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

If | A | = | kA |, where A is a square matrix of order 2, then sum of all possible values of k is ______.

Find the general solution of the differential equation:

(xy – x²) dy = y² dx

Read the following passage:

Based on the above, answer the following questions:

1. Show that (x² – y²) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
2. Solve the above equation to find its general solution. (2)

Write the number of vectors of unit length perpendicular to both the vectors `veca=2hati+hatj+2hatk and vecb=hatj+hatk`

Find : `int x^2/(x^4+x^2-2) dx`

If `veca=4hati-hatj+hatk` then find a unit vector parallel to the vector `veca+vecb`

If the function f : R → R be defined by f(x) = 2x − 3 and g : R → R by g(x) = x³ + 5, then find the value of (fog)⁻¹ (x).

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.