What are the definitions of a function, the value of f(-2) for f(x) = 4x + 3, the maximum number of roots for a quadratic equation, and an identity matrix?

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. It describes how one quantity depends on another.

2)

Given $f(x) = 4x + 3$.
To find $f(-2)$, substitute $x = -2$:

The value of $f(-2)$ is $\boxed{-5}$.

3)

A quadratic equation is of the form $ax^2 + bx + c = 0$, where $a \neq 0$. The number of roots depends on the discriminant $\Delta = b^2 - 4ac$.

• If $\Delta > 0$, there are two distinct real roots.
• If $\Delta = 0$, there is one real root (a repeated root).
• If $\Delta < 0$, there are two distinct complex roots.

A quadratic equation can have at most two roots (real or complex).
A quadratic equation can have $\boxed{2 \text{ roots}}$.

4)

An identity matrix is a square matrix where all the elements on the main diagonal are $1$s and all other elements are $0$s. It is denoted by $I$ or $I_n$, where $n$ is the dimension of the square matrix. When an identity matrix multiplies another matrix, the other matrix remains unchanged.

5)

Matrix A has order $2 \times 3$.
Matrix B has order $3 \times 1$.
For matrix multiplication AB to be possible, the number of columns in A must be equal to the number of rows in B.
Number of columns in A = 3.
Number of rows in B = 3.
Since $3 = 3$, multiplication AB is possible.

If A is an $m \times n$ matrix and B is an $n \times p$ matrix, then the product AB is an $m \times p$ matrix.
For A ($2 \times 3$) and B ($3 \times 1$), the order of AB will be $2 \times 1$.
Yes, AB is possible. The order of matrix AB will be $\boxed{2 \times 1}$.

6)

The slope $m$ of a line with inclination $\theta$ is given by $m = \tan \theta$.

Given inclination $\theta = 135^\circ$.

The slope of the line is $\boxed{-1}$.

7)

The coordinates of point P $(x, y)$ that divides the join of R $(x_1, y_1)$ and S $(x_2, y_2)$ in the ratio $m_1:m_2$ externally are given by:

The coordinates of point P are $\boxed{\left( \frac{m_1 x_2 - m_2 x_1}{m_1 - m_2}, \frac{m_1 y_2 - m_2 y_1}{m_1 - m_2} \right)}$.

8)

Given $\vec{a} = (1, 2)$ and $\vec{b} = (4, 5)$.
In column vector form:

The sum $\vec{a} + \vec{b}$ in column vector form is $\boxed{\begin{bmatrix} 5 \\ 7 \end{bmatrix}}$.

9)

The Quartile Deviation is given by the formula:

where $Q_3$ is the third quartile and $Q_1$ is the first quartile.

Given $Q.D. = 30$ and $Q_1 = 10$.

Multiply both sides by 2:

Add 10 to both sides:

The value of $Q_3$ is $\boxed{70}$.

10)

A box and whisker plot (or box plot) is a standardized way of displaying the distribution of data based on five key numbers: the minimum value, the first quartile ($Q_1$), the median ($Q_2$), the third quartile ($Q_3$), and the maximum value. The "box" represents the interquartile range ($Q_1$ to $Q_3$), and the "whiskers" extend to the minimum and maximum values within a certain range.

11)

Given $f(2x - 3) = 4x + 5$.
Let $y = 2x - 3$.
Then $y + 3 = 2x$, so $x = \frac{y+3}{2}$.
Substitute this expression for $x$ into the function:

Replacing $y$ with $x$, we get:

Substitute $x = -2$ into the expression for $f(x)$:

So, $f(x) = \boxed{2x + 11}$ and $f(-2) = \boxed{7}$.

12)

We need to divide $4x^3 + 2x^2 - 4x + 3$ by $2x + 3$.

        2x^2  - 2x   + 1
      _________________
2x + 3 | 4x^3 + 2x^2 - 4x + 3
        -(4x^3 + 6x^2)
        _________________
              -4x^2 - 4x
            -(-4x^2 - 6x)
            _________________
                    2x + 3
                  -(2x + 3)
                  _________
                          0

The quotient is $2x^2 - 2x + 1$.
The remainder is 0.
The quotient is $\boxed{2x^2 - 2x + 1}$ and the remainder is $\boxed{0}$.

13)

The point is $(4, 5)$, so $x = 4$ and $y = 5$.
The equation of the locus is $x^2 + y^2 + kx - 8y - 25 = 0$.
Substitute $x=4$ and $y=5$:

The value of $k$ is $\boxed{6}$.

14)

We need to construct a $2 \times 2$ matrix, so it will have elements $a_{11}, a_{12}, a_{21}, a_{22}$.
The rule for elements is $a_{ij} = 2i - 3j$.

For $a_{11}$ ($i=1, j=1$):

For $a_{12}$ ($i=1, j=2$):

For $a_{21}$ ($i=2, j=1$):

For $a_{22}$ ($i=2, j=2$):

The matrix is:

The constructed matrix is $\boxed{\begin{bmatrix} -1 & -4 \\ 1 & -2 \end{bmatrix}}$.

15)
This question was solved in the previous turn. The values are $a=1$ and $b=4$.

16)

Given $\sin 2\theta = \cos \theta$.
Use the double angle identity for sine: $\sin 2\theta = 2 \sin \theta \cos \theta$.

Factor out $\cos \theta$:

Case 1: $\cos \theta = 0$
For $0^\circ \le \theta \le 90^\circ$, the value of $\theta$ for which $\cos \theta = 0$ is $\theta = 90^\circ$.

Case 2: $2 \sin \theta - 1 = 0$

For $0^\circ \le \theta \le 90^\circ$, the value of $\theta$ for which $\sin \theta = \frac{1}{2}$ is $\theta = 30^\circ$.

The solutions are $\theta = 30^\circ$ and $\theta = 90^\circ$.
The solutions are $\boxed{30^\circ, 90^\circ}$.