SIMAK UI · Basic Mathematics (Matematika Dasar)

Functions, Matrices and Sequences

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If $f (x) = \frac{2 x + 1}{x - 3}$ for $x \neq = 3$ , determine the inverse function $f^{- 1} (x)$ .
Given matrices $A = (3512)$ and $B = (2 - 5 - 1 3)$ , find the product matrix $A B$ .
The sum of an infinite geometric series with first term $a = 12$ is $S = 16$ . Find the common ratio $r$ of the series.
If $f (x) = 3 x - 2$ and $(g \circ f) (x) = 9 x^{2} - 12 x + 7$ , determine the function $g (x)$ .
For which value of $k$ does the matrix $M = (k 12 3 k)$ fail to be invertible (singular)?
In an arithmetic progression, the 4th term is 17 and the 10th term is 41. Find the sum of the first 20 terms ( $S_{20}$ ).
If matrix $A = (1324)$ , compute the trace and determinant of matrix $A^{2}$ .
If $f (x) = x^{2} - 4 x + 5$ with domain $x \geq 2$ , find the inverse function $f^{- 1} (x)$ and its domain.
Three positive numbers form a geometric progression with common ratio $r > 1$ . If their product is 216 and their sum is 26, find the common ratio $r$ .
If matrix $P = (2134)$ , find the matrix $X$ satisfying $P X = (76)$ .
The sum to infinity of a geometric progression is 9 and the sum of the squares of its terms is 27. Find the first term $a$ and common ratio $r$ .
If $f (x) = 2 x - 3$ and $g (x) = \frac{x + 4}{2}$ , evaluate $(f^{- 1} \circ g^{- 1}) (3)$ .
Determine the determinant of the $3 \times 3$ matrix $K = 100240356$ .
If the sequence $a_{n}$ satisfies the recurrence $a_{1} = 3$ and $a_{n + 1} = 2 a_{n} + 1$ for all $n \geq 1$ , find the explicit formula for $a_{n}$ .
If $A = (cos θ sin θ - sin θ cos θ)$ , what is matrix $A^{n}$ for any positive integer $n$ ?
The sum of the first $n$ terms of an arithmetic progression is $S_{n} = 2 n^{2} + 5 n$ . Find the 10th term ( $U_{10}$ ).
If $f (x)$ is an odd function and $g (x)$ is an even function, which of the following composite or product functions is necessarily an ODD function?
Given matrix $A = (21 - 1 3)$ , compute the determinant of $3 A$ .
In a geometric sequence, $U_{2} = 6$ and $U_{5} = 162$ . Find the 7th term ( $U_{7}$ ).
If $f (x + 1) = 2 x^{2} + 3 x - 1$ , find the expression for $f (x - 1)$ .
If matrix $M = (a c b d)$ satisfies $M^{2} = 0$ , which condition MUST hold?
Find the sum of all integers between 100 and 300 that are divisible by 7.
If $f (x) = lo g_{2} x$ , solve the equation $f (x) + f (x - 2) = 3$ .
Given eigenvalues $λ_{1} = 2$ and $λ_{2} = 5$ for a $2 \times 2$ matrix $B$ , what is the value of $det (B^{- 1})$ ?
Determine the value of the infinite sum $\sum_{n = 1}^{\infty} \frac{1}{n ( n + 1 )}$ .

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