# CBSE Class 12 Maths –Chapter 4 Determinants- Study Materials

### Determinants Notes Class 12 Maths Chapter 4

1. DETERMINANT OF A SQUARE MATRIX

(i) If A = $$\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right]$$, then det. A = $$\left|\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right|$$ = a11a22</sub – a21a12</sub

(ii) If A = $$\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$$, then det. A = a11$$\left|\begin{array}{ll} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array}\right|$$ – a12 $$\left|\begin{array}{ll} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array}\right|$$ + a13 $$\left|\begin{array}{ll} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right|$$
= a11a22a33 – a23a32a11 – a12a21a33 + a12a23a11 + a13a21a32 – a13a31a22.

2. MINOR AND CO-FACTOR
(i) The minor of an element aij is a determinant, which is obtained by supressing die ith row and jth column. The minor of an element aij is denoted by Mij.

(ii) The co-factor of an element is its minor with proper sign. The co-factor of an element aij is denoted by Aij
Aij =(-1)i+jMij

3. PROPERTIES

(i) Reflection Property. The value of the determinant remains unaltered by interchanging its rows and columns.
(ii) Switching Property. If two adjacent rows (or columns) of a determinant are interchanged, then the sign of the determinant is changed.
(iii) Repetition Property. If two rows (or columns) of a determinant are identical, then its value is zero.
(iv) Scalar Multiple Property. If each element of a row (or column) of a determinant is multiplied f
by a constant ‘k’ then its value gets multiplied by the scalar ‘k’
(v) Sum Property. If each element of a row (or column) of a determinant is expressed as the sum
of two or more terms, then the determinant can be expressed as the sum of two or more determinants.
(vi) Invariance Property. If to any row (or column) of a determinant, a multiple of another row (or column) is added, the value of the determinant remains the same.
(vii) Factor Property. If a determinant Δ vanishes when for x is put a in those elements of Δ, which are polynomials in x, then (x – a) is a factor of Δ.

4. AREA OF A TRIANGLE

Area of a triangle whose vertices are (x1, y1), (x2, y2), (x3, y3) is given by:
D = $$\frac{1}{2}\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|$$
When the area of the triangle is zero, then the points are collinear.

Let A = $$\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$$, then adj, A = $$\left[\begin{array}{lll} \mathrm{A}_{11} & \mathrm{~A}_{21} & \mathrm{~A}_{31} \\ \mathrm{~A}_{12} & \mathrm{~A}_{22} & \mathrm{~A}_{32} \\ \mathrm{~A}_{13} & \mathrm{~A}_{23} & \mathrm{~A}_{33} \end{array}\right]$$, where capital letters are co-factors of corresponding small letters.

6. INVERSE OF A MATRIX

Invertible Matrix. Any n-rowed square matrix A is said to be invertible if there exists an n-rowed matrix B such that
AB = BA = In
B is called the inverse of A and is denoted as A-1.

Theorems.
(i) Inverse of every square matrix, if it exists, is unique.
(ii) A is invertible iff |A| ≠ 0
(iii) A-1 = $$\frac{\operatorname{adj} . \mathrm{A}}{|\mathrm{A}|}$$, if | A | ≠ 0.

PROPERTIES:

(i) (AB)-1 =B-1 A-1
(ii) (A’)-1 = (A-1)’
(iii) (Ak)-1 =(A-1)k, where k is any positive integer.

