lambda z = 0$

Ex. 6.5 | Q 5 | Page 89

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.

#### Chapter 6: Determinants solutions [Pages 90 - 95]

Q 1 | Page 90

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

Q 2 | Page 90

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

$\begin{bmatrix}5 - x & x + 1 \\ 2 & 4\end{bmatrix}$

Q 3 | Page 90

Write the value of the determinant
$\begin{bmatrix}2 & 3 & 4 \\ 2x & 3x & 4x \\ 5 & 6 & 8\end{bmatrix} .$

Q 4 | Page 90

State whether the matrix
$\begin{bmatrix}2 & 3 \\ 6 & 4\end{bmatrix}$ is singular or non-singular.

Q 5 | Page 90

Find the value of the determinant
$\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}$

Q 6 | Page 90

Find the value of the determinant
$\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}$

Q 7 | Page 90

Write the value of the determinant

$\begin{vmatrix}a & 1 & b + c \\ b & 1 & c + a \\ c & 1 & a + b\end{vmatrix} .$

Q 8 | Page 90

If $A = \begin{bmatrix}0 & i \\ i & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ , find the value of |A| + |B|.

Q 9 | Page 90

If $A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}$ , find |AB|.

Q 10 | Page 90

Evaluate $\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}$

Q 11 | Page 90

If w is an imaginary cube root of unity, find the value of $\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}$

Q 12 | Page 90

If $A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|$

Q 13 | Page 90

If $A = \left[ a_{ij} \right]$   is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.

Q 14 | Page 90

If A = [aij] is a 3 × 3 scalar matrix such that a11 = 2, then write the value of |A|.

Q 15 | Page 90

If I3 denotes identity matrix of order 3 × 3, write the value of its determinant.

Q 16 | Page 90

A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?

Q 17 | Page 90

On expanding by first row, the value of the determinant of 3 × 3 square matrix
$A = \left[ a_{ij} \right]\text{ is }a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}$ , where [Cij] is the cofactor of aij in A. Write the expression for its value on expanding by second column.

Q 18 | Page 90

Let A = [aij] be a square matrix of order 3 × 3 and Cij denote cofactor of aij in A. If |A| = 5, write the value of a31 C31  +  a32 C32 a33 C33.

Q 19 | Page 90

In question 18, write the value of a11 C21 + a12 C22 + a13 C23.

Q 20 | Page 90

Write the value of

$\begin{vmatrix}\sin 20^\circ & - \cos 20^\circ\\ \sin 70^\circ& \cos 70^\circ\end{vmatrix}$
Q 21 | Page 90

If A is a square matrix satisfying AT A = I, write the value of |A|.

Q 22 | Page 91

If A and B are square matrices of the same order such that |A| = 3 and AB = I, then write the value of |B|.

Q 23 | Page 91

A is a skew-symmetric of order 3, write the value of |A|.

Q 24 | Page 91

If A is a square matrix of order 3 with determinant 4, then write the value of |−A|.

Q 25 | Page 91

If A is a square matrix such that |A| = 2, write the value of |A AT|.

Q 26 | Page 91

Find the value of the determinant $\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .$

Q 27 | Page 91

Write the value of the determinant $\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .$

Q 28 | Page 91

If the matrix $\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}$  is singular, find the value of x.

Q 29 | Page 91

If A is a square matrix of order n × n such that  $|A| = \lambda$ , then write the value of |−A|.

Q 30 | Page 91

Find the value of the determinant $\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}$.

Q 31 | Page 91

If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.

Q 32 | Page 91

A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.

Q 33 | Page 91

If A and B are square matrices of order 3 such that |A| = − 1, |B| = 3, then find the value of |3 AB|.

Q 34 | Page 91

Write the value of  $\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .$

Q 35 | Page 91

Write the cofactor of a12 in the following matrix $\begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix} .$

Q 36 | Page 91

If $\begin{vmatrix}2x + 5 & 3 \\ 5x + 2 & 9\end{vmatrix} = 0$

Q 37 | Page 91

Find the value of x from the following : $\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0$

Q 38 | Page 91

Write the value of the determinant $\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}$

Q 39 | Page 91

If |A| = 2, where A is 2 × 2 matrix, find |adj A|.

