Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Read the following passage:

Based on the above, answer the following questions:

1. Show that (x² – y²) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
2. Solve the above equation to find its general solution. (2)

Find the angle between the following two lines:

`vecr = 2hati - 5hatj + hatk + λ(3hati + 2hatj + 6hatk)`

`vecr = 7hati - 6hatk + μ(hati + 2hatj + 2hatk)`

Using the matrix method, solve the following system of linear equations:

`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.

If `y=sin^-1(3x)+sec^-1(1/(3x)), `  find dy/dx

Differentiate `tan^(-1)(sqrt(1-x^2)/x)` with respect to `cos^(-1)(2xsqrt(1-x^2))` ,when `x!=0`

If y = x^x, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`

Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`

Evaluate: `int(5x-2)/(1+2x+3x^2)dx`

Evaluate : ` int x^2/((x^2+4)(x^2+9))dx`

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`

Find the derivative of the following function f(x) w.r.t. x, at x = 1 : 

`f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x`

Show that the following two lines are coplanar:

`(x−a+d)/(α−δ)= (y−a)/α=(z−a−d)/(α+δ) and (x−b+c)/(β−γ)=(y−b)/β=(z−b−c)/(β+γ)`

if `y = sin^(-1)[(6x-4sqrt(1-4x^2))/5]` Find `dy/dx `.

Find:

`int(x^3-1)/(x^3+x)dx`

If the function f(x)=2x³−9mx²+12m²x+1, where m>0 attains its maximum and minimum at p and q respectively such that p²=q, then find the value of m.

Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`

Prove that `y=(4sintheta)/(2+costheta)-theta `

Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b