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.

5. ADJOINT OF A MATRIX

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:
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 
  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 ∆
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 
 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 
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 
🔷 3. Third Order Determinant 
  Can be determined by expressing it in terms of second order determinants
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 
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,
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
  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:
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
  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 :
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
  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:
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 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

 
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 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:

 

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 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

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 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

 12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths


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

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  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

 12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths


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
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths

12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 
 Theorem 1 
  If A be any given square matrix of order n, 
 Then A (adj A) = (adj A) A = A I,
 Where I is the identity matrix of order n 
  Verification:
 
12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths
 Similarly, we can show (adj A) A = AI
Hence A (adj A) = (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

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 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)

12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths


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

12 class Maths Notes Chapter 4 Determinants free PDF| Quick revision Notes class 12 maths

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.

Important Questions for CBSE Class 12 Maths Expansion of Determinants

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Previous Years Examination Questions

1 Mark Questions

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Determinants Important Questions for CBSE Class 12 Maths Properties of Determinants

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Previous Years Examination Questions

1 Mark Questions
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4 Mark Questions
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CBSE Class 12 Maths Inverse of a Matrix and Application of Determinants and Matrix

important-questions-for-class-12-maths-cbse-inverse-of-a-matrix-and-application-of-determinants-and-matrix-t-3-1
important-questions-for-class-12-maths-cbse-inverse-of-a-matrix-and-application-of-determinants-and-matrix-t-3-2

Previous Years Examination Questions

6 Marks Questions
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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

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

Answer

Answer: (c) 134


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

Answer

Answer: (c) 0


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

Answer

Answer: (a) 0


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

Answer

Answer: (a) 0


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

Answer

Answer: (a) -9


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

Answer

Answer: (b) ±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)

Answer

Answer: (c) 0


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

Answer

Answer: (a) ±3


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

Answer

Answer: (a) 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

Answer

Answer: (c) 0


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

Answer

Answer: (c) 4xyz


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)

Answer

Answer: (c) ±√3


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

Answer

Answer: (c) ±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

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

Answer

Answer: (b) 3


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

Answer

Answer: (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

Answer

Answer: (c) 1


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

Answer

Answer: (a) 0


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

Answer

Answer: (a) 0


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

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

Answer

Answer: (c) f(0) = 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

Answer

Answer: (d) None of these


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

Answer

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

Answer

Answer: (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

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

Answer

Answer: (c) -4


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

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)

Answer

Answer: (b) x(5x + 8)


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

Answer

Answer: (b) 1


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

Answer

Answer: (a) 0


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

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

Answer

Answer: (c) 2


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

Answer

Answer: (b) 1


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

Answer

Answer: (b) -2


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

Answer

Answer: (c) -1


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

Answer

Answer: (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

Answer

Answer: (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

Answer

Answer: (a) 0


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

Answer

Answer: (a) α


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

Answer

Answer: (c) -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

Answer

Answer: (a) 0


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

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

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


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