Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`

Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t²)

If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.

If f(x) = (ax² + b)³, then the function g such that f(g(x)) = g(f(x)) is given by ____________.

If f : R → R, g : R → R and h : R → R is such that f(x) = x², g(x) = tanx and h(x) = logx, then the value of [ho(gof)](x), if x = `sqrtpi/2` will be ____________.

Let f : N → R : f(x) = `((2"x"−1))/2` and g : Q → R : g(x) = x + 2 be two functions. Then, (gof) `(3/2)` is ____________.

If f : R → R, g : R → R and h : R → R are such that f(x) = x², g(x) = tan x and h(x) = log x, then the value of (go(foh)) (x), if x = 1 will be ____________.

If f(x) = `(3"x" + 2)/(5"x" - 3)` then (fof)(x) is ____________.

Let f : R → R be the functions defined by f(x) = x³ + 5. Then f^-1(x) is ____________.

Let f : R – `{3/5}`→ R be defined by f(x) = `(3"x" + 2)/(5"x" - 3)` Then ____________.

If f(x) = (ax² – b)³, then the function g such that f{g(x)} = g{f(x)} is given by ____________.

Which one of the following functions is not invertible?

The inverse of the function `"y" = (10^"x" - 10^-"x")/(10^"x" + 10^-"x")` is ____________.

If f : R → R defind by f(x) = `(2"x" - 7)/4` is an invertible function, then find f^-1.

Consider the function f in `"A = R" - {2/3}` defiend as `"f"("x") = (4"x" + 3)/(6"x" - 4)` Find f^-1.

If f is an invertible function defined as f(x) `= (3"x" - 4)/5,` then f^-1(x) is ____________.

If f : R → R defined by f(x) `= (3"x" + 5)/2` is an invertible function, then find f^-1.

Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).

If ` abs((1 + "a"^2 "x", (1 + "b"^2)"x", (1 + "c"^2)"x"),((1 + "a"^2) "x", 1 + "b"^2 "x", (1 + "c"^2) "x"), ((1 + "a"^2) "x", (1 + "b"^2) "x", 1 + "c"^2 "x"))`, then f(x) is apolynomial of degree ____________.

`abs (("a"^2, 2"ab", "b"^2),("b"^2, "a"^2, 2"ab"),(2"ab", "b"^2, "a"^2))` is equal to ____________.