1. Solve the equation $6x+2=24$. Step 1: Subtract 2 from both sides: $$6x=24-2$$ Step 2: Simplify the right side: $$6x=22$$ Step 3: Divide both sides by 6: $$x=\frac{22}{6}=\frac{11}{3}$$ Answer: $x=\frac{11}{3}$. 2. Write the formula for total surface area of a cone and explain terms. The total surface area $$A$$ of a cone is given by: $$A=\pi r^2 + \pi rl$$ Where: - $r$ is the radius of the base of the cone, - $l$ is the slant height of the cone (the distance from the base edge to the apex along the side), - $\pi$ is approximately 3.1416. The first term $\pi r^2$ is the area of the circular base. The second term $\pi rl$ is the lateral surface area (the side surface). 3. Check if $x^3 + x^2 - (2 + \sqrt{2})x - \sqrt{2}$ is divisible by $x + x - 2$ or not. Note: $x+x-2=2x-2=2(x-1)$. We consider divisibility by $x-1$ (ignoring constant factor 2). Use Polynomial Remainder Theorem for $x-1$: Substitute $x=1$ into the polynomial: $$1^3 + 1^2 - (2+\sqrt{2})(1) - \sqrt{2} = 1 + 1 - 2 - \sqrt{2} - \sqrt{2} = 2 - 2 - 2\sqrt{2} = -2\sqrt{2} \neq 0$$ Since remainder is not zero, the polynomial is not divisible by $x-1$, hence not divisible by $x+x-2$. 4. Find $p(-1)$, $p(0)$, $p(2)$ for $p(x) = x^3 - 2x^2 - 3$, then find mean. Calculate: $p(-1) = (-1)^3 - 2(-1)^2 - 3 = -1 - 2 - 3 = -6$ $p(0) = 0 - 0 - 3 = -3$ $p(2) = 8 - 8 - 3 = -3$ Mean = $\frac{-6 + (-3) + (-3)}{3} = \frac{-12}{3} = -4$ 5. Find the three angles of a triangle given as $(2x)^\circ$, $(3x+10)^\circ$, and $(5x-20)^\circ$. Sum of angles in triangle: $$2x + 3x + 10 + 5x - 20 = 180$$ Simplify: $$ (2x + 3x + 5x) + (10 - 20) = 180 $$ $$ 10x - 10 = 180 $$ Add 10 to both sides: $$ 10x = 190 $$ Divide by 10: $$ x = 19 $$ Find each angle: $$ 2x = 2 \times 19 = 38^\circ $$ $$ 3x + 10 = 3 \times 19 + 10 = 57 + 10 = 67^\circ $$ $$ 5x - 20 = 5 \times 19 - 20 = 95 - 20 = 75^\circ $$ Check sum: $38 + 67 + 75 = 180^\circ$ (valid). Final answers: 1. $x=\frac{11}{3}$ 2. $A=\pi r^2 + \pi rl$ 3. Not divisible 4. Mean = $-4$ 5. Angles = $38^\circ$, $67^\circ$, $75^\circ$