Let me analyze the uploaded image and provide solutions to the mathematical problems. Hold on a moment while I process the content.Here’s the detailed solution for each question in the image:

Section A:

Each question carries 1 mark (1×10=10).

1. If a number is divisible by 10, its last digit is always ___?

   • Answer: 0

2. The mean proportional between 16 and 25 is ___?

   • Formula for mean proportional: $\sqrt{a \cdot b}$
   • $\sqrt{16 \cdot 25} = \sqrt{400} = 20$
   • Answer: 20

3. The number of odd numbers in a single-digit range is ___?

   • Odd numbers between 1 and 9: $1, 3, 5, 7, 9$ (5 numbers)
   • Answer: 5

4. The radian measure of $160^\circ$ is ___?

   • Formula: $\text{Radian} = \frac{\pi}{180} \times \text{degree}$
   • $\frac{\pi}{180} \times 160 = \frac{8\pi}{9}$
   • Answer: $\frac{8\pi}{9}$

5. If $\sin(\beta - 60^\circ) = \frac{1}{2}$, then $\cos \beta$ is ___?

   • Solving, $\beta - 60 = 30 \Rightarrow \beta = 90^\circ$
   • $\cos 90^\circ = 0$
   • Answer: 0

6. $\sin 12^\circ \cdot \cos 78^\circ =$ ___?

   • Using identity: $\sin x \cos (90^\circ - x) = \sin x \cdot \sin x = \sin^2 x$
   • $\sin 12^\circ \cdot \cos 78^\circ = \sin^2 12^\circ$
   • Answer: $\sin^2 12^\circ$

7. Find $x$, if the numbers $16, 15, 16, 15, x, 19, 17, 14$ have a mean of 15.

   • Sum of numbers: $15 \times 8 = 120$
   • $16 + 15 + 16 + 15 + x + 19 + 17 + 14 = 120$
   • $x = 120 - (16 + 15 + 16 + 15 + 19 + 17 + 14) = 120 - 112 = 8$
   • Answer: 8

8. If $\tan^2\alpha + \cot^2\alpha = 2$, then find $\tan^{12}\alpha + \cot^{12}\alpha$.

   • Using $\tan^2\alpha + \cot^2\alpha = (\tan^2\alpha + \cot^2\alpha)^3 = 2^3 = 8$
   • Answer: 8

9. If $A + B = 90^\circ$ and $\tan A = \frac{3}{4}$, then $\cot B =$ ___?

   • $B = 90^\circ - A$, $\cot B = \tan A = \frac{3}{4}$
   • Answer: $\frac{3}{4}$

10. If $X:Y = 2:3$, and $SP = 60$, find the respective amounts of $X$ and $Y$.

• Total parts: $2 + 3 = 5$
• $X = \frac{2}{5} \cdot 60 = 24$, $Y = \frac{3}{5} \cdot 60 = 36$
• Answer: $X = 24, Y = 36$

Section B:

Each question carries 3 marks (3×15=45).

1. Find the simple interest of 960 at $8\frac{1}{3}\%$ per annum for 2 years and 3 months.

   • Interest Rate: $8\frac{1}{3} = \frac{25}{3}\%$
   • Time: $2 \frac{3}{12} = \frac{27}{12} = \frac{9}{4}$ years
   • Simple Interest: $SI = \frac{P \cdot R \cdot T}{100} = \frac{960 \cdot \frac{25}{3} \cdot \frac{9}{4}}{100}$
   • $SI = \frac{960 \cdot 75}{1200} = 60$
   • Answer: 60

2. Find the interest on 600 at 8% annually if the total is 168.

   • $SI = 168 - 600 = 68$
   • Answer: 68

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