This Paper has

**48**answerable questions with**0**answered.C—4(BMS)Revised Syllabus | |

Time Allowed : 3 Hours | Full Marks : 100 |

The figures in the margin on the right side indicate full marks. |

(Notations and symbols have their usual meanings) |

Section A |

ARITHMETIC (15 marks)Answer Question No. 1 (compulsory – 5 marks) and any one (10 marks) from the rest. |

Marks |

1. | (a) | If the two numbers 20 and x + 2 are in the ratio 2 : 3, find x. | 1 | (0) | |||||||||||||||

(b) | If 15% of a particular number is 45, find the number. | 1 | (0) | ||||||||||||||||

(c) |
| 1 | (0) | ||||||||||||||||

(d) | What principal will be increased to Rs. 4,600 after 3 years at the rate of 5% per annum simple interest? | 2 | (0) | ||||||||||||||||

2. | (a) |
| 5 | (0) | |||||||||||||||

(b) | The average score of boys is 60, that of girls is 70 and that of all the candidates is 64 appearing in Mathematics of annual examination. Find the ratio of number of boys and number of girls there. If the total number of candidates appearing in Mathematics is 150, find the number of boys there. | 5 | (0) | ||||||||||||||||

3. | (a) | A dealer mixed two varieties of teas having costs Rs. 1,200 and Rs. 2,500 each per kg in such a way that he can gain 20% by selling the resultant mixture at Rs. 1,800 per kg. Find the proportion in which the two types of tea are mixed. | 5 | (0) | |||||||||||||||

(b) | If the Banker’s gain on a bill, due in 4 months at the rate of 6% per annum, be Rs. 200, find the bill value. Banker’s discount and true discount of the bill. | 5 | (0) |

Section B |

ALGEBRA (25 marks) |

Answer Question No.4 (compulsory – 5 marks) and any two (10 x 2 = 20 marks) from the rest. |

4. | Answer any five of the following: | 1x5 | ||||

(a) | Express 2√2 as a surd of sixth order. | (0) | ||||

(b) | Find the modulus of the complex number 3 + 4i. | (0) | ||||

(c) | If x varies as y then show that x^{2} + y^{2} varies as x^{2} – y^{2}. | (0) | ||||

(d) | Find the logarithm of 125 to the base 5√5. | (0) | ||||

(e) | Find the quadratic equation whose one root is 3 + √2. | (0) | ||||

(f) | Prove that ^{10}P_{10} x ^{22}C_{12} = ^{22}P_{10}. | (0) | ||||

(g) | If S be the set of all prime numbers and M = {0, 1, 2, 3}, find S ∩ M. | (0) | ||||

5. | (a) | For what value of m, 4x^{2} – 10mx + 4 = 0 and 4x^{2} – 9x + 2 = 0 have a common root? | 5 | (0) | ||

(b) | Out of 5 gentlemen and 3 ladies, a committee of 6 persons is to be selected. Find the number of committees to be formed (i) when there are 4 gents: (ii) when there is a majority of gents. | 5 | (0) | |||

6. | (a) | Publisher of a book pays a lump sum plus an amount for every copy for every copy he sold, to the author. If 1000 copies were sold the author would receive Rs. 2,500 and if 2700 copies were sold the author would receive Rs. 5,900. How much the author would receive if 5000 copies were sold? | 5 | (0) | ||

(b) | Construct a truth table for (p ∧ q) ∧ ∼ (p ∧ q). | 5 | (0) | |||

7. | (a) | Determine the time period by which a sum of money would be three times of itself at 8% p.a. C.I. [Given log _{10}3 = 0.4771, log_{10}1.08 = 0.0334] | 5 | (0) | ||

(b) | Solve: x(x + y + z) = 6; y(x + y + z) = 12; z(x + y + z) = 18. | 5 | (0) |

Section C |

MENSURATION (30 marks) |

Answer Question No.8 (compulsory – 10 marks) and any two (10 x 2 = 20 marks) from the rest. |

8. | Answer any five of the following: | 2x5=10 | ||||

(a) | For an isosceles right angled triangle with length 5 cm of one side adjacent to the right angle, find the length of the perimeter of the triangle. | (0) | ||||

(b) | The circumference of a circle is 88 cm. Determine the length of the perimeter of its semi–circle. | (0) | ||||

