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

If a, b and c are real numbers, and triangle =`|(b+c, c+a, a+b),(c+a,a+b, b+c),(a+b, b+c, c+a)|` = 0 Show that either a + b + c = 0 or a = b = c.

Solve the equations `|(x+a,x,x),(a,x+a,x),(x,x,x+a)| = 0, a != 0`

Prove that `|(a^2, bc, ac+c^2),(a^2+ab, b^2, ac),(ab, b^2+bc, c^2)| = 4a^2b^2c^2`

Choose the correct answer.

If a, b, c, are in A.P., then the determinant

`|(x+2, x+3,x +2a),(x+3,x+4,x+2b),(x+4,x+5,x+2c)|`

A. 0

B. 1

C. x

D. 2x

Prove that the function f(x) = xⁿ is continuous at x = n, where n is a positive integer.

Is the function f defined by f(x) = `{(x", if"  x<=1),(5", if"  x > 1):}`  continuous at x = 0? At x = 1? At x = 2?

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(2x + 3", if"  x<=2),(2x - 3", if"  x > 2):}`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(|x|+3", if"  x<= -3),(-2x", if" -3 < x < 3),(6x + 2", if"  x >= 3):}`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(|x|/x", if"  x != 0),(0", if"  x = 0):}`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x/|x|", if"  x<0),(-1", if"  x >= 0):}`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x+1", if"  x>=1),(x^2+1", if"  x < 1):}`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x^3 - 3", if"  x <= 2),(x^2 + 1", if"  x > 2):}`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x^10 - 1", if"  x<=1),(x^2", if"  x > 1):}`

Is the function defined by f(x) = `{(x+5", if"  x <= 1),(x -5", if"  x > 1):}` a continuous function?

Show that the function defined by g(x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Find the points of discontinuity of f, where f(x) = `{(sinx/x", if"  x<0),(x + 1", if"  x >= 0):}`.

Determine if f defined by f(x) = `{(x^2 sin  1/x", if"  x != 0),(0", if"  x = 0):}` is a continuous function?

Examine the continuity of f, where f is defined by:

f(x) = `{(sin x - cos x", if"  x != 0),(-1", if"  x = 0):}`

Using mathematical induction prove that  `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.