1. **Factorize** $25 - 16x^2$. This is a difference of squares: $a^2 - b^2 = (a - b)(a + b)$. Here, $25 = 5^2$ and $16x^2 = (4x)^2$. So, $$25 - 16x^2 = (5 - 4x)(5 + 4x).$$ 2. **Condition for congruence of triangles $\triangle ABC$ and $\triangle CDE$**. Given that $AB = DE$ and $BC = CD$, and they share vertex $C$. Triangles are congruent if they satisfy any of the criteria: SSS, SAS, ASA, AAS, or RHS. Here, if $\angle B = \angle D$ or $\angle ABC = \angle CDE$, then by SAS (Side-Angle-Side), $\triangle ABC \cong \triangle CDE$. 3. **Find $x^2 + y^2$ given $x + y = 7$ and $xy = 5$**. Use the identity: $$ (x + y)^2 = x^2 + 2xy + y^2 $$ Rearranged: $$ x^2 + y^2 = (x + y)^2 - 2xy $$ Substitute values: $$ x^2 + y^2 = 7^2 - 2 \times 5 = 49 - 10 = 39. $$ 4. **If 5 men finish a task in 8 days, how many extra days for 4 men?** Work done is constant. Total work = men $\times$ days = $5 \times 8 = 40$ man-days. For 4 men, days needed = $\frac{40}{4} = 10$ days. Extra days = $10 - 8 = 2$ days. 5. **Find $x$ in the circle with center $O$ given angle $43^\circ$ opposite $x$**. In a circle, angles subtended by the same chord are equal. So, $x = 43^\circ$. 6. **Shade $(A \cap B)'$ in the Venn diagram**. $(A \cap B)'$ is the complement of the intersection of $A$ and $B$. Shade all regions outside the overlap of $A$ and $B$. 7. **Find LCM of $4x^2$, $3x^2 - 9x$, and $6(x - 3)^2$**. Factor each: $4x^2 = 2^2 x^2$ $3x^2 - 9x = 3x(x - 3)$ $6(x - 3)^2 = 2 \times 3 (x - 3)^2$ LCM takes highest powers: Numbers: $2^2$, $3$, variables: $x^2$, $(x - 3)^2$ So, LCM = $2^2 \times 3 \times x^2 \times (x - 3)^2 = 4 \times 3 \times x^2 (x - 3)^2 = 12x^2 (x - 3)^2$. 8. **Find price without customs duty if price after 35% duty is 70200**. Let original price be $P$. Price after duty = $P + 0.35P = 1.35P = 70200$. So, $$P = \frac{70200}{1.35} = 52000.$$ 9. **True or False for parallelogram statements:** - If a quadrilateral has one pair of equal sides and another pair parallel, it is not necessarily a parallelogram. **False**. - In a rhombus, diagonals are not equal in length. **False**. - Diagonals of a rectangle bisect each other perpendicularly? No, they bisect but not perpendicularly. **False**. 10. **Simplify $\frac{3x - 6}{5} \div \frac{x^2 - 4}{x + 2}$**. Rewrite division as multiplication by reciprocal: $$ \frac{3x - 6}{5} \times \frac{x + 2}{x^2 - 4} $$ Factor: $3x - 6 = 3(x - 2)$ $x^2 - 4 = (x - 2)(x + 2)$ So, $$ \frac{3(x - 2)}{5} \times \frac{x + 2}{(x - 2)(x + 2)} = \frac{3(x - 2)(x + 2)}{5 (x - 2)(x + 2)} = \frac{3}{5} $$ Cancel $(x - 2)$ and $(x + 2)$. 11. **Solve inequality $2 - 3x > -7$ and find max integer $x$**. Subtract 2: $$ -3x > -9 $$ Divide by -3 (reverse inequality): $$ x < 3 $$ Maximum integer $x$ is 2. 12. **Find 3rd quartile of data: 10, 11, 12, 12, 13, 14, 15, 13, 17, 18**. Sort data: 10, 11, 12, 12, 13, 13, 14, 15, 17, 18 Number of data points $n=10$. 3rd quartile $Q_3$ position = $\frac{3(n+1)}{4} = \frac{3 \times 11}{4} = 8.25$. $Q_3$ lies between 8th and 9th data points: 8th = 15, 9th = 17 Interpolate: $$ Q_3 = 15 + 0.25(17 - 15) = 15 + 0.5 = 15.5 $$ 13. **Find $\angle ABC$ in circle with center $O$ and $\angle ADC = 116^\circ$**. $\angle ABC$ and $\angle ADC$ subtend the same arc. Angles subtending same arc are equal. So, $$\angle ABC = 116^\circ.$$ 14. **Find $x$ in parallelogram $ABCD$ with angles $(4x + 12)^\circ$ and $(3x + 14)^\circ$**. Opposite angles in parallelogram are equal: $$4x + 12 = 3x + 14$$ Solve: $$4x - 3x = 14 - 12$$ $$x = 2$$ 15. **Find radius of cone with height 3 cm and volume $16\pi$ cm³**. Volume formula: $$ V = \frac{1}{3} \pi r^2 h $$ Substitute: $$16\pi = \frac{1}{3} \pi r^2 \times 3$$ Simplify: $$16\pi = \pi r^2$$ Divide both sides by $\pi$: $$16 = r^2$$ So, $$r = 4 \text{ cm}.$$