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

`int (dx)/(x(x^2 + 1))` equals:

Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`

Consider f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`. Show that f is invertible with `f^(-1) (y) ((sqrt(y + 6)-1)/3)`

Hence Find

1) `f^(-1)(10)`

2) y if `f^(-1) (y) = 4/3`

where R₊ is the set of all non-negative real numbers.

Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`

Find : 

`∫ sin(x-a)/sin(x+a)dx`

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g (x) = αx + β, then what value should be assigned to α and β

Let f: R → R be defined by f(x) = 3x 2 – 5 and g: R → R by g(x) = `x/(x^2 + 1)` Then gof is ______.

Let f: A → B and g: B → C be the bijective functions. Then (g o f)^–1 is ______.

Let f: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",",  "if"  x  "is rational"),(1 - x",",  "if"  x  "is irrational"):}`. Then (f o f) x is ______.

Let f: N → R be the function defined by f(x) = `(2x - 1)/2` and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) `3/2` is ______.

Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.

Let f: R → R be the function defined by f(x) = sin (3x+2) ∀ x ∈ R. Then f is invertible.

The composition of functions is commutative.

The composition of functions is associative.

Every function is invertible.

Verify the following using the concept of integration as an antiderivative

`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`

Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x² = t

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`

Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin x)`