Class12 ISC Maths Chapter Determinants Exercise 4.2

Class 12 ISC Maths Chapter Determinants Exercise 4.2

Explore our extensive collection of questions on the topic of "Determinants," designed to help you understand and apply this fundamental concept in linear algebra. Our question bank covers key topics such as properties of determinants, cofactor expansion, minors, applications in solving systems of linear equations, and finding the inverse of matrices. These practice questions are perfect for students aiming to build a strong foundation in determinants and excel in exams. Whether you're revising concepts or looking to enhance your problem-solving skills, our comprehensive collection provides the ideal resource for mastering "Determinants."

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Determinants

a+b	2a+b	3a+b
2a+b	3a+b	4a+b
4a+b	5a+b	6a+b

102	18	36
1	3	4
17	3	6

x	a	x+a
y	b	y+b
z	c	z+c

a	b	c
a+2x	b+2y	c+2z
x	y	z

Class 12 ISC Maths Chapter Determinants Exercise 4.2

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Q1 Show that |A| = |B|, where A= 1 -3 2 2 3 -1 0 5 -4 and B= 1 2 0 -3 3 5 2 -1 -4

Q2 Without expanding, find the value of : a+b 2a+b 3a+b 2a+b 3a+b 4a+b 4a+b 5a+b 6a+b

Q3 Without expanding, find the values of

(i) 102 18 36 1 3 4 17 3 6

(ii) 2 7 65 3 8 75 5 9 86

Q4 Without expanding, find the values of

(i) x a x+a y b y+b z c z+c

(ii) a b c a+2x b+2y c+2z x y z

Q5 Without expanding, find the values of

(i) a-b b-c c-a x-y y-z z-x p-q q-r r-p

(ii) 1 bc a(b+c) 1 ca b(c+a) 1 ab c(a+b)

Q6(i) If A is a square matrix of order 2 and |A| = -5, find the value of |3A|

Q6(ii) If A is a square matrix of order 3 and |A| = 4, find the value of |2A|

Q6(iii) If A is a square matrix of order 3 and |A| = 5, find the value of |3A|

Q6(iv) If A is a square matrix of order 3 and |A| = -2, find the value of |-5A|

Q6(v) If A is a matrix of order 3x3 and its determinant is 4, then find |3A|

Q6(vi) If A is a 3x3 matrix and |3A|=K|A|, then write the value of k

Q7 If A = 4 3 2 5 and B= -1 1 3 6 ,then find the value of |AB|

Q8 Using the properties of determinants,solve the following equation for x: x+a x x x x+a x x x x+a =0

Q9 Show that one root of the equation x+a b c b x+c a c a x+b =0 is -(a+b+c)

Q10(i) Without expanding ,prove that a b c x y z p q r = x y z p q r a b c = y b q x a p z c r

Q10(ii) Without expanding ,prove that a a2 bc b b2 ca c c2 ab = 1 a2 a3 1 b2 b3 1 c2 c3

Q11 Use properties of determinants to solve for x:

(i) x+a b c c x+b a a b x+c = 0

(ii) 3-x -1 1 -1 5-x -1 1 -1 3-x = 0

Q12(i) If s=a + b + c, then prove that s+c a b c s+a b c a s+b = 2s3

Q12(ii) If a, b and c are in A.P ,then find the value of 2y+4 5y+7 8y+a 3y+5 6y+8 9y+b 4y+6 7y+9 10y+c

Q12(iii) Using the properties of determinant, prove that x-3 x-4 x-α x-2 x-3 x-β x-1 x-2 x-γ; = 0 ,where α,β and γ are in A.P

Q13 Using the properties of determinant, prove that a+b b+c c+a b+c c+a a+b c+a a+b b+c = 2 a b c b c a c a b

Using the properties of determinant, prove the following (14 to 27)identities:

Q14(i) 1+a b c a 1+b c a b 1+c = 1+a+b+c

Q14(ii) a+x y z x a+y z x y a+z = a2(a+x+y+z)

Q14(iii) y+k y y y y+k y y y y+k = k2(3y+k)

Q14(iv) x+4 2x 2x 2x x+4 2x 2x 2x x+4 = (5x+4)(4-x)2

Q15 a+b+nc na-a nb-b nc-c b+c+na nb-b nc-c na-a c+a+nb = n(a+b+c)3

Q16 1 x yz 1 y zx 1 z xy = 1 x x2 1 y y2 1 z z2 = (x-y)(y-z)(z-x)

