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

Evaluate:

`int x^2/(x^4+x^2-2)dx`

Let f : W → W be defined as

`f(n)={(n-1, " if n is odd"),(n+1, "if n is even") :}`

Show that f is invertible a nd find the inverse of f. Here, W is the set of all whole
numbers.

Find: `I=intdx/(sinx+sin2x)`

Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f · g)oh = (foh)·(goh)

Find gof and fog, if f(x) = |x| and g(x) = |5x – 2|.

Find gof and fog, if f(x) = 8x³ and `g(x) = x^(1/3)`.

If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3` show that fof(x) = x, for all `x ≠ 2/3`. What is the inverse of f?

State with reason whether following functions have inverse

f: {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}

State with reason whether following functions have inverse 

g: {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

State with reason whether following functions have inverse 

h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Show that f: [–1, 1] → R, given by f(x) = `x/(x + 2)`  is one-one. Find the inverse of the function f: [–1, 1] → Range f.

(Hint: For y in Range f, y = `f(x) = x/(x + 2)` for some x in [–1, 1] i.e., `x = (2y)/(1 - y)`)

Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Consider f: R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f⁻¹ of given f by `f^(-1)(y) = sqrt(y - 4)`, where R₊ is the set of all non-negative real numbers.

Consider f: R₊ → [–5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is invertible with `f^(-1)(y) = ((sqrt(y + 6) - 1)/3)`.

Let f: X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g₁ and g₂ are two inverses of f. Then for all y ∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f⁻¹ and show that (f⁻¹)⁻¹ = f.

Let f: X → Y be an invertible function. Show that the inverse of f⁻¹ is f, i.e., (f⁻¹)⁻¹ = f.

If f: R → R be given by `f(x) = (3 - x^3)^(1/3)`, then fof(x) is ______.

Let `f: R - {-4/3} → R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of f is map g: Range `f → R - {-4/3}` given by