Q 40 | Page 91

What is the value of the determinant $\begin{vmatrix}0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{vmatrix} ?$

Q 41 | Page 91

For what value of x is the matrix  $\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}$  singular?

Q 42 | Page 91

A matrix A of order 3 × 3 is such that |A| = 4. Find the value of |2 A|.

Q 43 | Page 92

Evaluate: $\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}$

Q 44 | Page 92

If $A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}$. Write the cofactor of the element a32.

Q 45 | Page 92
If $\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}$, then write the value of x.
Q 46 | Page 92

If $\begin{vmatrix}2x & x + 3 \\ 2\left( x + 1 \right) & x + 1\end{vmatrix} = \begin{vmatrix}1 & 5 \\ 3 & 3\end{vmatrix}$, then write the value of x.

Q 47 | Page 92

If $\begin{vmatrix}3x & 7 \\ - 2 & 4\end{vmatrix} = \begin{vmatrix}8 & 7 \\ 6 & 4\end{vmatrix}$ , find the value of x.

Q 48 | Page 92

If $\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}$ , write the value of x.

Q 49 | Page 92

If A is a 3 × 3 matrix, $\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|$  then write the value of k.

Q 50 | Page 92

Write the value of the determinant $\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix}$

Q 51 | Page 92

Write the value of the determinant $\begin{vmatrix}x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix}$

Q 52 | Page 92

If $A = \begin{bmatrix}\cos\theta & \sin\theta \\ - \sin\theta & \cos\theta\end{bmatrix}$ , then for any natural number, find the value of Det(An).

Q 53 | Page 95

Find the maximum value of $\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}$

Q 54 | Page 95

If x ∈ N and $\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}$  = 8, then find the value of x.

Q 55 | Page 95

If $\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8$ , write the value of x.

Q 56 | Page 95

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k.

#### Chapter 6: Determinants solutions [Pages 93 - 96]

Q 1 | Page 93

If A and B are square matrices of order 2, then det (A + B) = 0 is possible only when

• det (A) = 0 or det (B) = 0

• det (A) + det (B) = 0

• det (A) = 0 and det (B) = 0

•  A + B = O

Q 2 | Page 93

Which of the following is not correct?

• $|A| = | A^T |,\text{ where }A = \left[ a_{ij} \right]_{3 \times 3}$

• $|kA| = | k^3 |,\text{ where }A = \left[ a_{ij} \right]_{3 \times 3}$

• If A is a skew-symmetric matrix of odd order, then |A| = 0

• $\begin{vmatrix}a + b & c + d \\ e + f & g + h\end{vmatrix} = \begin{vmatrix}a & c \\ e & g\end{vmatrix} + \begin{vmatrix}b & d \\ f & h\end{vmatrix}$

Q 3 | Page 93

If $A = \begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}$  and Cij is cofactor of aij in A, then value of |A| is given

• a11 C31 + a12 C32 + a13 C33

• a11 C11 + a12 C21 + a13 C31

• a21 C11 + a22 C12 + a23 C13

•  a11 C11 + a21 C21 + a13 C31

Q 4 | Page 93

Which of the following is not correct in a given determinant of A, where A = [aij]3×3.

• Order of minor is less than order of the det (A)

• Minor of an element can never be equal to cofactor of the same element

• Value of determinant is obtained by multiplying elements of a row or column by  corresponding cofactors

• Order of minors and cofactors of elements of A is same

Q 5 | Page 93

Let $\begin{vmatrix}x & 2 & x \\ x^2 & x & 6 \\ x & x & 6\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e$
Then, the value of $5a + 4b + 3c + 2d + e$ is equal to

• 0

• - 16

• 16

• none of these

Q 6 | Page 93

The value of the determinant

$\begin{vmatrix}a^2 & a & 1 \\ \cos nx & \cos \left( n + 1 \right) x & \cos \left( n + 2 \right) x \\ \sin nx & \sin \left( n + 1 \right) x & \sin \left( n + 2 \right) x\end{vmatrix}\text{ is independent of}$

• n

• a

• x

• none of these

Q 7 | Page 93

If  $∆_1 = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{vmatrix}, ∆_2 = \begin{vmatrix}1 & bc & a \\ 1 & ca & b \\ 1 & ab & c\end{vmatrix},\text{ then }$}

• $∆_1 + ∆_2 = 0$

• $∆_1 + 2 ∆_2 = 0$

• $∆_1 = ∆_2$

• none of these

Q 8 | Page 93
If $D_k = \begin{vmatrix}1 & n & n \\ 2k & n^2 + n + 2 & n^2 + n \\ 2k - 1 & n^2 & n^2 + n + 2\end{vmatrix} and \sum^n_{k = 1} D_k = 48$, then n equals

• 4

• 6

• 8

•  none of these

Q 9 | Page 93

Let $\begin{vmatrix}x^2 + 3x & x - 1 & x + 3 \\ x + 1 & - 2x & x - 4 \\ x - 3 & x + 4 & 3x\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e$
be an identity in x, where abcde are independent of x. Then the value of e is

• 4

• 0

• 1

•  none of these

Q 10 | Page 94

Using the factor theorem it is found that a + bb + c and c + a are three factors of the determinant

$\begin{vmatrix}- 2a & a + b & a + c \\ b + a & - 2b & b + c \\ c + a & c + b & - 2c\end{vmatrix}$
The other factor in the value of the determinant is

• 4

• 2

• a + b + c

• none of these

Q 11 | Page 94

If a, b, c are distinct, then the value of x satisfying $\begin{vmatrix}0 & x^2 - a & x^3 - b \\ x^2 + a & 0 & x^2 + c \\ x^4 + b & x - c & 0\end{vmatrix} = 0\text{ is }$

• c

• a

•  b

•  0

Q 12 | Page 94

If the determinant $\begin{vmatrix}0 & x^2 - a & x^3 - b \\ x^2 + a & 0 & x^2 + c \\ x^4 + b & x - c & 0\end{vmatrix} = 0 \text{ is }$

• a, b, c are in H . P

• $\alpha\text{ is a root of 4a} x^2 + 12bx + 9c = 0\text{ or a, b, c are in G . P .}$

• a, b, c are in G . P . only

• a, b, c are in A . P .

Q 13 | Page 94

If ω is a non-real cube root of unity and n is not a multiple of 3, then  $∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}$

• 0

• ω

• ω2

• 1

Q 14 | Page 94

If $A_r = \begin{vmatrix}1 & r & 2^r \\ 2 & n & n^2 \\ n & \frac{n \left( n + 1 \right)}{2} & 2^{n + 1}\end{vmatrix}$ , then the value of $\sum^n_{r = 1} A_r$ is

• n

• 2n

• − 2n

•  n2

• None of these

Q 15 | Page 94

If a > 0 and discriminant of ax2 + 2bx + c is negative, then
$∆ = \begin{vmatrix}a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0\end{vmatrix} is$

• positive

• $\left( ac - b^2 \right) \left( a x^2 + 2bx + c \right)$

• negative

• 0

Q 16 | Page 94

The value of $\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}$

• 52

• 0

• 513

• 59

Q 17 | Page 94

$\begin{vmatrix}\log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9\end{vmatrix} \times \begin{vmatrix}\log_2 3 & \log_8 3 \\ \log_3 4 & \log_3 4\end{vmatrix}$

• 7

• 10

• 1

• 17

Q 18 | Page 94

If a, b, c are in A.P., then the determinant
$\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}$

• 0

• 1

• x

• 2x

Q 19 | Page 95

If $A + B + C = \pi$, then the value of $\begin{vmatrix}\sin \left( A + B + C \right) & \sin \left( A + C \right) & \cos C \\ - \sin B & 0 & \tan A \\ \cos \left( A + B \right) & \tan \left( B + C \right) & 0\end{vmatrix}$  is equal to

• 0

• 1

• 2 sin B tan A cos C

• none of these

Q 20 | Page 95

The number of distinct real roots of $\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0$  lies in the interval
$- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}$