Q17 x y z x2 y2 z2 x3 y3 z3 = xyz(x-y)(y-z)(z-x)

Q18(i) x y 1 α x 1 α β 1 = (x-α) (x-β)

Q18(ii) 1 x y 1 x+y y 1 x x+y = xy

Q18(iii) y+z x y z+x z x x+y y z = (x+y+z) (x-z)2

Q19 -a2 ab ac ab -b2 bc ca bc -c2 = 4a2b2c2

Q20 1 α α2+βγ 1 β β2+γα 1 γ γ2+αβ = 2(α-β)(β-γ) (γ-α)

Q21 a2+1 ab ac ab b2+1 bc ac bc c2+1 = a2+1 b2 c2 a2 b2+1 c2 a2 b2 c2+1 = 1+a2+b2+c2

Q22 a b c a2 b2 c2 bc ca ab = 1 a2 a3 1 b2 b3 1 c2 c3 =(a-b)(b-c)(c-a)(ab+bc+ca)

Q23 a2 a2-(b-c)2 bc b2 b2-(c-a)2 ca c2 c2-(a-b)2 ab =(a-b)(b-c)(c-a)(a+b+c)(a2+b2+c2)

Q24 (b+c)2 a2 bc (c+a)2 b2 ca (a+b)2 c2 ab =(a-b)(b-c)(c-a)(a+b+c)(a2+b2+c2)

Q25(i) a+b+c -c -b -c a+b+c -a -b -a a+b+c = 2(a+b)(b+c)(c+a)

Q25(ii) y+z x+y x z+x y+z y x+y z+x z = x3+y3+z3-3xyz

Q26 a b-c c-b a-c b c-a a-b b-a c = (a+b-c)(b+c-a)(c+a-b)

Q27 1 bc+ad b2c2+a2d2 1 ca+bd c2a2+b2d2 1 ab+cd a2b2+c2d = (a-b)(a-c)(a-d)(b-c)(b-d)(c-d)

Q28 Without expanding any of the determinants given below, prove that a2 b2 c2 (a+1)2 (b+1)2 (c+1)2 (a-1)2 (b-1)2 (c-1)2 =4 a2 b2 c2 a b c 1 1 1

Q29 Using properties of determinants, solve the following equations for x:

(i) 3x-8 3 3 3 3x-8 3 3 3 3x-8 =0

(ii) x+1 3 5 2 x+2 5 2 3 x+4 =0

(iii) x -6 -1 2 -3x x-3 -3 2x x+2 =0

(iv) 15-2x 11 10 11-3x 17 16 7-x 14 13 =0

Q30 If a, b, c are all different and a a3 a4-1 b b3 b4-1 c c3 c4-1 =0 , prove that abc(ab+bc+ca)=a+b+c

Q31 In a triangle ABC, if 1 1 1 1+cosA 1+cosB 1+cosC cosA+cos2A cosB+cos2B cosC+cos2C =0,then prove that ABC is an isosceles triangle

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Q1 Show that |A| = |B|, where
A=

1 -3 2

2 3 -1

0 5 -4

and B=

1 2 0

-3 3 5

2 -1 -4

Q2 Without expanding, find the value of :

a+b 2a+b 3a+b

2a+b 3a+b 4a+b

4a+b 5a+b 6a+b

(i)

102 18 36

1 3 4

17 3 6

(ii)

2 7 65

3 8 75

5 9 86

(i)

x a x+a

y b y+b

z c z+c

(ii)

a b c

a+2x b+2y c+2z

x y z

(i)

a-b b-c c-a

x-y y-z z-x

p-q q-r r-p

(ii)

1 bc a(b+c)

1 ca b(c+a)

1 ab c(a+b)

Q7 If
A =

4 3

2 5

and B=

-1 1

3 6

,then find the value of |AB|

Q8 Using the properties of determinants,solve the following equation for x:

x+a x x

x x+a x

x x x+a

=0

Q9 Show that one root of the equation

x+a b c

b x+c a

c a x+b

=0

is -(a+b+c)

Q10(i) Without expanding ,prove that

a b c

x y z

p q r

=

x y z

p q r

a b c

=

y b q

x a p

z c r

Q10(ii) Without expanding ,prove that

a a² bc

b b² ca

c c² ab

=

1 a² a³

1 b² b³

1 c² c³

(i)

x+a b c

c x+b a

a b x+c

= 0

(ii)

