Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

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`

Use elementary column operation C₂ → C₂ + 2C₁ in the following matrix equation :

`[[2,1],[2,0]] = [[3,1],[2,0]] [[1,0],[-1,1]]`

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

For what values of k, the system of linear equations

x + y + z = 2
2x + y – z = 3
3x + 2y + kz = 4

has a unique solution?

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 `

If `A=|[2,0,-1],[5,1,0],[0,1,3]|` , then find A^-1 using elementary row operations

Using the properties of determinants, solve the following for x:

`|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=0`

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).

Prove that  `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.