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

Consider f: R₊ → [–5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is invertible with `f^(-1)(y) = ((sqrt(y + 6) - 1)/3)`.

Let f: X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g₁ and g₂ are two inverses of f. Then for all y ∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f⁻¹ and show that (f⁻¹)⁻¹ = f.

Let f: X → Y be an invertible function. Show that the inverse of f⁻¹ is f, i.e., (f⁻¹)⁻¹ = f.

If f: R → R be given by `f(x) = (3 - x^3)^(1/3)`, then fof(x) is ______.

Let `f: R - {-4/3} → R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of f is map g: Range `f → R - {-4/3}` given by

Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

If f: R → R is defined by f(x) = x2 − 3x + 2, find f(f(x)).

Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.

Examine the consistency of the system of equations.

x + 2y = 2

2x + 3y = 3

Examine the consistency of the system of equations.

2x − y = 5

x + y = 4

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Examine the consistency of the system of equations.

x + y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

Solve the system of linear equations using the matrix method.

5x + 2y = 4

7x + 3y = 5

Solve the system of linear equations using the matrix method.

2x – y = –2

3x + 4y = 3

Solve the system of linear equations using the matrix method.

4x – 3y = 3

3x – 5y = 7

Solve the system of linear equations using the matrix method.

5x + 2y = 3

3x + 2y = 5

Solve the system of linear equations using the matrix method.

2x + y + z = 1

x – 2y – z = `3/2`

3y – 5z = 9