7. SINGULAR AND NON-SINGULAR MATRICES
A square matrix is said to be singular if |A| = 0 and non-singular if |A| ≠ 0.

8. SOLUTIONS OF EQUATIONS BY MATRIX METHOD To solve the equations :
$$\begin{array}{l} a_{11} x_{1}+a_{12} x_{2}+\ldots \ldots+a_{1 n} x_{n}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}+\ldots \ldots+a_{2 n} x_{n}=b_{2} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\ a_{m 1} x_{1}+a_{m 2} x_{2}+\ldots \ldots+a_{m n} x_{n}=b_{m} . \end{array}$$
Here X = A-1B,
where A = $$\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots \ldots \ldots & a_{1 n} \\ a_{21} & a_{22} & \ldots \ldots \ldots & a_{2 n} \\ \ldots & \ldots \ldots \ldots \ldots & \\ a_{m 1} & a_{m 2} \ldots \ldots \ldots . & a_{m n} \end{array}\right]$$, X = $$\left[\begin{array}{c} x_{1} \\ x_{2} \\ \cdots \\ x_{n} \end{array}\right]$$, B = $$\left[\begin{array}{c} b_{1} \\ b_{2} \\ \ldots \\ b_{m} \end{array}\right]$$

(i) If |A| ≠ 0, then the system is consistent and has a unique solution.
(ii) If | A | = 0 and (adj. A) B = O, (O being a zero matrix) then the system is consistent and has infinitely many solutions.
(iii) If | A | = 0 and (adj. A) B ≠ O, then the system is inconsistent and has no solution.

9. SOLUTION OF HOMOGENEOUS EQUATIONS
To solve the equations :
a1x + b1y + c1z = 0
a2x + b2y + c1z = 0
a3x + b3y + c3z = 0.

Here AX = 0, where A = $$\left[\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right]$$ and X = $$\left[\begin{array}{l} x \\ y \\ z \end{array}\right]$$

(i) If |A| ≠ 0, then system has only trivial solution.
(ii) If |A| = 0, the system has infinitely many solutions.

### CBSE Revision Notes for CBSE Class 12 Mathematics Determinants

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

System of algebraic equations can be expressed in the form of matrices.
• Linear Equations Format
a1x+b1y=c1
a2x+b2y=c2
• Matrix Format:

The values of the variables satisfying all the linear equations in the system, is called solution of system of linear equations.
If the system of linear equations has a unique solution. This unique solution is called determinant of Solution or det A
Applications of Determinants

👉 Engineering
👉 Science
👉 Economics
👉 Social Science, etc.

Determinant
A determinant is defined as a (mapping) unction from the set o square matrices to the set of real numbers
Every square matrix A is associated with a number, called its determinant
Denoted by det (A) or |A| or ∆

Only square matrices have determinants.
The matrices which are not square do not have determinants
For matrix A, |A| is read as determinant of A and not modulus of A.
Types of Determinant

1. First Order Determinant

Let A = [a ] be the matrix of order 1, then determinant of A is defined to be equal to a
If A = [a], then det (A) = |A| = a
🔷 2. Second Order Determinant

🔷 3. Third Order Determinant
Can be determined by expressing it in terms of second order determinants

The below method is explained for expansion around Row 1
The value of the determinant, thus will be the sum of the product of element in line parallel to the diagonal minus the sum of the product of elements in line perpendicular to the line segment. Thus,

The same procedure can be repeated for Row 2, Row 3, Column 1, Column 2, and Column 3
🔷 Note
Expanding a determinant along any row or column gives same value.
This method doesn’t work for determinants of order greater than 3.
For easier calculations, we shall expand the determinant along that row or column which contains maximum number of zeros
In general, if A = kB where A and B are square matrices of order n, then | A| = kⁿ |B |, where n = 1, 2, 3
Properties of Determinants
Helps in simplifying its evaluation by obtaining maximum number of zeros in a row or a column.
These properties are true for determinants of any order.

Property 1
The value of the determinant remains unchanged if its rows and columns are interchanged
Verification:

Expanding ∆₁ along first column, we get
∆₁ =a₁ (b₂ c₃ – c₂ b₃) – a₂(b₁ c₃ – b₃ c₁) + a₃ (b₁ c₂ – b₂ c₁)
Hence ∆ = ∆₁
🔷 Note:
It follows from above property that if A is a square matrix,
Then det (A) = det (A’), where A’ = transpose of A
If Ri = ith row and Ci = ith column, then for interchange of row and
columns, we will symbolically write Ci⇔Ri
Property 2
If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
Verification :

Property 3
If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.
Verification:

