Home
Class 11
MATHS
The value of |(a(1) x(1) + b(1) y(1),a(1...

The value of `|(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) +b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))|`, is

A

`a_(1) a_(2) a_(3) b_(1) b_(2) b_(3)`

B

`x_(1) x_(2) x_(3) y_(1) y_(2) y_(3)`

C

0

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
We have,
`|(a_(1),b_(1),0),(a_(2),b_(2),0),(a_(3),b_(3),0)||(x_(1),y_(1),0),(x_(2),y_(2),0),(x_(3),y^(3),0)| = 0 xx 0 = 0`
`rArr |(a_(1) x_(1) + b_(1) y_(1) + 0,a_(1) x_(2) + b_(1) y_(2) + 0,a_(1) x_(3) + b_(1) y_(3) + 0),(a_(2) x_(1) + b_(2) y_(1) + 0,a_(2) x_(2) + b_(2) y_(2) + 0,a_(2) x_(3) + b_(2) y_(3) + 0),(a_(3) x_(1) + b_(3) y_(1) + 0,a_(3) x_(2) + b_(3) y_(2) + 0,a_(3) x_(3) + b_(3) y_(3) + 0)| = 0` [Using row by-row multiplication]
`rArr |(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) + b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))| = 0`
Promotional Banner

Topper's Solved these Questions

  • DETERMINANTS

    OBJECTIVE RD SHARMA|Exercise Section I - Solved Mcqs|74 Videos

  • DETERMINANTS

    OBJECTIVE RD SHARMA|Exercise Section II - Assertion Reason Type|6 Videos

  • CONIC SECTIONS

    OBJECTIVE RD SHARMA|Exercise Section I - Solved Mcqs|1 Videos

  • DISCRETE PROBABILITY DISTRIBUTIONS

    OBJECTIVE RD SHARMA|Exercise Exercise|40 Videos

Similar Questions

Explore conceptually related problems

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =5 , then the value of Delta = |(b_(2) c_(3) - b_(3) c_(2),a_(3) c_(2) - a_(2) c_(3),a_(2) b_(3) -a_(3) b_(2)),(b_(3) c_(1) - b_(1) c_(3),a_(1) c_(3) - a_(3) c_(1),a_(3) b_(1) - a_(1) b_(3)),(b_(1) c_(2) - b_(2) c_(1),a_(2) c_(1) - a_(1) c_(2),a_(1) b_(2) - a_(2) b_(1))| is

Prove that |{:(a_(1)alpha_(1)+b_(1)beta_(1),a_(1)alpha_(2)+b_(1)beta_(2),a_(1)alpha_(3)+b_(1)beta_(3)),(a_(2)alpha_(1)+b_(2)beta_(1),a_(2)alpha_(2)+b_(2)beta_(2),a_(2)alpha_(3)+b_(2)beta_(3)),(a_(3)alpha_(1)+b_(3)beta_(1),a_(3)alpha_(2)+b_(3)beta_(2),a_(3)alpha_(3)+b_(3)beta_(3)):}| =0.

det[[2a_(1)b_(1),a_(1)b_(2)+a_(2)b_(1),a_(1)b_(3)+a_(3)b_(1)a_(1)b_(2)+a_(2)b_(1),2a_(2)b_(2),a_(2)b_(3)+a_(3)b_(2)a_(1)b_(3)+a_(3)b_(1),a_(3)b_(2)+a_(2)b_(3),2a_(3)b_(3)]]=

The value of the determinant Delta = |(1 + a_(1) b_(1),1 + a_(1) b_(2),1 + a_(1) b_(3)),(1 + a_(2) b_(1),1 + a_(2) b_(2),1 + a_(2) b_(3)),(1 + a_(3) b_(1) ,1 + a_(3) b_(2),1 + a_(3) b_(3))| , is

The determinant |(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

if quad /_=[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

the value of the determinant |{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}| is

OBJECTIVE RD SHARMA-DETERMINANTS-Chapter Test
  1. The value of |(a(1) x(1) + b(1) y(1),a(1) x(2) + b(1) y(2),a(1) x(3) +...

    Text Solution

    |

  2. Consider the system of equations a(1) x + b(1) y + c(1) z = 0 a(2)...

    Text Solution

    |

  3. If D = |(1,1,1),(1,1 +x,1),(1,1,1 +y)| " for " x!= 0, y != 0, then D i...

    Text Solution

    |

  4. If |(x +a,b,c),(a,x +b,c),(a,b,x +c)| = 0, then x equals

    Text Solution

    |

  5. |(sin^(2) x,cos^(2) x,1),(cos^(2) x,sin^(2) x,1),(- 10,12,2)| =

    Text Solution

    |

  6. The system of linear equations x + y + z = 2 2x + y -z = 3 3x + ...

    Text Solution

    |

  7. The roots of the equation |(3x^(2),x^(2) + x cos theta + cos^(2) the...

    Text Solution

    |

  8. |(bc,bc'+b'c,b'c'),(ca,ca'+c'a,c'a'),(ab,ab'+a'b,a'b')| is equal to

    Text Solution

    |

  9. If alpha, beta, gamma are the cube roots of 8 , then |(alpha,beta,gamm...

    Text Solution

    |

  10. One root of the equation |(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 is

    Text Solution

    |

  11. If a,b,c are non-zero real numbers then D=|[b^2 c^2, bc, b+c] , [c^2a^...

    Text Solution

    |

  12. If x, y , z are in A.P., then the value of the det (A) is , where A ...

    Text Solution

    |

  13. The value of |(b +c,a,a),(b,c +a,b),(c,c,a +b)|, is

    Text Solution

    |

  14. If a , b , c are non-zero real numbers and if the system of equations ...

    Text Solution

    |

  15. If a!=6,b,c satisfy|[a,2b,2c],[3,b,c],[4,a,b]|=0 ,then abc =

    Text Solution

    |

  16. The value of Delta = |(1^(2),2^(2),3^(2)),(2^(2),3^(2),4^(2)),(3^(2),4...

    Text Solution

    |

  17. The value of Delta = |(a,a +b,a +2b),(a +2b,a,a +b),(a +b,a +2b,a)| is...

    Text Solution

    |

  18. If all the elements in a square matrix A of order 3 are equal to 1 or ...

    Text Solution

    |

  19. The roots of the equation |(1,4,20),(1,-2,5),(1,2x,5x^(2))| = 0 are

    Text Solution

    |

  20. If f(x) = |(sin x,cos x,tan x),(x^(3),x^(2),x),(2x,1,1)|, " then " lim...

    Text Solution

    |

  21. If A, B and C are the angles of a triangle and |(1,1,1),(1 + sin A,1...

    Text Solution

    |