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Factorise the determinant |[x^3-a^3,x^2,...

Factorise the determinant `|[x^3-a^3,x^2,x],[b^2-a^3,b^2,b],[c^3-a^3,c^2,c]|` without expanding.

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Prove [[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[c^3-x^3,c^2,c]] = (a-b)(a-c)(b-c)(abc-x^3)

Prove that the following. [[a,a^2,a^3],[b,b^2,b^3],[c,c^2,c^3]] = abc(a-b)(b-c)(c-a)

Prove the following: [[a,b,c],[b,c,a],[c,a,b]]=3abc-a^3-b^3-c^3

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

If 2s=a+b+c show that [[a^2,(s-a)^2,(s-a)^2],[(s-b)^2,b^2,(s-b)^2],[(s-c)^2,(s-c)^2,c^2]] = 2s^3(s-a)(s-b)(s-c)

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):}|

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]|

Prove that |[a-b-c,2a,2a],[2b, b-c-a, 2b],[2c, 2c, c-a-b]|= (a+b+c)^3 .

Without expanding,show that the following determinant vanishes. abs((b^2c^2,bc,b+c),(c^2a^2,ca,c+a),(a^2b^2,ab,a+b))

Prove that the following. [[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=(a+b+c)^3

SHARAM PUBLICATION-MODEL QUESTION PAPER 12-EXERCISE
  1. If the matrix A is such that [[1,-1],[2,3]]A=[[-4,1],[7,7]], find A.

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  2. Verify that A=[[a,b],[c,d]] satisfies the equation A^2-(a+d)A+(ad-bc...

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  3. Factorise the determinant |[x^3-a^3,x^2,x],[b^2-a^3,b^2,b],[c^3-a^3,c^...

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  4. Factorize the following. [[x+a,b,c],[b,x+c,a],[c,a,x+b]]

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  5. If A=[{:(3,1),(-1,2):}] then prove that A^(2)-5A+7I=O

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  6. Find the value of k if the function f(x) defined by f(x)={(2x-1,whenx<...

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  7. If sin y = x sin (a + y) then show that (dy)/(dx)=(sin^2(a+y))/(sina)

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  8. If y^2cot x = x^2 coty then find (dy)/(dx)

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  9. Find the intervals where function is increasing function y = cos x + s...

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  10. Find the equation of the normal to the curve 5x^2+3y^2=23 at (2,-1)

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  11. Integrate int sec xtanx.sqrt(tan^2x-3).dx

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  12. int(xe^x)/(1+x^2)dx

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  13. Evaluate int0^1[tan^-1x+x/(1+x^2)]dx

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  14. Find the area of region enclosed by y^2 = 4ax and x^2 = 4ay.

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  15. Find the particular solution of the following differential equation : ...

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  16. If the sum of two unit vectors is a unit vectors find the magnitude of...

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  17. If hati +hatj+hatk and 2hati -alphahatj+3hatk are orthogonal to each o...

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  18. Find the equation of the line through the point (2, 3, 5) and parallel...

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  19. Prove that the lines (x+4)/3=(y+6)/5=(z-1)/(-2) and 3x-2y+z+5=0=2x+3y...

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  20. Solve : tan^(-1)x+tan^(-1)((2x)/(1-x^2))=pi/2

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