If we interchange the identical rows (or columns) of the determinant ∆, then ∆ does not change.
However, by Property 2, it follows that ∆ has changed its sign
Therefore ∆ = -∆ or ∆ = 0
Property 4
If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k

Verification

Property 5

If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

Verification:

Property 6

If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation

Property 7

If each element of a row (or column) of a determinant is zero, then its value is zero

Property 8

In a determinant, If all the elements on one side of the principal diagonal are Zero’s , then the value of the determinant is equal to the product of the elements in the principal diagonal

Area of a Triangle

Let (x₁,y₁), (X₂, y₂), and (x₃, y₃) be the vertices of a triangle, then

Note

Area is a positive quantity, we always take the absolute value of the determinant .
If area is given, use both positive and negative values of the determinant for caleulation.
The area of the triangle formed by three collinear points is zero.

Minors and Cofactors

Minor

If the row and column containing the element a₁₁ (i.e., 1st row and 1st column)are removed, we get the second order determinant which is called the Minor of element a₁₁
Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column which element aij lies.

Minor of an element aij is denoted by Mij
Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order n-1

Eg: Find Minor o the element 6 in the determinant A given

Cofactor

If the minors are multiplied by the proper signs we get cofactors
The cofactor of the element aij is Cij = (-1) Mij
The signs to be multiplied are given by the rule

Cofactor of 4 is A₁₂ =(-1) M₁₂ =(-1)³(4) =-4

Adjoint and Inverse of a Matrix

Adjoint of matrix is the transpose of the matrix of cofactors of the given matrix

Theorem 1
If A be any given square matrix of order n,
Where I is the identity matrix of order n
Verification:

Similarly, we can show (adj A) A = AI

Singular & No Singular Matrix:

A square matrix A is said to be singular if |A| = o
A square matrix A is said to be non-singular if |A | 0

Theorem 2

If A and B are non-singular matrices of the same order, then AB and BA are also non- singular matrices of the same order.

Theorem 3

The determinant of the product of matrices is equal to product of their respective determinants, that is, AB =|A| |B| , where A and B are square matrices of the same order

Theorem 4

A square matrix A is invertible if and only if A is non-singular matrix.

Verification

Let A be invertible matrix of order n and I be the identity matrix of order n. Then, there exists a square matrix B of order n such that AB = BA = I

Now AB = I. So |AB| = I or |A| |B| = 1 (since |I|= 1, |AB|=| A||B|). This gives |A|  0. Hence A is non-singular.

Conversely, let A be non-singular. Then |A| ≠ 0
Now A (adj A) = (adj A) A = |A| I (Theorem 1)

Applications of Determinants and Matrices

Used for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear equations.

🔷 Consistent system

A system of equations is said to be consistent if its solution (one or more) exists.

🔷 Inconsistent system

A system of equations is said to be inconsistent if its solution does not exist

Solution of system of linear equations using inverse of a matrix

Let the system of Equations be as below:

a₁x+b₁y +c₁z=d₁
a₂x +b₂y +c₂z=d₂
a₃x+b₃y+c₃z=d₃

Case I

If A is a non-singular matrix, then its inverse exists.

AX = B
A⁻¹(AX) = A⁻¹B (premultiplying by A⁻¹)
(A⁻¹A)X -A⁻¹B (by associative property)
1X = A⁻¹B
X = A⁻¹B

This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method

Case II

If A is a singular matrix, then |A| = 0.
In this case, we calculate (adj A) B.
If (adj A) B  O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent.
If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution

Summary

For a square matrix A in matrix equation AX = B

|A|  0, there exists unique solution
|A| = 0 and (adj A) B  0, then there exists no solution
|A| o and (adj A) B = 0, then system may or may not be consistent.

