PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}- 1 & 4 \\ 2 & 3\end{bmatrix}\]

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{bmatrix}\]

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{bmatrix}\]

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1\end{bmatrix}\]

 \[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}a & h & g \\ h & b & f \\ g & f & c\end{bmatrix}\]

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}2 & - 1 & 0 & 1 \\ - 3 & 0 & 1 & - 2 \\ 1 & 1 & - 1 & 1 \\ 2 & - 1 & 5 & 0\end{bmatrix}\]

The function f : R → R defined by

`f (x) = 2^x + 2^(|x|)` is 

Let the function

\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]

\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]

The function 

f : A → B defined by 

f (x) = - x² + 6x - 8 is a bijection if 

Let 

\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is 

Let

f : R → R be given by

\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]

where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
 

(d) one-one and onto

Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is

The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]

The range of the function

\[f\left( x \right) =^{7 - x} P_{x - 3}\]

A function f  from the set of natural numbers to integers defined by

`{([n-1]/2," when  n is  odd"   is ),(-n/2,when  n  is  even ) :}`

Let f be an injective map with domain {x, y, z} and range {1, 2, 3}, such that exactly one of the following statements is correct and the remaining are false.

\[f\left( x \right) = 1, f\left( y \right) \neq 1, f\left( z \right) \neq 2 .\]

The value of

\[f^{- 1} \left( 1 \right)\] is 

Which of the following functions form Z to itself are bijections?

Which of the following functions from

to itself are bijections?

Let

\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]