PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

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

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`

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.

Find the integrating factor of the differential equation.

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

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`

Prove that :

x + y + z + w = 2
x − 2y + 2z + 2w = − 6
2x + y − 2z + 2w = − 5
3x − y + 3z − 3w = − 3

2x − 3z + w = 1
x − y + 2w = 1
− 3y + z + w = 1
x + y + z = 1

2x − y = 5
4x − 2y = 7

Differentiate the following functions from first principles e^−x.

Differentiate the following functions from first principles e^3x.

Differentiate the following functions from first principles e^ax+b.

Differentiate the following functions from first principles e^cos x.

Differentiate the following functions from first principles  \[e^\sqrt{2x}\].

Differentiate the following functions from first principles log cos x ?

​Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .