The image presents several math problems. We will solve them one by one. The problems are: 1. a) Expand and simplify $(n+4)^2 + (n-2)^2$ b) At an electrical shop, out of 92 people, 45 bought cables, and 58 bought sockets, while 14 asked prices without buying. Represent the information on a Venn diagram. Calculate the number of people who bought both sockets and cables. c) A straight line passing through points $(3,2)$ and $(4,5)$ intersects the y-axis at -23. Find the equation of the line.

1. Problem Description

The image presents several math problems. We will solve them one by one. The problems are:

1. a) Expand and simplify $(n+4)^2 + (n-2)^2$

b) At an electrical shop, out of 92 people, 45 bought cables, and 58 bought sockets, while 14 asked prices without buying. Represent the information on a Venn diagram. Calculate the number of people who bought both sockets and cables.

c) A straight line passing through points $(3,2)$ and $(4,5)$ intersects the y-axis at -

2

3. Find the equation of the line.

2. a) Solve for $x$ in the equation $(\frac{1}{4})^x = 8^{3x+2}$.

b) Given $v^2 = u^2 + 2as$, calculate the final velocity $v$ of a moving car with initial velocity $u = 30 m/s$, acceleration $a = 5 m/s^2$, and distance $s = 320 m$.

c) A hunter on top of a tree sees an antelope at an angle of depression of $30^\circ$. The height of the tree is 8m. Illustrate the information in a diagram. Find the distance between the hunter and antelope.

3. a) Simplify $\frac{2x-1}{3} - \frac{x-4}{4} + \frac{2x+1}{8}$.

2. Solution Steps

1. a) Expand and simplify $(n+4)^2 + (n-2)^2$:

$(n+4)^2 = n^2 + 8n + 16$

$(n-2)^2 = n^2 - 4n + 4$

$(n+4)^2 + (n-2)^2 = (n^2 + 8n + 16) + (n^2 - 4n + 4) = 2n^2 + 4n + 20$

2. b) Let $C$ be the set of people who bought cables, and $S$ be the set of people who bought sockets.

Total number of people = 92

$|C| = 45$

$|S| = 58$

Number of people who didn't buy anything = 14

Number of people who bought at least one item = 92 - 14 = 78

$|C \cup S| = |C| + |S| - |C \cap S|$

$78 = 45 + 58 - |C \cap S|$

$|C \cap S| = 45 + 58 - 78 = 103 - 78 = 25$

The number of people who bought both sockets and cables is

2

5. Venn diagram: Draw two overlapping circles representing the sets $C$ and $S$. The intersection contains 25 elements. The part of $C$ not in $S$ contains $45-25 = 20$ elements. The part of $S$ not in $C$ contains $58-25=33$ elements. The area outside both circles contains 14 elements.

3. c) A straight line passing through points $(3,2)$ and $(4,5)$ intersects the y-axis at -

2

3. Find the equation of the line.

Slope of the line, $m = \frac{5-2}{4-3} = \frac{3}{1} = 3$.

The equation of the line in point-slope form is $y - y_1 = m(x - x_1)$.

Using point $(3,2)$, $y - 2 = 3(x - 3)$

$y - 2 = 3x - 9$

$y = 3x - 7$

The line intersects the y-axis at -7, not -23 as given in the problem.

Assuming the problem statement is correct. The equation has the form $y = mx + c$.

Since the line passes through $(3,2)$ and $(4,5)$, we have

$2 = 3m + c$

$5 = 4m + c$

Subtracting the equations gives $3 = m$. So, $m=3$.

Plugging this into $2=3m+c$, we get $2 = 3(3)+c$, so $2=9+c$, and $c = -7$.

However, the line intersects the y-axis at -

2

3. This is a contradiction. There is no linear equation that goes through $(3,2)$ and $(4,5)$ and has a y-intercept of -

2

3. Assuming the y-intercept of the line is -7, the equation is $y = 3x - 7$.

4. a) Solve for $x$ in the equation $(\frac{1}{4})^x = 8^{3x+2}$.

$(\frac{1}{4})^x = (4^{-1})^x = 4^{-x} = (2^2)^{-x} = 2^{-2x}$

$8^{3x+2} = (2^3)^{3x+2} = 2^{9x+6}$

$2^{-2x} = 2^{9x+6}$

$-2x = 9x + 6$

$-11x = 6$

$x = -\frac{6}{11}$

5. b) Given $v^2 = u^2 + 2as$, calculate the final velocity $v$ of a moving car with initial velocity $u = 30 m/s$, acceleration $a = 5 m/s^2$, and distance $s = 320 m$.

$v^2 = u^2 + 2as$

$v^2 = (30)^2 + 2(5)(320) = 900 + 3200 = 4100$

$v = \sqrt{4100} = \sqrt{100 \cdot 41} = 10\sqrt{41} \approx 64.03 m/s$

6. c)

Let $h$ be the height of the tree, $h=8$ m. Let $d$ be the horizontal distance between the base of the tree and the antelope.

Since the angle of depression is $30^\circ$, we have $\tan(30^\circ) = \frac{8}{d}$.

$d = \frac{8}{\tan(30^\circ)} = \frac{8}{\frac{1}{\sqrt{3}}} = 8\sqrt{3}$

Let $x$ be the distance between the hunter and the antelope.

$x^2 = 8^2 + (8\sqrt{3})^2 = 64 + 64 \cdot 3 = 64 + 192 = 256$

$x = \sqrt{256} = 16$ meters

7. a) Simplify $\frac{2x-1}{3} - \frac{x-4}{4} + \frac{2x+1}{8}$.

The common denominator is

2

4. $\frac{2x-1}{3} - \frac{x-4}{4} + \frac{2x+1}{8} = \frac{8(2x-1)}{24} - \frac{6(x-4)}{24} + \frac{3(2x+1)}{24}$

$= \frac{16x - 8 - (6x - 24) + 6x + 3}{24} = \frac{16x - 8 - 6x + 24 + 6x + 3}{24}$

$= \frac{(16 - 6 + 6)x + (-8 + 24 + 3)}{24} = \frac{16x + 19}{24}$

3. Final Answer

1. a) $2n^2 + 4n + 20$

b) 25 people bought both sockets and cables. Venn diagram described above.

c) If we ignore the y-axis intersection at -23, the line is $y = 3x - 7$.

2. a) $x = -\frac{6}{11}$

b) $v = 10\sqrt{41} \approx 64.03$ m/s

c) The distance between the hunter and the antelope is 16 meters.

3. a) $\frac{16x + 19}{24}$