CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant

-2aa+ba+cb+a-2bb+cc+ac+b-2c

The other factor in the value of the determinant is
(a) 4
(b) 2
(c) a + b + c
(d) none of these


Solution

(a) 4

Δ=-2a       a + b      a + cb + a   -2b         b + cc + a     c + b    -2c Let a + b = 2C, b + c = 2A and c + a = 2B.a + b + b + c + c + a = 2A + 2B + 2C2a + b + c  = 2A + B + Ca + b + c = A + B + CAlso,a = a + b + c  - b + c = A + B + C - 2A = B + C - ASimilarly,b = C + A - B,  c = A + B - CΔ = 2A - 2B - 2C             2C                        2B      2C                  2B-2C-2A                 2A      2B                       2A                 2C-2A-2B = 8 × A - B - C           C                      B      C             B - C - A              A     B                    A                C - A - B    taking out 2 common from R1 R2  R3=8 ×  A - B            C + B               B B - A           B - C               A B + A          C - B        C-A-B      Applying C1C1+C2 , C2C2 + C3=8 ×  A - B    C + B            B  0              2B           A + B 2B             0            C - B                    Applying R2 R1+ R2, R3 R2 +R3=8 × A - B    2B           A + B  0             C - B +   2B ×   C + B       B    2B       A + B          Expanding along C1=16 BA - BC - B + C + BA + B - 2B2= 32 ABC=32b + c2c + a2a + b2=4a + bb + cc + aHence, 4 is the other factor of the determinant. 

Mathematics
RD Sharma XII Vol 1 (2015)
Standard XII

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
Same exercise questions
View More



footer-image