** Ex 4.2, 11**

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Ex 4.2, 11 By using properties of determinants, show that: (i) a−b−c2a2a2bb−c−a2b2c2cc−a−b = (a + b + c)3 Taking L.H.S = a−b−c2a2a2bb−c−a2b2c2cc−a−b Applying R1 → R1 + R2 + R3 = a−b−c+2𝑏+2𝑐2a+b−c−a+2c2a+2𝑏+𝑐−𝑎−𝑏2bb−c−a2b2c2cc−a−b = 𝐚+𝐛+𝐜𝐚+𝐛+𝐜𝐚+𝒃+𝒄2bb−c−a2b2c2cc−a−b Taking common (a + b + c) from R1 = (a+b+c) 1112b𝑏−𝑐−𝑎2𝑏2c2𝑐c−a−b Applying C1 → C1 – C2 = (a+b+c) 𝟏−𝟏112b−(b−c−a)b−c−a2𝑏2c−2c2𝑐𝑐−𝑎−𝑏 = (a+b+c) 𝟎11b+c+ab−c−a2𝑏02𝑐𝑐−𝑎−𝑏 Applying C2 → C2 – C3 = (𝑎+𝑏+𝑐) 0𝟏−𝟏1𝑏+𝑐+𝑎𝑏−𝑐−𝑎−2𝑏2𝑏02𝑐−(𝑐−𝑎−𝑏)𝑐−𝑎−𝑏 = (𝑎+𝑏+𝑐) 0𝟎1𝑏+𝑐+𝑎−𝑏−𝑐−𝑎2𝑏0𝑎+𝑏+𝑐𝑐−𝑎−𝑏 Taking (a + b + c) common from C1 = (𝑎+𝑏+𝑐)(𝑎+𝑏+𝑐) 0011−𝑏−𝑐−𝑎2𝑏0(𝑎+𝑏+𝑐)𝑐−𝑎−𝑏 = 𝑎+𝑏+𝑐2 0011−(𝒂+𝒃+𝒄)2𝑏0(𝒂+𝒃+𝒄)𝑐−𝑎−𝑏 Taking common (a + b + c) from C2 = 𝑎+𝑏+𝑐2(a+b+c) 0011−12𝑏01𝑐−𝑎−𝑏 Expanding determinant along R1 = (a + b + c)3 0 −12b1c−a−b−0 12b0c−a−b+1 1−101 = (a + b + c)3 0−0+1(1−0) = (a + b + c)3 1 = (a + b + c)3 Hence proved Ex 4.2, 11 By using properties of determinants, show that: (ii) x+y+2zxyzy+z+2xyzxz+x+2y = 2(x+y+z)3 Taking L.H.S x+y+2zxyzy+z+2xyzxz+x+2y Applying C1 → C1 + C2 + C3 = 𝑥+𝑦+2𝑧+𝑥+𝑦𝑥𝑦z+y+z+2x+yy+𝑧+2𝑥yz+x+z+x+2yxz+x+2y = 𝟐(𝒙+𝒚+𝒛)𝑥𝑦𝟐(𝒙+𝒚+𝒛)y+𝑧+2𝑥y𝟐(𝒙+𝒚+𝒛)xz+x+2y Taking common 2(𝑥+𝑦+𝑧) from C1 = 𝟐(𝐱+𝐲+𝐳) 1𝑥𝑦1y+𝑧+2𝑥y1xz+x+2y Applying R2 → R2 – R3 = 2 x+y+z 1𝑥𝑦𝟏−𝟏y+𝑧+2𝑥−𝑥y−(𝑧+𝑥+2𝑦)1xz+x+2y = 2(x+y+z) 1𝑥𝑦𝟎𝑥+𝑦+𝑧−𝑥−𝑦−𝑧1xz+x+2y = 2(x+y+z) 1𝑥𝑦0(𝒙+𝒚+𝒛)−(𝒙+𝒚+𝒛)1xz+x+2y Taking common (𝑥+𝑦+𝑧) from 2nd Row = 2(x+y+z)(x+y+z) 1𝑥𝑦01−11xz+x+2y Applying R3 → R3 – R1 = 2 x+y+z2 1𝑥𝑦01−1𝟏−𝟏x−𝑥z+x+2y−y = 2 x+y+z2 1𝑥𝑦01−1𝟎0x+y+z Taking common (𝑥+𝑦+𝑧) Common from 3rd Row = 2 x+y+z2 x+y+z 1𝑥𝑦01−1001 Expanding Determinant along C1 = 2 x+y+z3 1 1−101−0 𝑥𝑦01+0 xy1−1 = 2 x+y+z3 1 1−101−0+0 = 2 x+y+z3 1 1−0−𝑥 0+𝑦(0) = 2 x+y+z3 1 = 2 x+y+z3 = R.H.S Hence proved

Ex 4.1, 7
Important

Example 14 Important

Example 15 Important

Example 16 Important

Ex 4.2, 7 Important

Ex 4.2, 8 Important

Ex 4.2, 11 Important You are here

Ex 4.2, 12 Important

Ex 4.2, 13 Important

Ex 4.2, 14 Important

Ex 4.2, 15 Important

Example 18 Important

Ex 4.3, 2 Important

Ex 4.3, 3 Important

Example 24 Important

Example 26 Important

Ex 4.5, 10 Important

Ex 4.5, 15 Important

Ex 4.5, 18 Important

Ex 4.6, 13 Important

Ex 4.6, 15 Important

Ex 4.6, 16 Important

Example 32 Important

Example 34 Important

Misc. 2 Important

Misc 11 Important

Misc. 15 Important

Misc. 16 Important

Misc. 19 Important

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.