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

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Find the differential equation representing the curve y = cx + c².

Solve for x : tan^-1 (x - 1) + tan^-1x + tan^-1 (x + 1) = tan^-1 3x

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

If y = P e^ax + Q e^bx, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

If A = `[(alpha, beta),(gamma, -alpha)]` is such that A2 = I then ______.

Differentiate the function with respect to x:

`(5x)^(3cos 2x)`

Give an example of a function which is one-one but not onto ?

Give an example of a function which is not one-one but onto ?

Give an example of a function which is neither one-one nor onto ?

Which of the following functions from A to B are one-one and onto?
 f₁ = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}

Which of the following functions from A to B are one-one and onto?

 f₂ = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}

 Which of the following functions from A to B are one-one and onto ?  

f₃ = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}. 

Prove that the function f : N → N, defined by f(x) = x² + x + 1, is one-one but not onto

Let A = {−1, 0, 1} and f = {(x, x²) : x ∈ A}. Show that f : A → A is neither one-one nor onto.

Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x²

Classify the following function as injection, surjection or bijection :  f : Z → Z given by f(x) = x²

Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x³

Classify the following function as injection, surjection or bijection :  f : Z → Z given by f(x) = x³