1 Mark Questions

### Determinants Class 12 MCQs Questions with Answers

Question 1.
$$\left[\begin{array}{ccc} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right]$$
(a) (x – y) (y + z)(z + x)
(b) (x + y) (y – z)(z – x)
(c) (x – y) (y – z)(z + x)
(d) (x – y) (y – z) (z – x)

Answer: (d) (x – y) (y – z) (z – x)

Question 2.
The value of the determinant
$$\left[\begin{array}{ccc} 3 & 1 & 7 \\ 5 & 0 & 2 \\ 2 & 5 & 3 \end{array}\right]$$
(a) 124
(b) 125
(c) 134
(d) 144

Question 3.
If a, b, c are in A.P. then the determinant
$$\left[\begin{array}{ccc} x+2 & x+3 & x+2a \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c \end{array}\right]$$
(a) 1
(b) x
(c) 0
(d) 2x

Question 4.
If w is a non-real root of the equation x² – 1 = 0. then
$$\left[\begin{array}{ccc} 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \\ ω^{2} & 1 & ω \end{array}\right]$$ =
(a) 0
(b) 1
(c) ω
(d) ω²

Question 5.
If Δ = $$\left[\begin{array}{cc} 10 & 2 \\ 30 & 6 \end{array}\right]$$ then A =
(a) 0
(b) 10
(c) 12
(d) 60

Question 6.
If 7 and 2 are two roots of the equation $$\left[\begin{array}{ccc} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right]$$ then the third root is
(a) -9
(b) 14
(c) $$\frac{1}{2}$$
(d) None of these

Question 7.
If $$\left[\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right]$$ = $$\left[\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right]$$ x is equal to
(a) 6
(b) ±6
(c) -1
(d) -6

Question 8.
$$\left[\begin{array}{ccc} 1 & a & a^{2}-bc \\ 1 & b & b^{2}-ca \\ 1 & c & c^{2}-ab \end{array}\right]$$ is equal to
(a) abc
(b) ab + bc + ca
(c) 0
(d) (a – b)(b – c)(c – a)

Question 9.
A = $$\left[\begin{array}{ll} \alpha & q \\ q & \alpha \end{array}\right]$$ |A³| = 125 then α =
(a) ±3
(b) ±2
(c) ±5
(d) 0

Question 10.
If a ≠ 0 and $$\left[\begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+a & 1 \\ 1 & 1 & 1+a \end{array}\right]$$ = 0 then a =
(a) a = -3
(b) a = 0
(c) a = 1
(d) a = 3

Question 11.
If x > 0 and x ≠ 1. y > 0. and y ≠ 1, z > 0 and z ≠ 1 then
$$\left[\begin{array}{ccc} 1 & log_{x}y & log_{x}z \\ log_{y}x & 1 & log_{y}z \\ log_{z}x & log_{z}y & 1 \end{array}\right]$$ is equal to
(a) 1
(b) -1
(c) 0
(d) None of these

Question 12.
$$\left[\begin{array}{ccc} y+z & z & x \\ y & z+x & y \\ z & z & x+y \end{array}\right]$$ is equal to
(a) 6xyz
(b) xyz
(c) 4xyz
(d) xy + yz + zx

Question 13.
If $$\left[\begin{array}{cc} 2 & 4 \\ 5 & 1 \end{array}\right]$$ = $$\left[\begin{array}{cc} 2x & 4 \\ 6 & x \end{array}\right]$$ then the value of x is
(a) ±2
(b) ±$$\frac{1}{3}$$
(c) ±√3
(d) ± (0.5)

Question 14.
If $$\left[\begin{array}{cc} 2x & 5 \\ 8 & x \end{array}\right]$$ = $$\left[\begin{array}{cc} 6 & -2 \\ 7 & 3 \end{array}\right]$$ then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6

Question 15.
The value of determinant $$\left[\begin{array}{ccc} a-b & b+c & a \\ b-c & c+a & b \\ c-a & a+b & c \end{array}\right]$$
(a) a³ + b³ + c ³
(b) 3bc
(c) a³ + b³ + c³ – 3abc
(d) None of these

Answer: (c) a³ + b³ + c³ – 3abc

Question 16.
The area of a triangle with vertices (-3, 0) (3, 0) and (0, k) is 9 sq. units. The value of k will be
(a) 9
(b) 3
(c) -9
(d) 6

