1. **Problem statement:** In the circle with center O, given $\angle ACB = 36^\circ$ and line DC tangent at C, find: (i) $\angle ABC$ (ii) $\angle BAC$ (iii) $\angle BCD$ 2. **Formula and rules:** - The angle at the center is twice the angle at the circumference subtending the same arc. - Tangent to a circle is perpendicular to the radius at the point of contact. - The sum of angles in a triangle is $180^\circ$. 3. **Step-by-step solution:** (i) Find $\angle ABC$: - $\angle ACB = 36^\circ$ is an angle at the circumference. - The arc AB subtends $\angle ACB$ at the circumference and $\angle AOB$ at the center. - By the circle theorem, $\angle AOB = 2 \times \angle ACB = 2 \times 36^\circ = 72^\circ$. - Triangle OBC is isosceles with $OB = OC$ (radii). - Let $\angle OBC = \angle OCB = x$. - Sum of angles in $\triangle OBC$: $x + x + 72^\circ = 180^\circ$ so $2x = 108^\circ$ and $x = 54^\circ$. - $\angle ABC$ subtends the same arc as $\angle OBC$, so $\angle ABC = 54^\circ$. (ii) Find $\angle BAC$: - In $\triangle ABC$, sum of angles is $180^\circ$. - Known angles: $\angle ACB = 36^\circ$, $\angle ABC = 54^\circ$. - So $\angle BAC = 180^\circ - 36^\circ - 54^\circ = 90^\circ$. (iii) Find $\angle BCD$: - Line DC is tangent at C, so $\angle OCB = 90^\circ$ (radius perpendicular to tangent). - $\angle BCD = 90^\circ - \angle ACB = 90^\circ - 36^\circ = 54^\circ$. --- 4. **Problem statement:** Each interior angle of a regular polygon is $162^\circ$. Find: (i) the size of each exterior angle (ii) the number of sides 5. **Formula and rules:** - Exterior angle = $180^\circ - $ interior angle - Sum of exterior angles of any polygon = $360^\circ$ - Number of sides $n = \frac{360^\circ}{\text{exterior angle}}$ 6. **Step-by-step solution:** (i) Exterior angle = $180^\circ - 162^\circ = 18^\circ$ (ii) Number of sides $n = \frac{360^\circ}{18^\circ} = 20$ --- 7. **Problem statement:** (a) Factorize completely: $3a^2b - 6ab + 5ab^2$ 8. **Step-by-step solution:** - Factor out common factor $ab$: $$3a^2b - 6ab + 5ab^2 = ab(3a - 6 + 5b)$$ - Simplify inside parentheses: $$3a - 6 + 5b$$ - Factor 3 from $3a - 6$: $$3a - 6 = 3(a - 2)$$ - So expression is: $$ab(3(a - 2) + 5b)$$ - No further factorization possible. --- 9. **Problem statement:** (b) Simplify $2 - 5(y - 7) + 3y$ 10. **Step-by-step solution:** - Expand $-5(y - 7)$: $$-5y + 35$$ - Expression becomes: $$2 - 5y + 35 + 3y$$ - Combine like terms: $$2 + 35 + (-5y + 3y) = 37 - 2y$$ --- **Final answers:** (i) $\angle ABC = 54^\circ$ (ii) $\angle BAC = 90^\circ$ (iii) $\angle BCD = 54^\circ$ (b)(i) Exterior angle = $18^\circ$ (b)(ii) Number of sides = 20 (11)(a) Factorized form: $ab(3a - 6 + 5b)$ (11)(b) Simplified form: $37 - 2y$