Determine the value of q if a=2, b=-1 and c=-4. Leave your answer in simplest surd form. State whether q is rational or irrational. Between which TWO consecutive integers does q lie?

Here are the solutions for Question 1 from the first section of the image (DBE NOV 19 Q1):

1.1 Factorise the following expressions fully:

1.1.1 $3y^2+y$ Step 1: Identify the common factor. $3y^2+y = y(3y+1)$ The fully factorised expression is $\boxed{y(3y+1)}$.

1.1.2 $x^2-10x-24$ Step 1: Find two numbers that multiply to $-24$ and add to $-10$. These numbers are $2$ and $-12$. Step 2: Factor the quadratic trinomial. $x^2-10x-24 = (x+2)(x-12)$ The fully factorised expression is $\boxed{(x+2)(x-12)}$.

1.1.3 $9x^2-y^2+10y-25$ Step 1: Group the terms involving $y$ and factor out $-1$. $9x^2-(y^2-10y+25)$ Step 2: Recognise the perfect square trinomial $y^2-10y+25 = (y-5)^2$. $9x^2-(y-5)^2$ Step 3: Apply the difference of squares formula $A^2-B^2=(A-B)(A+B)$, where $A=3x$ and $B=(y-5)$. $(3x-(y-5))(3x+(y-5))$ Step 4: Simplify the terms inside the parentheses. $(3x-y+5)(3x+y-5)$ The fully factorised expression is $\boxed{(3x-y+5)(3x+y-5)}$.

1.2 Simplify the following expressions fully:

1.2.1 $(4+\frac{1}{x})(2-\frac{3}{x})$ Step 1: Expand the expression by multiplying each term. $4(2) + 4\left(-\frac{3}{x}\right) + \frac{1}{x}(2) + \frac{1}{x}\left(-\frac{3}{x}\right)$ Step 2: Perform the multiplications. $8 - \frac{12}{x} + \frac{2}{x} - \frac{3}{x^2}$ Step 3: Combine like terms. $8 - \frac{10}{x} - \frac{3}{x^2}$ The simplified expression is $\boxed{8 - \frac{10}{x} - \frac{3}{x^2}}$.

1.2.2 $\frac{5x-5}{5x}$ Step 1: Factor out the common factor in the numerator. $\frac{5(x-1)}{5x}$ Step 2: Cancel out the common factor $5$. $\frac{x-1}{x}$ The simplified expression is $\boxed{\frac{x-1}{x}}$.

1.2.3 $\frac{3^{n+1}+3^n}{27 \cdot 3^{n-1}}$ Step 1: Factor out $3^n$ from the numerator. $\frac{3^n \cdot 3^1 + 3^n}{27 \cdot 3^{n-1}} = \frac{3^n(3+1)}{27 \cdot 3^{n-1}}$ Step 2: Simplify the numerator and express $27$ as a power of $3$. $\frac{4 \cdot 3^n}{3^3 \cdot 3^{n-1}}$ Step 3: Apply the exponent rule $a^m \cdot a^p = a^{m+p}$ to the denominator. $\frac{4 \cdot 3^n}{3^{3+(n-1)}} = \frac{4 \cdot 3^n}{3^{n+2}}$ Step 4: Apply the exponent rule $\frac{a^m}{a^p} = a^{m-p}$. $4 \cdot 3^{n-(n+2)} = 4 \cdot 3^{n-n-2} = 4 \cdot 3^{-2}$ Step 5: Evaluate $3^{-2}$. $4 \cdot \frac{1}{3^2} = 4 \cdot \frac{1}{9} = \frac{4}{9}$ The simplified expression is $\boxed{\frac{4}{9}}$.