BE Automobile Engineering सत्र १ (इंजीनियरिंग) - University of Mumbai Important Questions

Find all values of `(1+i)^(1/3)` & show that their continued
Product is (1+i).

If `y=2^xsin^2x cosx` find `y_n`

If u=`f((y-x)/(xy),(z-x)/(xz)),"show that"  x^2 (del_u)/(del_x)+y^2 (del_u)/(del_y)+x^2 del_u/del_z=0`

Show that the roots of x⁵ =1 can be written as 1, `alpha^1,alpha^2,alpha^3,alpha^4` .hence show that `(1-alpha^1) (1-alpha^2) (1-alpha^3)(1-alpha^4)=5.`

If `u=x^2+y^2+z^2` where `x=e^t, y=e^tsint,z=e^tcost`

Prove that `(du)/(dt)=4e^(2t)`

Expand `2x^3+7x^2+x-6` in powers of (x-2)

Using De Moivre’s theorem prove that]

`cos^6theta-sin^6theta=1/16(cos6theta+15cos2theta)`

If `tan(x/2)=tanh(u/2),"show that" u = log[(tan(pi/4+x/2))] `

Solve  `x^4-x^3+x^2-x+1=0.`

Show that all roots of `(x+1)^6+(x-1)^6=0` are given by -icot`((2k+1)n)/12`where k=0,1,2,3,4,5.

Prove that `sin^5theta=1/16[sin5theta-5sin3theta+10sintheta]`

Expand `2x^3+7x^2+x-1` in powers of x - 2

Find all values of `(1 + i)^(1/3` and show that their continued product is (1+ 𝒊 ).

By using De Moivre's Theorem obtain tan 5θ in terms of tan θ and show that `1-10 tan^2(pi/10)+5tan^4(pi/10)=0`.

If y=(x+√x²-1 ,Prove that

`(x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0`

Find the n^th derivative of `x^3/((x+1)(x-2))`

Find the nth derivative of cos 5x.cos 3x.cos x.

If `y=e^(tan^(-1)x)`.Prove that

`(1+x^2)y_(n+2)+[2(n+1)x-1]y_(n+1)+n(n+1)y_n=0`

Evaluate `lim_(x->0) sinx log x.`

If U = `e^(xyz) f((xy)/z)` prove that `x(delu)/(delx)+z(delu)/(delx)2xyzu` and `y(delu)/(delx)+z(delu)/(delz)=2xyzu` and hence show that `x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`