Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

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

Evaluate:

`int((x+3)e^x)/((x+5)^3)dx`

Show that lines: 

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar 

Also, find the equation of the plane containing these lines.

Let f : N→N be a function defined as f(x)=`9x^2`+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find `f^-1`(43) and` f^−1`(163).

If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x²≤1, then find dy/dx.