Home
Class 12
MATHS
If m is a positive integer and Dr=|2r...

If `m` is a positive integer and `D_r=|2r-1\ ^m C_r1m^2-1 2^m m+1s in^2(m^2)s in^2(m)s in^2(m+1)|` . Prove that `sum_(r=0)^m D_r=0` .

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • DERIVATIVES AS A RATE MEASURER

    RD SHARMA ENGLISH|Exercise All Questions|168 Videos

  • DIFFERENTIABILITY

    RD SHARMA ENGLISH|Exercise All Questions|135 Videos

Similar Questions

Explore conceptually related problems

If m is a positive integer and D_r=|(2r-1,\ ^m C_r,1),(m^2-1, 2^m ,m+1),(sin^2(m^2),sin^2(m),sin^2(m+1))| . Prove that sum_(r=0)^m D_r=0 .

Let m be a positive integer and Deltar = |[2r-1,.^mC_r,1],[m^2-1,2^m,m+1],[sin^2(m^2),sin^2m,sin(m^2)]| . Then the value of sum_(r=0)^m Deltar

If k a n d n are positive integers and s_k=1^k+2^k+3^k++n^k , then prove that sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+1)dot

If m in N and mgeq2 prove that: |1 1 1\ ^m C_1\ ^(m+1)C_1\ ^(m+2)C_1\ ^m C_2\ ^(m+1)C_2\ ^(m+2)C_2|=1 .

In Figure, /_1=60^0a n d\ /_2=(2/3)^(r d) of a right angle. Prove that l || m

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

In the coefficients of rth, (r+1)t h ,a n d(r+2)t h terms in the binomial expansion of (1+y)^m are in A.P., then prove that m^2-m(4r+1)+4r^2-2=0.

In the coefficients of rth, (r+1)t h ,a n d(r+2)t h terms in the binomial expansion of (1+y)^m are in A.P., then prove that m^2-m(4r+1)+4r^2-2=0.

Prove that sum_(r=1)^(m-1)(2r^2-r(m-2)+1)/((m-r)^m C_r)=m-1/mdot

If p+r=mq and (1)/(q)+(1)/(s)=(m)/(r ), then prove that : p:q=r:s .

RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c...

    Text Solution

    |

  2. Without expanding, show that the value of each of the determinants is ...

    Text Solution

    |

  3. If m is a positive integer and Dr=|2r-1\ ^m Cr1m^2-1 2^m m+1s in^2(...

    Text Solution

    |

  4. Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y...

    Text Solution

    |

  5. Let Deltar=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3...

    Text Solution

    |

  6. If Deltar=|[2^(r-1) , 2.3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 ,...

    Text Solution

    |

  7. Find a quadratic polynomial phi(x) whose zeros are the maximum and mi...

    Text Solution

    |

  8. Let f(x)=|[secx,cosx,sec^2x+cotxcosecx],[cos^2x,cos^2x,cosec^2x],[1,co...

    Text Solution

    |

  9. Show that: |[a,b-c,c+b],[a+c,b,c-a],[a-b,b+a,c]|=(a+b+c)(a^2+b^2+c^2).

    Text Solution

    |

  10. If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(...

    Text Solution

    |

  11. Solve: |a+x a-x a-x a-x a+x a-x a-x a-x a+x|=0

    Text Solution

    |

  12. Show that: |[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]|=3(a+b+c)(a b...

    Text Solution

    |

  13. 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-...

    Text Solution

    |

  14. Solve : |[x-2,2x-3,3x], [-4x-4,2x-9,3x-16], [x-8,2x-27,3x-64]| = 0

    Text Solution

    |

  15. If x+y+z=0 , prove that |[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y ...

    Text Solution

    |

  16. If a , b , c are all positive and are pth ,qth and rth terms of a G.P....

    Text Solution

    |

  17. Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-...

    Text Solution

    |

  18. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

    Text Solution

    |

  19. Prove that: |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|=2a...

    Text Solution

    |

  20. Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2

    Text Solution

    |