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Prove [[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[...

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)`

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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: [[(b+c)^2,a^2,bc],[(c+a)^2,b^2,ca],[(a+b)^2,c^2,ab]] = (a^2+b^2+c^2)(a+b+c)(b-c)(c-a)(a-b)

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.

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 the following: [[1,1,1],[a,b,c],[a^3,b^3,c^3]] =(b-c)(c-a)(a-b)(a+b+c)

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

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

Show that [[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]] = 3(a+b+c)(ab+bc+ca)

Prove that: |[1, 1, 1],[a, b, c],[a^2, b^2, c^2]|=(a-b)(b-c)(c-a)

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

MODERN PUBLICATION-TEST PAPER 8-EXERCISE
  1. If y=e^(ax)sinbx show that y2-2ay1+(a^2+b^2)y=0.

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  2. Prove the inequality x^2e^(-x^2) le e^(-1),x in R.

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  3. Find the interval in which y=(x-1)^2(x+2) is : (i) increasing (ii)...

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  4. Evaluate : int dx/(x(x^4+1))

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  5. Evaluate : int dx/sqrt (x^2+8x)

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  6. Evaluate : int(0)^(pi/2) e^x cos x dx .

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  7. From the differential equation whose general solution is y=asint+be^t.

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  8. Find the unit vector perpendicular to the plane ABC, where the positio...

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  9. Show that (vec(a)-vec(b))xx(vec(a)+vec(b))=2(vec(a)xxvec(b)).

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  10. The direction cosines of a straight line in two neighbouring positions...

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  11. Find the equation of the plane Passing through the intersection of the...

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  12. Find the equation of the straight line perpendicular to the line (x-2)...

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  13. Exaniming consistency and solvability, solve the following equations b...

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  14. 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...

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  15. (i) If x=sin t, y=sin 2t then prove that (1-x^2)d^2y/dx^2-xdy/dx+4y=0....

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  16. Show that the radius of the right cicular cylinder of greatest curved ...

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  17. Find the solution of the differential equation x sin (y/x)dy =(y sin...

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  18. Find the area of the region included between the parabola y^(2)=2x and...

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  19. A variable plane is at a constant distance p from the origin and meets...

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  20. (i) If the vectors vec a+vec b+vec c = 0 |vec a|=3,|vec b|=5 and |vec ...

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