Home
Class 12
MATHS
If a^2+b^2+c^2=1, then prove that |[a^...

If `a^2+b^2+c^2=1,` then prove that `|[a^2 +(b^2+c^2)cosθ, ab(1−cosθ), ac(1−cosθ)],[ba(1−cosθ),b^2(c^2+a^2)cosθ,bc(1−cosθ)],[ca(1−cosθ),cb(1−cosθ),c^2+(a^2+b^2)cosθ]|` independent of a,b,c?

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • DETERMINANT

    CENGAGE ENGLISH|Exercise Multiple Correct Answer|5 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE ENGLISH|Exercise Multiple correct answers type|11 Videos

Similar Questions

Explore conceptually related problems

Prove that : (cosecθ−cotθ)^2 = (1+cosθ)/(1−cosθ) ​

abs([-a^2,ab,ac],[ba,-b^2,bc],[ca,cb,-c^2]) = 4a^2.b^2.c^2

Prove the identity: |[b^2+c^2,ab, ac],[ba,c^2+a^2,bc],[ca, cb ,a^2+b^2]|=4a^2b^2c^2

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove that: |-a^2 ab ac ba -b^2 bc ac bc c^2| =4a^2b^2c^2 .

Prove the identities: |[b^2+c^2,ab, ac],[ba,c^2+a^2,bc],[ca, cb ,a^2+b^2]|=4a^2b^2c^2

If a, b, c are in A.P., then prove that : (i) ab+bc=2b^(2) (ii) (a-c)^(2)=4(b^(2)-ac) (iii) a^(2)+c^(2)+4ca=2(ab+bc+ca).

Using properties of determinants, prove that |[a^2, bc, ac+c^2] , [a^2+ab, b^2, ac] , [ab, b^2+bc, c^2]| = 4a^2b^2c^2

(sinθ+cosθ)(1−sinθcosθ) can be written as :

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

CENGAGE ENGLISH-DETERMINANTS-All Questions
  1. If x ,y ,z are different from zero and "Delta"=|[a, b-y ,c-z],[ a-x, b...

    Text Solution

    |

  2. If A , B ,C are angles of a triangle, then the value of |[e^(2i A),e^(...

    Text Solution

    |

  3. If a^2+b^2+c^2=1, then prove that |[a^2 +(b^2+c^2)cosθ, ab(1−cosθ), ...

    Text Solution

    |

  4. For the equation |[1,x,x^2],[x^2, 1,x],[x,x^2, 1]|=0, There are exa...

    Text Solution

    |

  5. Let Deltar=|[r-1,n,6],[(r-1)^2,2n^2,4n-2],[(r-1)^3,3n^3,3n^2-3n]|. Sho...

    Text Solution

    |

  6. Let m be a positive integer and Deltar=|[2r-1,.^mCr,1],[m^2-1,2^m,m+1]...

    Text Solution

    |

  7. a^(−1)+b^(−1)+c^(−1)=0 such that |[1+a,1,1,],[1,1+b,1,],[1,1,1+c,]| =△...

    Text Solution

    |

  8. Find the area of a triangle whose vertices are A(3,2),B(11 ,8) and C(8...

    Text Solution

    |

  9. If Dk=1nn2k n^2+n+1n^2+n2k-1n^2n^2+n+1a n dsum(k=1)^n Dk=56. then n eq...

    Text Solution

    |

  10. If a!=p ,b!=q ,c!=r and |[p, b, c ],[a, q, c ],[a ,b, r]|=0, then find...

    Text Solution

    |

  11. If x1,x2,x3 as well as y1, y2, y3 are in G.P. with same common rati...

    Text Solution

    |

  12. If the lines a1x+b1y+1=0,a2x+b2y+1=0 and a3x+b3y+1=0 are concurrent,...

    Text Solution

    |

  13. For a fixed positive integer n , if =|n !(n+1)!(n+2)!(n+1)!(n+2)!(n+3)...

    Text Solution

    |

  14. Find the value of a for which the lines 2x+y-1=0, a x+3y-3=0, 3x+2...

    Text Solution

    |

  15. if f(alpha,beta)=|[cos alpha,-sin alpha,1],[sin alpha,cos alpha,1],[co...

    Text Solution

    |

  16. If the lines a x+y+1=0,x+b y+1=0 and x+y+c=0(a ,b ,c being distinct...

    Text Solution

    |

  17. If (1+x+x^(2))^(25)=a(0)+a(1)x+a(2)x^(2)+...+a(50).x^(50), then a(0)+a...

    Text Solution

    |

  18. Find the value of lambda for which the homogeneous system of equations...

    Text Solution

    |

  19. If f,g,a n d \ h are differentiable functions of x and d(x)=|[f,g,h]...

    Text Solution

    |

  20. Let a ,b ,c be positive and not all equal. Show that the value of the ...

    Text Solution

    |