• 1

• 2

• 3

• 0

Q 21 | Page 95

Let $A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}$

• $Det \left( A \right) = 0$

• $Det \left( A \right) \in \left( 2, \infty \right)$

• $Det \left( A \right) \in \left( 2, 4 \right)$

• $Det \left( A \right) \in \left[ 2, 4 \right]$

Q 22 | Page 95

If $\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}$ , then x =

•  3

• ± 3

• ± 6

• 6

Q 23 | Page 95

If$f\left( x \right) = \begin{vmatrix}0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0\end{vmatrix}$

• f(a) = 0

• f(b) = 0

• f(0) = 0

• f(1) = 0

Q 24 | Page 95

The value of the determinant

$\begin{vmatrix}a - b & b + c & a \\ b - c & c + a & b \\ c - a & a + b & c\end{vmatrix}$

• $a^3 + b^3 + c^3$

• 3bc

• $a^3 + b^3 + c^3 - 3abc$

• none of these

Q 25 | Page 95

If xyare different from zero and $\begin{vmatrix}1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z\end{vmatrix} = 0$ , then the value of x−1 + y−1 + z−1 is

• xyz

•  x−1 y−1 z−1

• − x − y − z

• − 1

Q 26 | Page 95

The determinant  $\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}$

• $abc\left( b - c \right)\left( c - a \right)\left( a - b \right)$

• $\left( b - c \right)\left( c - a \right)\left( a - b \right)$

• $\left( a + b + c \right)\left( b - c \right)\left( c - a \right)\left( a - b \right)$

• none of these

Q 27 | Page 95

If $x, y \in \mathbb{R}$, then the determinant

$∆ = \begin{vmatrix}\cos x & - \sin x & 1 \\ \sin x & \cos x & 1 \\ \cos\left( x + y \right) & - \sin\left( x + y \right) & 0\end{vmatrix}$

• $\left[ - \sqrt{2}, \sqrt{2} \right]$

• $\left[ - 1, 1 \right]$

• $\left[ - \sqrt{2}, 1 \right]$

• $\left[ - 1, - \sqrt{2} \right]$

Q 28 | Page 96

The maximum value of  $∆ = \begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1\end{vmatrix}$ is (θ is real)

• 1/2

• sqrt3/2

• sqrt2

• -sqrt3/2

Q 29 | Page 96

The value of the determinant $\begin{vmatrix}x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x\end{vmatrix}$ is

• 9x2(x + y)

• 9y2(x + y)

• 3y2(x + y)

• 7x2(x + y)

Q 30 | Page 96

Let $f\left( x \right) = \begin{vmatrix}\cos x & x & 1 \\ 2\sin x & x & 2x \\ \sin x & x & x\end{vmatrix}$ $\lim_{x \to 0} \frac{f\left( x \right)}{x^2}$  is equal to

• 0

• -1

• 2

• 3

Q 31 | Page 96

There are two values of a which makes the determinant  $∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix}$  equal to 86. The sum of these two values is

• 4

• 5

• - 4

• 9

Q 32 | Page 96

If $\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16$ , then the value of $\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}$ is

• 4

• 8

• 16

• 32

Q 33 | Page 96

The value of $\begin{vmatrix}1 & 1 & 1 \\ {}^n C_1 & {}^{n + 2} C_1 & {}^{n + 4} C_1 \\ {}^n C_2 & {}^{n + 2} C_2 & {}^{n + 4} C_2\end{vmatrix}$ is

• 2

• 4

• 8

•  n2

## Chapter 6: Determinants

Ex. 6.1Ex. 6.2Ex. 6.3Ex. 6.4Ex. 6.5Others

## RD Sharma solutions for Class 12 Mathematics chapter 6 - Determinants

RD Sharma solutions for Class 12 Maths chapter 6 (Determinants) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 6 Determinants are Applications of Determinants and Matrices, Elementary Transformations, Adjoint and Inverse of a Matrix, Properties of Determinants, Determinant of a Square Matrix, Determinants of Matrix of Order One and Two, Determinant of a Matrix of Order 3 × 3, Rule A=KB, Introduction of Determinant, Area of a Triangle, Minors and Co-factors.

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