wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Show that:
(i) 11+xa-b+11+xb-a=1

(ii) xaa-bxaa+b÷xbb-axbb+aa+b=1

(iii) x1a-b1a-cx1b-c1b-ax1c-a1c-b=1

(iv) xa2+b2xaba+bxb2+c2xbcb+cxc2+a2xaca+c=x2a3+b3+c3

(v) xa-ba+bxb-cb+cxc-ac+a=1

(vi) xa-a-11a-1aa+1=x

(vii) ax+1ay+1x+yay+2az+2y+zaz+3ax+3z+x=1

(viii) 3a3ba+b3b3cb+c3c3ac+a=1

Open in App
Solution

(i)
11+xa-b+11+xb-a
=11+xaxb+11+xbxa=xbxb+xa+xaxa+xb=xb+xaxa+xb=1

(ii)
xaa-bxaa+b÷xbb-axbb+aa+b=xaa-bxaa+b×xbb+axbb-aa+b=xa2-abxa2+ab×xb2+abxb2-aba+b
=xa2-ab-a2-ab×xb2+ab-b2+aba+b=x-2ab×x2aba+b=x-2ab+2aba+b=x0a+b=1

(iii)
x1a-b1a-cx1b-c1b-ax1c-a1c-b=x1a-b×1a-cx1b-c×1b-ax1c-a×1c-b=x1a-b×1a-c+1b-c×1b-a+1c-a×1c-b=xb-c-a-c+a-ba-ba-cb-c=x0=1

(iv)
xa2+b2xaba+bxb2+c2xbcb+cxc2+a2xaca+c=xa2+b2-aba+bxb2+c2-bcb+cxc2+a2-aca+c=xa+ba2+b2-abxb+cb2+c2-bcxa+cc2+a2-ac=xa3+b3xb3+c3xa3+c3=x2a3+b3+c3

(v)
xa-ba+bxb-cb+cxc-ac+a=xa-ba+bxb-cb+cxc-ac+a=xa2-b2xb2-c2xc2-a2=xa2-b2+b2-c2+c2-a2=x0=1

(vi)
xa-a-11a-1aa+1=xa-1a1a-1×aa+1=xa2-1aaa2-1=xa2-1a×aa2-1=x1=x

(vii)
ax+1ay+1x+yay+2az+2y+zaz+3ax+3z+x=ax+1-y-1x+yay+2-z-2y+zaz+3-x-3z+x=ax-yx+yay-zy+zaz-xz+x=ax2-y2+y2-z2+z2-x2=a0=1

(viii)
3a3ba+b3b3cb+c3c3ac+a=3a-ba+b3b-cb+c3c-ac+a=3a2-b2+b2-c2+c2-a2=30=1

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon