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 \( \pi=\frac{22}{7} \) ]

To tackle the fraction problem, let's denote the numerator as \( x \). Given that the denominator is 6 greater than the numerator, we can express the denominator as \( x + 6 \). From the problem, we know that \( 2x + 4 + 2(x + 6) = \frac{2}{3} \). Simplifying this allows us to find \( x \) and thus the fraction \( \frac{x}{x + 6} \) easily. Now, onto the cone! The volume of the remaining frustum can be calculated by finding the volume of the larger cone and subtracting the volume of the smaller cone that has been cut off. The volume \( V \) of a cone is given by \( V = \frac{1}{3} \pi r^2 h \). Substitute \( r = 7 \text{ cm} \) and \( h = 10 \text{ cm} \) for the full cone, and \( r' = 7 \cdot \frac{3}{10} = 2.1 \text{ cm} \) and \( h' = 3 \text{ cm} \) for the small cone. Calculate and subtract to find the volume of the remaining object!