Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`

Evaluate :   `∫1/(cos^4x+sin^4x)dx`

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the two-headed coin?

Evaluate :

`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`

Find the distance between the planes 2x - y +  2z = 5 and 5x - 2.5y + 5z = 20

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 co-domain being 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 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.