Home
Class 12
MATHS
If a,b, and c are all different and if ...

If a,b, and c are all different and if
`|{:(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),1+c^(3)):}|`=0 Prove that abc =-1.

Text Solution

Verified by Experts

The correct Answer is:
`-1`
Given that `|{:(a,, a^(2),, 1+a^(3)), (b,,b^(2),,1+b^(3)),(c,,c^(2),,1+c^(3)):}|=0`
`|{:(a,,a^(2),,1),(b,,b^(2),,1),(c,, c^(2),,1):}|+ abc|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|=0`
Operating `C_2 harr C_3 and ` then `C_1 harr C_2` in first determinant
`|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|+abc |{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}| =0`
either 1 + abc =0 or `|{:(1,,a ,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|=0`
Also given that vectors `vecA, vecB and vecC` are non-coplanar.
`|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}| ne 0`
So we must have `1+ abc =0 or abc =-1`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • INTRODUCTION TO VECTORS

    CENGAGE PUBLICATION|Exercise TRUE OR FALSE|1 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE PUBLICATION|Exercise ARCHIVES SUBJECTIVE TYPE|9 Videos

  • INTEGRALS

    CENGAGE PUBLICATION|Exercise All Questions|762 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|541 Videos

Similar Questions

Explore conceptually related problems

if a, b, c, are all distinct and |{:(a,a^3,a^4-1),(b,b^3,b^4-1),(c,c^3,c^4-1):}|=0 , show that abc(ab+bc+ca)=a+b+c.

If a , b ,c are all different and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-1]|=0, show that a b c(a b+b c+c a)=a+b+c

prove that , |{:(a,a^2,a^3+bc),(b,b^2,b^3+ca),(c,c^2,c^3+ab):}|=(a-b)(b-c)(c-a)(abc+bc+ca+ab)

Without expanding the determinant, prove that {:|( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) |:} ={:|( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) |:}

Using properties of determinants prove that, |{:((b+c)^(2),a^(2),a^(2)),(b^(2),(c+a)^(2),b^(2)),(c^(2),c^(2),(a+b)^(2)):}|=2abc(a+b+c)^(3)

Prove |{:(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ab,c^3):}|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

|{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

If a^(1/3)+b^(1/3)+c^(1/3)=0 show that (a+b+c)^(3)=27 abc

If a, b, c are in G.P., show that a^(2)b^(2)c^(2)((1)/(a^(3))+(1)/(b^(3))+(1)/(c^(3))) = a^(3) + b^(3) + c^(3) .

|{:(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=abc(a-b)(b-c)(c-a)

Home
Profile