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Without expanding prove abs((bc, a, a^2)...

Without expanding prove `abs((bc, a, a^2),(ca, b, b^2),(ab, c, c^2))=abs((1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3))`

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Without expanding the determinants 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):}|

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

Prove without expanding that |[bc,a,a^2],[ca,b,b^2],[ab,c,c^2]|=|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|

Show that: abs((1,a,a^2),(1,b,b^2),(1,c,c^2))=(a-b)(b-c)(c-a)

Prove that |(1, a, a^3),(1, b, b^3),(1, c, c^3)| = (a-b)(b-c)(c-a)(a+b+c).

Show that: abs((a,a^2,a^3),(b,b^2,b^3),(c,c^2,c^3))=abc(a-b)(b-c)(c-a)

Prove that abs((a,b,c),(a^2,b^2,c^2),(bc,ca,ab))=(a-b)(b-c)(c-a)(ab+bc+ca)

Show that without expanding at any stage |{:(1/a,a^2,bc),(1/b,b^2,ca),(1/c,c^2,ab):}|=0

Using properties of determinants, prove the following abs{:(a^2, bc, ac +c^2 ),(a^(2) + ab, b^(2),ac ),(ab, b^(2) + bc,c^(2) ):}=4a^(2) b^(2) c^(2) .

USHA PUBLICATION-MODEL QUESTION SET-MODEL QUESTION SET
  1. Construct an example to show that f(A nn B )!= f(A) nn f(B) where A nn...

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  2. Express [(1,2,3),(4,0,1),(-1,5,-2)] as a sum of a symmetric and a skew...

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  3. Without expanding prove abs((bc, a, a^2),(ca, b, b^2),(ab, c, c^2))=ab...

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  4. Find the inverse of the matrix [[1,2],[3,4]]

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  5. Find A and B where 2A+B=[[2,2,5],[5,4,3],[1,1,4]]"and" A-2B=[[1,6,5],...

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  6. Solve for x, |{:(15-2x,11,10),(11-3x,17,16),(7-x,14,13):}|=0

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  7. Find absolute maximum and absolute minimum of f(x)={((x+1)^2,xle0),((x...

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  8. If x^7y^3=(x+y)^10, then find (d^2y)/(dx^2)

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  9. If y=e^(ax)sinbx show that y2-2ay1+(a^2+b^2)y=0.

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  10. Find the equation of tangent to the curve x=y^2-2 at the points where ...

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  11. Find the intervals in which the function y=(lnx)/(x) is increasing and...

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  12. Solve dy/dx=x+y

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  13. int(2sinx+3cosx)/(3sinx+4cosx)dx=?

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  14. Solve (dy)/(dx)=(y^2)/(xy-x^2).

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  15. Integrate int(-3//5)^(3//5)[2x+1]dx

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  16. Evaluate int0^1(x^5)/sqrt(1-x^2) dx

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  17. Find the co-ordinates of the point where the perpendicular from the or...

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  18. Resolve the vector vec(b) = hat(i) + hat(j) + hat(k) into vectors para...

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  19. Find the equation of the plane passing through the line x = y = z and ...

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  20. Find the equation of the line through (-1,0,1) and perpendicular to th...

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