Secrios B [60 marks] Answer five questions only from this section. All questions carry equal marks. The denominator of a fraction is 6 greater than its numerator. If 4 is added to twice the numerator and twice the denominator of the same fraction, the result is $$ \frac{2}{3} $$. Find the fraction. The top of a cone of height 10 cm and base radius 7 cm is cut off by a plane parallel to the base. If the distance between the plane and the base is 3 cm , calculate the volume of the remaining object. [Take $$ \pi=\frac{22}{7} $$ ]

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### **Question 1**

**Problem:**
The denominator of a fraction is 6 greater than its numerator. If 4 is added to twice the numerator and twice the denominator of the same fraction, the result is $$ \frac{2}{3} $$. Find the fraction.

**Solution:**

1. **Let’s denote:**
   - Numerator = $$ x $$
   - Denominator = $$ y $$

2. **Given relationships:**
   - $$ y = x + 6 $$ (since the denominator is 6 greater than the numerator)
   - $$ \frac{2x + 4}{2y + 4} = \frac{2}{3} $$ (after adding 4 to twice the numerator and denominator)

3. **Substitute $$ y = x + 6 $$ into the equation:**
   $$
   \frac{2x + 4}{2(x + 6) + 4} = \frac{2}{3}
   $$
   $$
   \frac{2x + 4}{2x + 12 + 4} = \frac{2}{3}
   $$
   $$
   \frac{2x + 4}{2x + 16} = \frac{2}{3}
   $$

4. **Cross-multiply to solve for $$ x $$:**
   $$
   3(2x + 4) = 2(2x + 16)
   $$
   $$
   6x + 12 = 4x + 32
   $$
   $$
   6x - 4x = 32 - 12
   $$
   $$
   2x = 20
   $$
   $$
   x = 10
   $$

5. **Find $$ y $$:**
   $$
   y = x + 6 = 10 + 6 = 16
   $$

6. **The fraction is:**
   $$
   \frac{x}{y} = \frac{10}{16} = \frac{5}{8} \quad \text{(simplified)}
   $$

**Answer:** The fraction is $$ \frac{5}{8} $$.

---

### **Question 2**

**Problem:**
The top of a cone of height 10 cm and base radius 7 cm is cut off by a plane parallel to the base. If the distance between the plane and the base is 3 cm, calculate the volume of the remaining object. (Take $$ \pi = \frac{22}{7} $$)

**Solution:**

1. **Understand the Geometry:**
   - Original cone height ($$ H $$) = 10 cm
   - Original cone base radius ($$ R $$) = 7 cm
   - Plane cuts the cone 3 cm from the base, so the height of the remaining frustum ($$ h $$) = 3 cm.

2. **Determine the Radius of the Upper Base ($$ r $$):**
   - Using similar triangles:
     $$
     \frac{r}{R} = \frac{h}{H} \Rightarrow \frac{r}{7} = \frac{3}{10} \Rightarrow r = \frac{21}{10} = 2.1 \, \text{cm}
     $$

*However, this contradicts an earlier solution. Let's re-evaluate:*

**Correct Similarity Approach:**
   - Total height ($$ H $$) = 10 cm
   - Height from the apex to the cutting plane ($$ H' $$) = 10 cm - 3 cm = 7 cm
   - Therefore:
     $$
     \frac{r}{R} = \frac{H'}{H} = \frac{7}{10} \Rightarrow r = 7 \times \frac{7}{10} = 4.9 \, \text{cm}
     $$

3. **Volume of the Frustum:**
   - Formula:
     $$
     V = \frac{1}{3} \pi h (R^2 + Rr + r^2)
     $$
   - Plugging in the values:
     $$
     V = \frac{1}{3} \times \frac{22}{7} \times 3 \times (7^2 + 7 \times 4.9 + 4.9^2)
     $$
     $$
     V = \frac{22}{7} \times (49 + 34.3 + 24.01)
     $$
     $$
     V = \frac{22}{7} \times 107.31
     $$
     $$
     V \approx \frac{22}{7} \times 107.31 \approx 337.26 \, \text{cm}^3
     $$

**Answer:** The volume of the remaining object is approximately **337.26 cm³**.
**Question 1:** The fraction is $$ \frac{5}{8} $$. **Question 2:** The volume of the remaining object is approximately **337.26 cm³**.

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