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

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.

Using elementary transformations, find the inverse of the matrix A =  `((8,4,3),(2,1,1),(1,2,2))`and use it to solve the following system of linear equations :

8x + 4y + 3z = 19

2x + y + z = 5

x + 2y + 2z = 7

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.

If the function f : R → R be given by f[x] = x² + 2 and g : R ​→ R be given by  `g(x)=x/(x−1)` , x≠1, find fog and gof and hence find fog (2) and gof (−3).

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.

Using properties of determinants, prove that :

`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`