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|{:(1,a,bc),(1,b,ca),(1,c,ab):}|...

`|{:(1,a,bc),(1,b,ca),(1,c,ab):}|`

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`= ( a-b)(b-c) (c-a) `
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|{:(1,1,1),(a,b,c),(bc,ca,ab):}|=(a-b)(b-c)(c-a)

Without expanding prove that, |{:(1,bc,b+c),(1,ca,c+a),(1,ab,a+b):}|=|{:(1,a,a^(2)),(1,b,b^(2)),(1,c,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,b,1),(b,c,1),(c,a,1):}|=2010 and if |{:(c-a,c-b,ab),(a-b,a-c,bc),(b-c,b-a,ca):}|-|{:(c-a,c-b,c^(2)),(a-b,a-c,a^(2)),(b-c,b-a,b^(2)):}|=p , then the number of positive divisors of p is

|{:(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b)):}|=0

Evalute |{:(0,c,b),(c,0,a),(b,a,0):}| and hence show that, |{:(" "-a^2," "ab," "ac),(" "ab," "-b^2," "bc),(" "ca," "bc," "-c^2):}|=4a^2b^2c^2

|{:(1,bc,bc(b+c)),(1,ca,ca(c+a)),(1,ab,ab(a+b)):}|=0

If D=|(1,a,a^(2)-bc),(1,b,b^(2)-ca),(1,c,c^(2)-ab)| , then D is -

Without expanding prove that , |{:(1,ab,1/a+1/b),(1,bc,1/b+1/c),(1,ca,1/c+1/a):}|=0

Without expanding prove that |{:(1,a,a^(2)-bc),(1,b,b^(2)-ca),(1,c,c^(2)-ab):}|=0

NCERT BANGLISH-DETERMINANTS -Miscellaneous Exercises on Chapter 4
  1. |{:(1,a,bc),(1,b,ca),(1,c,ab):}|

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  2. Prove that the determinant {:|( x, sin theta ,cos theta ),( -sin thet...

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  3. Without expanding the determinant, prove that {:|( a, a ^(2), bc ),( ...

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  4. Evaluate {:|( cos alpha cos beta , cos alpha sin beta , -sin alpha ),...

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  5. If a,b and c are real numbers, and Delta ={:|( b+c,C+a,a+b),( c+a,a...

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  6. Solve the equation {:|( x+a,x,x),(x,x+a,x),(x,x,x+a) |:}=0,ane 0

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  7. Prove that {:|( a^(2) , bc, ac+c^(2)),( a^(2) +ab,b^(2) ,ac),( ab,b^(2...

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  8. If A^(-1) ={:[( 3,-1,1),(-15,6,-5),(5,-2,2) ]:}and B= {:[( 1,2,-2),( -...

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  9. Evaluate {:|( x,y , x+y),( y,x+y,x),( x+y,x,y)|:}

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  10. Evaluate {:|(1,x,y),( 1,x+y,y),( 1,x,x+y)|:}

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  11. Using properties of determinants in Exercises prove that : {:|( alph...

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  12. {:|( x,x^(2) , 1+ px^(3) ),( y,y^(2) , 1+ py^(2)),( z,z^(2) , 1+pz^(2)...

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  13. {:|( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) |:} = 3( a+b+c) (...

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  14. {:|( 1,1+p,1+p+q),( 2,3+2p,4+3p+2q),( 3,6+3p,10+6p+3q)|:}=1

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  15. {:|( sin alpha , cos alpha ,cos (alpha +delta) ),( sinbeta, cos beta,c...

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  16. Solve the system of equations (2)/( x) + ( 3)/( y) +(10 )/( z) = 4 ...

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  17. If a,b,c are in A.P. then the determinant {:|( x+2,x+3,x+2a),( x+3,...

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  18. If x,y,z are nonzero real number , then the inverse of matrix A= {:[( ...

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  19. Let A={:[( 1,sin theta , 1),( -sin theta , 1, sin theta ),( -1 ,-sin ...

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