### Group 'A'
1) Define inverse function.
An inverse function reverses the input-output relation of the original function. If [tex]\( f(x) \)[/tex] is a function, then its inverse [tex]\( f^{-1}(x) \)[/tex] satisfies the conditions:
[tex]\[
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
\][/tex]
2) If (x-2) is a factor of a polynomial [tex]\( f(x) \)[/tex] of degree 3, what is the value of the remainder?
According to the factor theorem, since [tex]\( (x-2) \)[/tex] is a factor of the polynomial [tex]\( f(x) \)[/tex], the value of the remainder is zero. Therefore, the remainder is:
[tex]\[
0
\][/tex]
3) State factor theorem.
The factor theorem states that a polynomial [tex]\( f(x) \)[/tex] has a factor [tex]\( (x - c) \)[/tex] if and only if [tex]\( f(c) = 0 \)[/tex]. In other words, [tex]\( c \)[/tex] is a root of the polynomial [tex]\( f(x) \)[/tex].
4) What is the determinant of unit matrix of the order 2x2?
A unit matrix (also known as an identity matrix) is a square matrix with ones on the main diagonal and zeros elsewhere. For a 2x2 unit matrix:
[tex]\[
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\][/tex]
The determinant of this 2x2 unit matrix is:
[tex]\[
1
\][/tex]
5) If matrix B is the inverse matrix of matrix A, what is the relation between A and B?
If matrix [tex]\( B \)[/tex] is the inverse of matrix [tex]\( A \)[/tex], then the product of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in either order (i.e., [tex]\( A \times B \)[/tex] and [tex]\( B \times A \)[/tex]) is the identity matrix [tex]\( I \)[/tex]:
[tex]\[
A \times B = B \times A = I
\][/tex]
6) If [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are the first term and last term of an A.P, having [tex]\( r \)[/tex] terms, what is the sum?
The sum [tex]\( S \)[/tex] of an arithmetic progression (A.P) is given by the formula:
[tex]\[
S = \frac{r}{2} \times (p + q)
\][/tex]
where [tex]\( p \)[/tex] is the first term, [tex]\( q \)[/tex] is the last term, and [tex]\( r \)[/tex] is the number of terms.
7) Give an example of arithmetic series.
An example of an arithmetic series with a common difference of 3 is:
[tex]\[
2, 5, 8, 11, 14
\][/tex]
8) If there are 5 arithmetic means between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], how many terms are there in the given A.P?
If there are 5 arithmetic means between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], then the total number of terms including [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
7
\][/tex]
