Let's solve the given problems step-by-step:
### Problem 3:
Given that [tex]\( r = \binom{4a}{3} \)[/tex], [tex]\( q = \binom{5}{-4} \)[/tex], and [tex]\( r \cdot q = -2 \)[/tex], we need to find the value of [tex]\( a \)[/tex].
1. Understanding the Binomial Coefficient [tex]\( q \)[/tex]:
[tex]\[
q = \binom{5}{-4}
\][/tex]
The binomial coefficient [tex]\( \binom{n}{k} \)[/tex] is defined for non-negative integers [tex]\( n \)[/tex] and [tex]\( k \)[/tex]. If [tex]\( k \)[/tex] is negative, the binomial coefficient is zero:
[tex]\[
\binom{5}{-4} = 0
\][/tex]
2. Analyzing the Given Equation:
Since [tex]\( q = 0 \)[/tex], we substitute [tex]\( q \)[/tex] into [tex]\( r \cdot q \)[/tex]:
[tex]\[
r \cdot q = r \cdot 0 = 0
\][/tex]
However, it is given that [tex]\( r \cdot q = -2 \)[/tex]. This leads to a contradiction because any number multiplied by zero should result in zero, not [tex]\(-2\)[/tex]. Therefore, there seems to be an inconsistency in the problem as stated, and it cannot be solved with the given information.
### Problem 4:
Given two vectors:
[tex]\[
y = \begin{pmatrix}
3 - x \\
5x - 2
\end{pmatrix}, \quad z = \begin{pmatrix}
-1 \\
-1
\end{pmatrix}
\][/tex]
we need to find the value of [tex]\( x \)[/tex] such that the dot product [tex]\( y \cdot z = -9 \)[/tex].
1. Compute the Dot Product:
The dot product of two vectors [tex]\( y \)[/tex] and [tex]\( z \)[/tex] is given by:
[tex]\[
y \cdot z = (3 - x)(-1) + (5x - 2)(-1)
\][/tex]
2. Expand and Simplify:
[tex]\[
y \cdot z = - (3 - x) - (5x - 2)
\][/tex]
[tex]\[
y \cdot z = -3 + x - 5x + 2
\][/tex]
[tex]\[
y \cdot z = -3 + 2 + x - 5x
\][/tex]
[tex]\[
y \cdot z = -1 - 4x
\][/tex]
3. Set Equal to the Given Value:
Given [tex]\( y \cdot z = -9 \)[/tex], we substitute and solve for [tex]\( x \)[/tex]:
[tex]\[
-1 - 4x = -9
\][/tex]
[tex]\[
-1 - 4x + 1 = -9 + 1
\][/tex]
[tex]\[
-4x = -8
\][/tex]
[tex]\[
x = 2
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{2}\)[/tex].
### Final Answers:
3. The value of [tex]\( a \)[/tex] cannot be determined due to an inconsistency in the provided information.
4. The value of [tex]\( x \)[/tex] is [tex]\(\boxed{2}\)[/tex].
