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

If y = (sec^-1 x )² , x > 0, show that 

`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`

If y = sin^-1 x + cos^-1x find  `(dy)/(dx)`.

Solve the differential equation: (1 + x²) dy + 2xy dx = cot x dx

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. 

If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`

If `"y" = (sin^-1 "x")^2, "prove that" (1 - "x"^2) (d^2"y")/(d"x"^2) - "x" (d"y")/(d"x") - 2 = 0`.

Solve the differential equation : `"x"(d"y")/(d"x") + "y" - "x" + "xy"cot"x" = 0; "x" != 0.`

If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`

Let R be the equivalence relation in the set Z of integers given by R = {(a, b): 2 divides a – b}. Write the equivalence class [0]

If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g

If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g

If A = {1, 2, 3} and f, g are relations corresponding to the subset of A × A indicated against them, which of f, g is a function? Why?
f = {(1, 3), (2, 3), (3, 2)}
g = {(1, 2), (1, 3), (3, 1)}

Let f: R → R be defined by f(x) = sin x and g: R → R be defined by g(x) = x 2 , then f o g is ______.

Let f, g: R → R be defined by f(x) = 2x + 1 and g(x) = x² – 2, ∀ x ∈ R, respectively. Then, find gof

If A = {a, b, c, d} and the function f = {(a, b), (b, d), (c, a), (d, c)}, write f^–1 

If the mappings f and g are given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write f o g.

If functions f: A → B and g: B → A satisfy gof = I_A, then show that f is one-one and g is onto