The figures in the margin indicate full marks for the questions.

1. Answer the following Questions as directed: (1 × 8 = 8)

• (a) Fill in the blank: The only even prime number is ________.
• (b) Write True or False: The set of integers obeys commutative law under subtraction.
• (c) Who is known as 'Father of Statistics'?
• (d) Fill in the blank: 10% profit on \(SP = \_\_\%\) profit on CP.
• (e) Write True or False: All facts numerically expressed are statistics.
• (f) Fill in the blank: Surface area of a sphere of radius r is ________.
• (g) Fill in the blank: Mailed questionnaire method of collecting primary data cannot be used if the informants are ________.
• (h) Define frequency density of a class interval.

2. Solve the following questions: (2 × 5 = 10)

(a) Prove that: \( 1 + \log_{10} 5 = 2 - \log ^{2}\)

(b) Simplify: \( \{(\sqrt[5]{64})^{-2}\}^{5/3} \)

(c) If \( (a-b):(a+b) = 5:13 \), then find \( a:b \).

(d) Convert the following number to vulgar fraction: \( 0.2\overline{77} \)

(e) Solve: \( |3x-1| < 7 \)

3. Answer the following questions: (3 × 5 = 15)

(a) If 1 km = 0.62 mile and the speed of a train is 50 km per hour, then how many miles will it cover in 3 hours?

(b) Prove that: \( \frac{1}{1+a^b+a^{-c}} + \frac{1}{1+b^c+b^{-a}} + \frac{1}{1+c^a+c^{-b}} = 1 \)

(c) Define mean proportion and third proportion with an example in each case.

(d) Find the area of a rhombus whose perimeter is 100 cm and one of its diagonal is 40 cm.

(e) Solve: \( \sqrt{\frac{x}{1-x}} + \sqrt{\frac{1-x}{x}} = \frac{13}{6} \) OR \( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = \frac{13}{6} \); \( xy = 36 \)

Section C: Descriptive Questions (5 Marks Each)

4. Water flows from a pipe of diameter 14 cm at the rate of 5 metres per minute. Find the volume of water flowing out of the pipe in 1 hour in litre.

5. What are the advantages of sampling over census method?

Write a note on the importance of statistics in business and commerce.

6. A cylindrical jar of radius 12 cm contains water to a depth of 18 cm. If 16 spheres each of radius 1.5 cm are immersed in it, the jar will completely filled with water. Find the height of the cylindrical jar.

7. Two mixtures P and Q contains milk and honey in the ratio 4:5 and 3:2 respectively. How many litres of P must be mixed with 45 litres of Q so that the resulting mixture may contain equal quantities of milk and honey?

8. Show that: \( 7 \log \frac{15}{16} + 6 \log \frac{8}{3} + 5 \log \frac{2}{5} + \log \frac{32}{25} - \log 3 = 0 \)

9. Write a brief note on different parts of a table.

In a sample study about food habit of students in two colleges, the following information was received:
College A: Girls 40%, total vegetarian were 45% and non-vegetarian boys were 20%.
College B: Boys were 50%, non-vegetarian boys were 30% and vegetarian girls were 15%.
Present the above in tabular form.

10. Show that the points (7, 3), (3, 0), (0, -4) and (4, -1) are the consecutive vertices of a rhombus.

Section D: Long Answer Questions

11. (a) Discuss the reasons for distrust in statistics. (4)
(b) What are the characteristics of good questionnaire? (4)

12. Following are the weight (in kg) of 50 persons. Prepare a frequency distribution taking the class intervals as 20-25, 25-30, etc. Find also the cumulative frequency and frequency density: (8)

13. (a) What are the advantages of diagram? (4)
(b) Draw a less than type of ogive from the following data: (4)

Income (Rs.)	No. of workers
500-600	15
600-700	37
800-900	26
900-1000	10

14. (a) Fill in the blanks: (1 × 3 = 3)
(i) ________ may be located graphically from histogram.
(ii) Row headings are called ________.
(iii) 1 acre = ________ sq. yards.
(b) If the points \( (a+1, 1) \), \( (2a+1, 3) \) and \( (2a+2, 2a) \) are collinear, then prove that \( a=2 \) or \( \frac{1}{2} \).