Simplify: sqrt(72) - sqrt(32), Solve: 2x2 − 3x − 2 = 0, Evaluate: 10(1000) - 10(10), Find the gradient of y = 3x2 − 2x at x = 1

The simplified expression is $\boxed{2\sqrt{2}}$.

2. Solve: $2x^2 - 3x - 2 = 0$

We look for two numbers that multiply to $2 \times (-2) = -4$ and add up to $-3$. These numbers are $-4$ and 1.

The solutions are $\boxed{x = -\frac{1}{2}, 2}$.

3. Evaluate: $\log_{10}(1000) - \log_{10}(10)$

The evaluated expression is $\boxed{2}$.

4. Find the gradient of $y = 3x^2 - 2x$ at $x = 1$

The gradient at $x=1$ is $\boxed{4}$.

5. Find the distance between points $(1,2)$ and $(4,6)$

Let $(x_1, y_1) = (1,2)$ and $(x_2, y_2) = (4,6)$.

The distance between the points is $\boxed{5 \text{ units}}$.

6. Find the determinant of matrix $\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$

Here, $a=3, b=2, c=1, d=4$.

The determinant of the matrix is $\boxed{10}$.

7. Solve: $\sin x = \sqrt{3}/2$ for $0^\circ \le x \le 360^\circ$

The angle whose sine is $\sqrt{3}/2$ is $60^\circ$. So, $x = 60^\circ$.

Sine is positive in the first and second quadrants.
The first quadrant solution is $x_1 = 60^\circ$.
The second quadrant solution is $x_2 = 180^\circ - 60^\circ = 120^\circ$.
The solutions in the given range are $\boxed{x = 60^\circ, 120^\circ}$.

8. Expand: $(x + 2)(x - 3)$

The expanded expression is $\boxed{x^2 - x - 6}$.

9. Find the mean of $2, 4, 6, 8$

The mean is $\boxed{5}$.

10. Integrate: $\int 2x \,dx$

Here, $a=2$ and $n=1$.

The integral is $\boxed{x^2 + C}$.

Section B

11. (a) Solve the equation: $x^2 + 5x + 6 = 0$

We look for two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3.

The solutions are $\boxed{x = -2, -3}$.

(b) Hence find the sum and product of the roots

Here, $a=1, b=5, c=6$.

Alternatively, using the roots from (a): $(-2) + (-3) = -5$.

Alternatively, using the roots from (a): $(-2) \times (-3) = 6$.
The sum of the roots is $\boxed{-5}$ and the product of the roots is $\boxed{6}$.

12. (a) Simplify: $(3x^2y)(2xy^3)$

The simplified expression is $\boxed{6x^3y^4}$.

(b) Express your answer in index form

The answer from part (a) is already in index form.
The expression in index form is $\boxed{6x^3y^4}$.

13. (a) Solve the simultaneous equations:
$x + y = 5$ (Equation 1)
$x^2 + y^2 = 13$ (Equation 2)

If $x = 2$:

If $x = 3$:

The solutions are $\boxed{(x=2, y=3) \text{ and } (x=3, y=2)}$.

(b) Find $xy$

For $(x=2, y=3)$:

For $(x=3, y=2)$:

In both cases, $xy$ is $\boxed{6}$.

14. (a) Find the equation of a line passing through $(2,3)$ and $(4,7)$

The equation of the line is $\boxed{y = 2x - 1}$.

(b) Find the midpoint of the line