3-x -1 1

-1 5-x -1

1 -1 3-x

= 0

Q12(i) If s=a + b + c, then prove that

s+c a b

c s+a b

c a s+b

= 2s³

Q12(ii) If a, b and c are in A.P ,then find the value of

2y+4 5y+7 8y+a

3y+5 6y+8 9y+b

4y+6 7y+9 10y+c

Q12(iii) Using the properties of determinant, prove that

x-3 x-4 x-α

x-2 x-3 x-β

x-1 x-2 x-γ;

= 0

,where α,β and γ are in A.P

Q13 Using the properties of determinant, prove that

a+b b+c c+a

b+c c+a a+b

c+a a+b b+c

= 2

a b c

b c a

c a b

Q14(i)

1+a b c

a 1+b c

a b 1+c

= 1+a+b+c

Q14(ii)

a+x y z

x a+y z

x y a+z

= a²(a+x+y+z)

Q14(iii)

y+k y y

y y+k y

y y y+k

= k²(3y+k)

Q14(iv)

x+4 2x 2x

2x x+4 2x

2x 2x x+4

= (5x+4)(4-x)²

Q15

a+b+nc na-a nb-b

nc-c b+c+na nb-b

nc-c na-a c+a+nb

= n(a+b+c)³

Q16

1 x yz

1 y zx

1 z xy

=

1 x x²

1 y y²

1 z z²

= (x-y)(y-z)(z-x)

Q17

x y z

x² y² z²

x³ y³ z³

= xyz(x-y)(y-z)(z-x)

Q18(i)

x y 1

α x 1

α β 1

= (x-α) (x-β)

Q18(ii)

1 x y

1 x+y y

1 x x+y

= xy

Q18(iii)

y+z x y

z+x z x

x+y y z

= (x+y+z) (x-z)²

Q19

-a² ab ac

ab -b² bc

ca bc -c²

= 4a²b²c²

Q20

1 α α²+βγ

1 β β²+γα

1 γ γ²+αβ

= 2(α-β)(β-γ) (γ-α)

Q21

a²+1 ab ac

ab b²+1 bc

ac bc c²+1

=

a²+1 b² c²

a² b²+1 c²

a² b² c²+1

= 1+a²+b²+c²

Q22

a b c

a² b² c²

bc ca ab

=

1 a² a³

1 b² b³

1 c² c³

=(a-b)(b-c)(c-a)(ab+bc+ca)

Q23

a² a²-(b-c)² bc

b² b²-(c-a)² ca

c² c²-(a-b)² ab

=(a-b)(b-c)(c-a)(a+b+c)(a²+b²+c²)

Q24

(b+c)² a² bc

(c+a)² b² ca

(a+b)² c² ab

=(a-b)(b-c)(c-a)(a+b+c)(a²+b²+c²)

Q25(i)

a+b+c -c -b

-c a+b+c -a

-b -a a+b+c

= 2(a+b)(b+c)(c+a)

Q25(ii)

y+z x+y x

z+x y+z y

x+y z+x z

= x³+y³+z³-3xyz

Q26

a b-c c-b

a-c b c-a

a-b b-a c

= (a+b-c)(b+c-a)(c+a-b)

Q27

1 bc+ad b²c²+a²d²

1 ca+bd c²a²+b²d²

1 ab+cd a²b²+c²d

= (a-b)(a-c)(a-d)(b-c)(b-d)(c-d)

Q28 Without expanding any of the determinants given below, prove that

a² b² c²

(a+1)² (b+1)² (c+1)²

(a-1)² (b-1)² (c-1)²

=4

a² b² c²

a b c

1 1 1

(i)

3x-8 3 3

3 3x-8 3

3 3 3x-8

=0

(ii)

x+1 3 5

2 x+2 5

2 3 x+4

=0

(iii)

x -6 -1

2 -3x x-3

-3 2x x+2

=0

(iv)

15-2x 11 10

11-3x 17 16

7-x 14 13

=0

Q30 If a, b, c are all different and

a a³ a⁴-1

b b³ b⁴-1

c c³ c⁴-1

=0

, prove that abc(ab+bc+ca)=a+b+c

Q31 In a triangle ABC, if

1 1 1

1+cosA 1+cosB 1+cosC

cosA+cos²A cosB+cos²B cosC+cos²C

=0,then prove that ABC is an isosceles triangle