Example 31
Last updated at Dec. 8, 2016 by Teachoo
Transcript
Example 31 (Method 1) If a, b, c, are in A.P, find value of 2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c Given a, b & c are in A.P Then b – a = c – b b – a – c + b = 0 2b – a – c = 0 Solving 2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c Multiply & Divide by 2 = 22 2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c Multiplying 2 to R2 = 12 2y+45y+78y+a𝟐(3y+5)𝟐(6y+8)𝟐(9y+b)4y+67y+910y+c = 12 2y+45y+78y+a6y+1012y+1618y+2b4y+67y+910y+c Applying R2 →R2 – R1 – R3 = 12 2y+45y+78y+a6y+10− 2𝑦+4−(4𝑦+6)12y+16− 5𝑦+7−(7𝑦+9)18y+2b− 8y+a−(10y+c)4y+67y+910y+c = 12 2y+45y+78y+a6y+10−2𝑦−4−4𝑦−612y+16−5𝑦−7−7𝑦−918y+2b−2𝑦−𝑎−10𝑦−𝑐4y+67y+910y+c = 12 2𝑦+45𝑦+78𝑦+𝑎00𝟐𝒃−𝒂−𝒄4𝑦+67𝑦+910𝑦+𝑐 = 12 2𝑦+45𝑦+78𝑦+𝑎00𝟎4𝑦+67𝑦+910𝑦+𝑐 If any row as column of determinant are zero, then value of determinant is also zero. = 12 × 0 = 0 Thus, the value of determinant is 0 Example 31(Method 2) If a, b, c, are in A.P, find value of 2𝑦+45𝑦+78𝑦+𝑎3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 We need to find value of 2𝑦+45𝑦+78𝑦+𝑎3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 Given a, b & C are in A.P Then b – a = c – b b + b = a + c 2b = a + c Consider 2𝑦+45𝑦+78𝑦+𝑎3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 Applying R1 →R1 + R3 – 2R2 = 2𝑦+4+ 4𝑦+6−2(3𝑦+5)5𝑦+7+ 7𝑦+9−2(6𝑦+8)8𝑦+𝑎+ 10𝑦+𝑐−2(9𝑦+𝑏)3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 = 2𝑦+4+4𝑦+6−6𝑦−105𝑦+7+7𝑦+9−12𝑦−168𝑦+𝑎+10𝑦+𝑐−18𝑦−2𝑏3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 = 6𝑦−6𝑦+10−1012𝑦−12𝑦+16−1618𝑦−18𝑦+𝑎+𝑐−2𝑏3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 = 00𝒂+𝒄−2𝑏3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 = 00𝟐𝒃−2𝑏3𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 = 0003𝑦+56𝑦+89𝑦+𝑏4𝑦+67𝑦+910𝑦+𝑐 If any row as column of determinant are zero, then value of determinant is also zero. = 0 Hence, value of determinant is 0
Examples
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Example 6
Example 7
Example 8
Example 9
Example 10
Example 11
Example 12
Example 13
Example 14 Important
Example 15 Important
Example 16 Important
Example 17
Example 18 Important
Example 19
Example 20
Example 21
Example 22
Example 23
Example 24 Important
Example 25
Example 26 Important
Example 27
Example 28
Example 29
Example 30
Example 31 You are here
Example 32 Important
Example 33
Example 34 Important
About the Author
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.