Question 17.
The determinant $$\left[\begin{array}{ccc} b^{2}-ab & b-c & bc-ac \\ ab-a^{2} & a-b & b^{2}-ab \\ bc-ac & c-a & ab-a^{2} \end{array}\right]$$ equals
(a) abc(b – c)(c -a)(a – b)
(b) (b – c)(c – a)(a – b)
(c) (a + b + c)(b – c)(c – a)(a – b)
(d) None of these

Question 18.
The number of distinct real roots of $$\left[\begin{array}{ccc} sin x & cos x & cos x \\ cos x & sin x & cos x \\ cos x & cos x & sin x \end{array}\right]$$ = 0 in the interval –$$\frac{π}{4}$$ ≤ x ≤ $$\frac{π}{4}$$ is
(a) 0
(b) 2
(c) 1
(d) 3

Question 19.
If A, B and C are angles of a triangle, then the determinant
$$\left[\begin{array}{ccc} -1 & cos C & cos B \\ cos C & -1 & cos A \\ cos B & cos A & -1 \end{array}\right]$$
(a) 0
(b) -1
(c) 1
(d) None of these

Question 20.
Let f(t) = $$\left[\begin{array}{ccc} cot t & t & 1 \\ 2 sin t & t & 2t \\ sin t & t & t \end{array}\right]$$ then $$_{t→0}^{lim}$$ $$\frac{f(t)}{t^2}$$ is equal to
(a) 0
(b) -1
(c) 2
(d) 3

Question 21.
The maximum value of $$\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+sin θ & 1 \\ 1+cos θ & 1 & 1 \end{array}\right]$$ is (θ is real number)
(a) $$\frac{1}{2}$$
(b) $$\frac{√3}{2}$$
(c) $$\frac{2√3}{4}$$
(d) √2

Answer: (a) $$\frac{1}{2}$$

Question 22.
If f(x) = $$\left[\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right]$$ then
(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0

Question 23.
If A = $$\left[\begin{array}{ccc} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right]$$ then A-1 exists if
(a) λ = 2
(b) λ ≠ 2
(c) λ ≠ -2
(d) None of these

Question 24.
If A and B are invertible matrices, then which of the following is not correct?
(b) det (a)-1 = [det (a)]-1
(c) (AB)-1 = B-1A-1
(d) (A + B)-1 = B-1 + A-1

Answer: (d) (A + B)-1 = B-1 + A-1

Question 25.
If x, y, z are all different from zero and
$$\left[\begin{array}{ccc} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{array}\right]$$ = 0, then value of x-1 + y-1 + z-1 is
(a) xyz
(b) x-1y-1z-1
(c) -x – y – z
(d) -1

Question 26.
The value of the determinant $$\left[\begin{array}{ccc} x & x+y & x+2y \\ x+2y & x & x+y \\ x+y & x+2y & x \end{array}\right]$$ is
(a) 9x² (x + y)
(b) 9y² (x + y)
(c) 3y² (x + y)
(d) 7x² (x + y)

Answer: (b) 9y² (x + y)

Question 27.
There are two values of a which makes determinant
Δ = $$\left[\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{array}\right]$$ = 86, then sum of these number is
(a) 4
(b) 5
(c) -4
(d) 9

Question 28.
Evaluate the determinant Δ = $$\left|\begin{array}{cc} log_{3}512 & log_{4}3 \\ log_{3}8 & log_{4}9 \end{array}\right|$$
(a) $$\frac{15}{2}$$
(b) 12
(c) $$\frac{14}{3}$$
(d) 6

Answer: (a) $$\frac{15}{2}$$

Question 29.
$$\left|\begin{array}{cc} x & -7 \\ x & 5 x+1 \end{array}\right|$$
(a) 3x² + 4
(b) x(5x + 8)
(c) 3x + 4x²
(d) x(3x + 4)

Question 30.
$$\left|\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \alpha \end{array}\right|$$
(a) 0
(b) 1
(c) 2
(d) 3

