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

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1 when x = 0.

If for any 2 x 2 square matrix A, `A("adj"  "A") = [(8,0), (0,8)]`, then write the value of |A|

Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`

Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R

Find `int dx/(5 - 8x - x^2)`

Show that the function f : R_* → R_* defined by f(x) = `1/x` is one-one and onto, where R_* is the set of all non-zero real numbers. Is the result true if the domain R_* is replaced by N, with the co-domain being the same as R?

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : R → R given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x³

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x³

Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Show that the modulus function f : R → R given by f(x) = |x| is neither one-one nor onto, where |x| is x if x is positive or 0 and |x| is − x if x is negative.

Show that the Signum Function f : R → R, given by `f(x) = {(1", if"  x > 0), (0", if"  x  = 0), (-1", if"  x < 0):}` is neither one-one nor onto.

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

In the following case, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 3 − 4x

In the following case, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 1 + x²

Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is a bijective function.

Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.

State whether the function f is bijective. Justify your answer.

Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by f(x) = `((x- 2)/(x -3))`. Is f one-one and onto? Justify your answer.