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QUESTION BANK mathematics (DETERMINANTS)
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THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER DETERMINANTS ALONG WITH THE ANSWER KEY AT THE END
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DeterminantsQUICK RECAP
DETERMINANT X If A = [a] be a matrix of order 1, then det (A) = a.
a a
8 If A = [aij] is a square matrix of order n, then X If A = 11 12 be a matrix of order 2,
a number (real or complex) associated to a21 a22
matrix A is called determinant of A. a11 a12
then A = = a11a22 − a21a12
It is denoted by det A or |A| or D. a21 a22
, a11 a12 a13 Note :
(i) Since area is always a positive quantity,
X If A = a21 a22 a23 be a matrix of order 3,
a31 a32 a33 therefore we always take the absolute
then value of the determinant for the area.
a11 a12 a13 (ii) Area of a triangle formed by three
a a23
A = a21 a22 a23 = (−1)1+1 a11 22 collinear points is always zero.
a32 a33
a31 a32 a33
MINOR OF AN ELEMENT
a a a21 a22
+(−1)1+2 a12 21 23 + (−1)1+3 a13 8 Minor of an element aij of the determinant
a31 a33 a31 a32
of matrix A is the determinant obtained by
= a11(a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23)
+ a13 (a21 a32 – a31 a22) deleting ith row and jth column. Minor of aij is
Note : |KA| = Kn|A|, where A is of order n. denoted by Mij.
Properties of Determinants X Minor of an element of a determinant of order
X The value of the determinant remains n(n ≥ 2) is a determinant of order n – 1.
unchanged if its rows and columns are
interchanged. COFACTOR OF AN ELEMENT
X If any two rows (or columns) of a determinant
8 Cofactor of an element aij of determinant
are interchanged, then the value of the
determinant is multiplied by –1. of matrix A is, Aij = (–1)i + j Mij.
X If any two rows (or columns) of a determinant X The determinant of a matrix A can also be
are identical then the value of determinant is obtained by using cofactors i.e., sum of
zero. product of elements of a row (or column)
X If the elements of a row (or column) of a
with their corresponding cofactors.
determinant are multiplied by a scalar, then
the value of the new determinant is equal \ |A| = a11 A11 + a12 A12 + a13 A13
to same scalar times the value of the original X If the elements of one row (or column) are
determinant. multiplied with the cofactors of elements of
X If each element of any row (or column) of any other row (or column), then their sum is
a determinant is the sum of two (or more)
zero i.e., a11A21 + a12A22 +a13A23 =0
terms then the determinant is expressible as
the sum of two (or more) determinants of the ADJOINT OF A MATRIX
same order.
X The value of a determinant does not change 8 The adjoint of a square matrix is the
when any row (or column) is multiplied transpose of the matrix of cofactors.
by a number or an expression and then
X Adjoint of A is denoted by adjA.
added to or subtracted from any other row
(or column). Remark : For a matrix A of order n,
A(adjA) = (adjA)A = |A|In
AREA OF A TRIANGLE
8 The area of a triangle whose vertices are (x1, y1), SINGULAR AND NON-SINGULAR MATRIX
(x2, y2) and (x3, y3) is given by 8 Let A be a square matrix, then A is called
x1 y1 1 X Singular matrix, iff |A| = 0
1
∆ = x2 y2 1 X Non-singular matrix, iff |A| ≠ 0
2
x3 y3 1 Note : If A and B are non-singular matrices
1 of same order, then AB and BA are also non-
= {x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 )}
2 singular matrices of same order.
,8 A square matrix A is invertible iff A is non- X For a system of equations, AX = B.
singular matrix and (i) If |A| ≠ 0, then the given system of
1 equations is consistent and has a unique
A−1 = (adjA)
A solution.
SOLUTION OF A SYSTEM OF LINEAR (ii) If |A| = 0 and (adjA)B ≠ O, then the
EQUATIONS solution does not exist and the given
system is inconsistent.
8 For a square matrix A, a system of equations
AX = B is said to be (iii) If |A| = 0 and (adjA)B = O, then the given
(i) Consistent, if it has one or more system may be either consistent or
solutions. inconsistent, according as the system
(ii) Inconsistent, if its solution doesn’t have either infinitely many solutions or
exist. no solution.
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
4.2 Determinant cos 15° sin 15°
11. Evaluate : . (AI 2011)
sin 75° cos 75°
VSA (1 mark)
1. Find the maximum value of 3 4
12. If A = , find the value of 3|A|.
1 1 1 1 2 (AI 2011C)
1 1 + sin θ 1 .(Delhi 2016)
1 1 1 + cos θ
4.3 Properties of Determinants
VSA (1 mark)
x + 3 −2
2. If x ∈ N and = 8 , then find the 13. If A is a square matrix of order 3 and |A| = 5,
−3x 2x
then the value of |2A′| is
value of x. (AI 2016)
(a) –10 (b) 10
x sin θ cos θ (c) –40 (d) 40 (2020)
3. If − sin θ −x 1 = 8, write the value of x. 14. If A is a skew-symmetric matrix of order 3,
cos θ 1 x then the value of |A| is
(Foreign 2016)
(a) 3 (b) 0
1 2 1 3 (c) 9 (d) 27 (2020)
4. If A = and B = , write the
3 −1 −1 1 15. If A is a 3 × 3 matrix such that |A| = 8, then
value of |AB|. (Delhi 2015C) |3A| equals
(a) 8 (b) 24
2x 5 6 −2 (c) 72 (d) 216 (2020)
5. If = , write the value of x.
8 x 7 3
(Delhi 2014) 16. If A and B are square matrices each of order
3 and |A| = 5, |B| = 3, then the value of |3AB|
3x 7 8 7
6. If = , find the value of x. is . (2020)
−2 4 6 4
(AI 2014) 17. If A and B are square matrices of the same
order 3, such that |A| = 2 and AB = 2I, write
7. Write the value of the determinant the value of |B|. (Delhi 2019)
p p +1
. (Delhi 2014C) x+y y+z z+x
p −1 p
18. Write the value of ∆ = z x y .
2 7 65 −3 −3 −3
8. Write the value of 3 8 75 . (AI 2014C) (AI 2015)
5 9 86 19. If A is a 3 × 3 matrix, |A| ≠ 0 and |3A| = k |A|,
then write the value of k. (Foreign 2014)
x+1 x −1 4 −1
9. If x − 3 x + 2 = 1 3 , then write the 20. Let A be a square matrix of order 3 × 3. Write
the value of |2A|, where |A| = 4.
value of x. (Delhi 2013)
(AI 2012, Delhi 2011C)
2x x +3 1 5
10. If = , then write the 21. The value of the determinant of a matrix A of
2(x + 1) x + 1 3 3 order 3 × 3 is 4. Find the value of |5A|.
value of x. (Delhi 2013C) (Delhi 2012C)
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