Question 31.
$$\left|\begin{array}{ll} \cos 15^{\circ} & \sin 15^{\circ} \\ \sin 75^{\circ} & \cos 75^{\circ} \end{array}\right|$$
(a) 0
(b) 5
(c) 3
(d) 7

Question 32.
$$\left|\begin{array}{cc} a+i b & c+i d \\ -c+i d & a-i b \end{array}\right|$$
(a) (a + b)²
(b) (a + b + c + d)²
(c) (a² + b² – c² – d²)
(d) a² + b² + c² + a²

Answer: (d) a² + b² + c² + a²

Question 33.
If $$\left|\begin{array}{lll} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right|$$ = $$k\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|$$ then k =
(a) 0
(b) 1
(c) 2
(d) 3

Question 34.
If $$\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|$$ = k (a + b + c)³ then k is
(a) 0
(b) 1
(c) 2
(d) 3

Question 35.
$$\left|\begin{array}{lll} a+1 & a+2 & a+4 \\ a+3 & a+5 & a+8 \\ a+7 & a+10 & a+14 \end{array}\right|$$ =
(a) 2
(b) -2
(c) 4
(d) -4

Question 36.
If abc ≠ 0 and $$\left|\begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{array}\right|$$ = 0 then $$\frac{1}{a}$$ + $$\frac{1}{b}$$ + $$\frac{1}{c}$$ =
(a) 1
(b) 2
(c) -1
(d) -3

Question 37.
$$\left|\begin{array}{ccc} 2 x y & x^{2} & y^{2} \\ x^{2} & y^{2} & 2 x y \\ y^{2} & 2 x y & x^{2} \end{array}\right|$$ =
(a) (x³ + y³)²
(b) (x² + y²)³
(c) -(x² + y²)³
(d) -(x³ + y³)²

Question 38.
$$\left|\begin{array}{ccc} b^{2} c^{2} & b c & b+c \\ c^{2} a^{2} & c a & c+a \\ a^{2} b^{2} & a b & a+b \end{array}\right|$$ =
(a) a7 + b7 + c7
(b) (a + b + c)7
(c) (a² + b² + c²) (a5 + b5 + c5)
(d) 0

Question 39.
If a, b, c are cube roots of unity, then
$$\left|\begin{array}{lll} e^{a} & e^{2 a} & e^{3 a}-1 \\ e^{b} & e^{2 b} & e^{3 b}-1 \\ e^{c} & e^{2 c} & e^{3 c}-1 \end{array}\right|$$ =
(a) 0
(b) e
(c) e²
(d) e³

Question 40.
The value of
$$\left|\begin{array}{ccc} \cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \end{array}\right|$$
is independent of
(a) α
(b) β
(c) α.β
(d) None of these

Question 41.
If x is a complex root of the equation
$$\left|\begin{array}{lll} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{array}\right|$$ + $$\left|\begin{array}{ccc} 1-x & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x \end{array}\right|$$ = 0
then x2007 + x-2007 =
(a) 1
(b) -1
(c) -2
(d) 2

Question 42.
$$\left|\begin{array}{lll} b-c & c-a & a-b \\ c-a & a-b & b-c \\ a-b & b-c & c-a \end{array}\right|$$ =
(a) 0
(b) 1
(c) 2
(d) 3

Question 43.
Let Δ = $$\left|\begin{array}{ccc} x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3} \end{array}\right|$$ then the value of Δ is
(a) (x – y) (y- z)(z – x)
(b) xyz
(c) x² + y² + z²)²
(d) xyz (x – y)(y – z) (z – x)

Answer: (d) xyz (x – y)(y – z) (z – x)

Question 44.
The value of the determinant $$\left|\begin{array}{ccc} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{array}\right|$$
(a) (α + β)(β + γ)(γ + α)
(b) (α – β)(β – γ) (γ – α) (α + β + γ)
(c) (α + β + γ)² (α – β – γ)²
(d) αβγ (α + β + γ)

Answer: (b) (α – β)(β – γ) (γ – α) (α + β + γ)

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