(c) | The diameter of the base of a cylindrical pillar is 7 m and its height is 18 m. Find the volume of the pillar. | (0) | ||||

(d) | The external and internal radii of a hollow sphere are 6 cm and 3 cm respectively. Find the volume of the sphere. | (0) | ||||

(e) | If the point (5, y) divides the line joining points (4, 2) and (7, 5) internally, find y. | (0) | ||||

(f) | Find the gradient of the line having equal intercepts of the axes in the first quadrant of the coordinate system. | (0) | ||||

(g) | Determine the equation of a circle with (2, 3) and (4, 5) as the extremeties of its diameter. | (0) | ||||

(h) | Find the eccentries of the ellipse whose minor axis is half of its major axis. | (0) | ||||

9. | (a) | A circle of diameter 14 meter is inscribed within a square touching the sides. Find the area of the fillets thus formed and the cost of developing one fillet at the rate of Rs. 200 per square meter. | 5 | (0) | ||

(b) | Find the equations of the sides of a triangle ABC whose vertices are A (–1, 8), B(4, –2) and C(–5, –3). | 5 | (0) | |||

10. | (a) | These base of a right pyramid is an equilateral triangle of side 10√3 cm each. The area of the total surface of the pyramid is 270√3 cm^{2}. Find the height of the pyramid. | 5 | (0) | ||

(b) | Show that the two circles x^{2} + y^{2} + 4x + 10y = 20 and x^{2} + y^{2} – 8x – 6y + 16 = 0 touch each other externally. Also determine the coordinates of the point of contact. | 5 | (0) | |||

11. | (a) | A conical tent is required to accommodate 11 persons. Each person must have 4 sq.m of space on the ground and 20 cu.m of air to breathe. Find the vertical height. | 5 | (0) | ||

(b) | Find the equation of the hyperbola whose focus is at (1, 2), directrix 2x + y = 1 and eccentricity √3. | 5 | (0) |

Section D |

ELEMENTARY STATISTICS (30 marks) |

Answer Question No. 12 (compulsory – 10 marks) and any two (10 x 2 = 20 marks) from the rest. |

12. | Answer any five of the following: | 2x5=10 | ||||||||||||||

(a) | Find the mean and mode of the 9 observations 9, 2, 5, 3, 5, 7, 5, 1, 8. | (0) | ||||||||||||||

(b) | If two groups have number of observations 10 and 5 and means 50 and 20 respectively, find the grouped mean. | (0) | ||||||||||||||

(c) |
45, find the mean of y. | (0) | ||||||||||||||

(d) | If 2x_{1} + 3y_{1} = 5 for i = 1, 2, ....., n and mean deviation of x_{1}, x_{2}, ......, x_{n} about their mean is 12, find the mean deviation of y_{1}, y_{2}, ..... y_{n} about their mean. | (0) | ||||||||||||||

(e) | If the mean and variance of a variable are 8 and 4 respectively, find the coefficient of variation of the variable in form of percentage. | (0) | ||||||||||||||

(f) | If the coefficient of skewness, mean and variance of a variable are –6, 80 and 4, find the mode of that variable. | (0) | ||||||||||||||

(g) | If the means of two groups of 30 and 50 observations are equal and their standard deviations are 8 and 4 respectively, find the grouped variance. | (0) | ||||||||||||||

(h) | For two observations a and b, show that standard deviation is half the distance between them. | (0) | ||||||||||||||

13. | (a) | Draw a blank table to show the number of students sexwise admitted in each of 3 streams. Arts, Science and Commerce in the years 2000 and 2001 in a college of Kolkata showing totals in each stream, sex and year. | 5 | (0) | ||||||||||||

(b) | Represent the following data of productions of a steel factory by a line diagram:
| 5 | (0) | |||||||||||||

14. | Marks obtained by 30 students in History of a Test Examination, 2004 of some school are as follows:
| 4+3+3 | (0) | |||||||||||||

15. | (a) | Fifty students appeared in an examination. The result of the passed students are given below:
The average marks of all students is 52. Find the average marks of the students who failed in the examination. | 5 | (0) | ||||||||||||

(b) | Find the standard deviation from the following frequency distribution:
| 5 | (0) |