This Paper has

**50**answerable questions with**2**answered.P—4(BMS)Syllabus 2008 | |

Time Allowed : 3 Hours | Full Marks : 100 |

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

Answer all questions. |

Notations and symbols have usual meanings. |

SECTION I (Arithmetic — 10 marks) |

1. | Answer any two of the following: Choose the correct option showing the proper reasons/calculations. | 3x2 | |||||||||||||||||

(a) | Two numbers are in the ratio of 3:4. If 10 is subtracted from both of them then the ratio becomes 1:3. The numbers are
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(b) | A person drove his car 50 km at an average speed of 20 km/h. He drove first 30 km of his journey at an average speed of 60 km/h. Then average speed of last 20 km is
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(c) | For a sum of money to become 2¼ times of itself in 5 years, the rate of interest is
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2. | Answer any one of the following. | 4x1 | |||||||||||||||||

(a) |
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(b) | The Bill Value (B.V.) of a bill is Rs. 1,01,000. Find the Banker’s Gain (B.G.) after 73 days at 5% p.a. | (0) |

SECTION II (Algebra — 15 marks) |

3. | Answer any three of the following: Choose the correct option showing proper reasons/calculations. | 3x3 | ||||||||||||

(a) | Solution of (3√2)^{2x + 7} = (4√2)^{7x + ⅔} is
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(b) | The number of ways can the letters of the word MONDAY be arranged to end with Y but not begin with M is
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(c) | Let A − k varies directly as B where k is constant. If A = 750 then B = 500. If A = 1175 then B = 1350. If A = 550 then B will be.
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(d) | If A = (1, 2, 3, 4), B = (2, 3, 5, 6) and C = (3, 4, 6, 7) then (A − B) ∩ (A − C) is
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(e) | Let p be the statement "the student is tall" and q be the statement "the student is intelligent" then symbolic form of the statement that "the student is neither tall nor intelligent" is
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4. | Answer any two of the following: | 3x2 | ||||||||||||

(a) | In how many ways can a committee of 2 ladies and 3 gentlemen be formed from a group of 5 ladies and 6 gentlemen? | (0) | ||||||||||||

(b) |
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(c) | lf w be an imaginary cube root of unity then show that (1 + w − w^{2} ) (1 − w + w^{2}) = 4. | (0) |

SECTION III (Mensuration — 15 marks) |

5. | Answer any three of the following: Choose the correct option showing proper reasons/calculations. | 3x3 | ||||||||||||||||||

(a) | Altitude of an equilateral triangle having a base of length 2 cm is
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(b) | How many times will wheel of a car rotate in a journey of 1925 metres if it is known that the radius of the wheel is 49 cm?
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(c) | The volume (in cu. cm) of a right triangular prism with sides as 10, 15 and 19 cm with altitude of prism as 8 cm is
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(d) | Three solid metal spheres of radii 3 cm, 4 cm and 5 cm are melted to form a new sphere. The radius of this new sphere is
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(e) | The volumes of two cones having equal radius of their bases are in the ratio 1 : 2. The ratio of their heights is
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6. | Answer any two of the following: | 3x2 | ||||||||||||||||||

(a) | The length, breadth and height of a cage made of wire are 6 m, 3 m and 2 m respectively. Find the length of the longest stick that can be placed in the cage. | (0) | ||||||||||||||||||

(b) | Curved surface area of a solid right circular cylinder having 10 cm as diameter of the base is 100 sq cm. Find the volume of this cylinder. | (0) | ||||||||||||||||||

(c) | If a circle and a square have the same perimeter then show that their areas are in the ratio 14 : 11.
| (0) |

SECTION IV (Co–ordinate Geometry — 10 marks) |

7. | Answer any two of the following: Choose the correct option showing the proper reasons/calculations. | 3x2 | ||||||||||||||||||||

(a) | The ratio in which the point (2, 3) divides the portion of a straight line joining the points (1, 2) and (4, 5) internally is
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(b) | A straight line passing through the point of intersection of lines 2x + y = 4 and x − y + 1 = 0 and parallel to the line 3x + 2y = 5 is
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(c) | The centre and radius of the circle (x − 2) (x − 4) + (y − 3) (y − 5) = 0 are
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(d) | The eccentricity of the ellipse 4x^{2} − 24x + 9y^{2} + 36y + 36 = 0 is
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8. | Answer any one of the following: | 4x1 | ||||||||||||||||||||

(a) | Find the equation of the parabola whose vertex and focus are at (3, 5) and (6, 5). | (0) | ||||||||||||||||||||

(b) | Given for a hyperbola, co-ordinates of the centre is (-3, 2), length of latus rectum is 9 and eccentricity is
| (0) |

SECTION V (Calculus — 15 marks) |

9. | Answer any three of the following: Choose the correct option showing proper reasons/calculations. | 3x3 | |||||||||||||||||||||||||||

(a) |
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(b) | The value of k for which f(x) = x + 2 for x ≤ 2 = k – x ^{2} for x > 2 is continuous at x = 2 is
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(c) |
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(d) | If u = x^{2} + y^{2} + z^{2}, the value of xu_{x} + yu_{y} + zu_{z} is
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(e) |
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10. | Answer any two of the following: | 3x2 | |||||||||||||||||||||||||||

(a) |
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(b) | Show that x^{3} - 6x^{2} + 9x - 10 is maximum at x = 1 but is minimum at x = 3. | (0) | |||||||||||||||||||||||||||

(c) |
| (0) |

SECTION VI (Statistical Methods — 35 marks) |

11. | Answer any seven of the following: Choose the correct option showing proper reasons/calculations. | 3x7 | |||||||||||||||||||||||||||||||

(a) |
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(b) | Geometric mean of first group of 4 observations is 8 and that of second group of 3 observations is 1024. Then geometric mean of all the 7 observations is
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(c) | The median of the following frequency distribution of x
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(d) | For a group of 10 items Σx = 60, Σx^{2} = 850 and mode = 5. Then the Pearson’s coefficient of skewness is
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(e) | If two variables x and y are related by 3x - 2y - 4 = 0 and arithmetic mean of x is 10, then the arithmetic mean of y is
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(f) | Mean deviation about median of 13, 84, 68, 24, 96, 139,84,27 is
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(g) | If 25 observations are each 1, 25 observations are each 3 and 50 observations are each 0, then variance of all 100 observations is
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(h) |
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(i) | If the variance of the first n natural numbers is 14, then the value of n is
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(j) | Arithmetic mean of a series of observations is 6 and its coefficient of variation is 50%, then the variance of the observations is
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12. | (a) | Answer any two of the following: | 5x2 | ||||||||||||||||||||||||||||||

(i) | Draw a simple bar chart to represent year-wise student strength (in thousands) in certain university from the following data:
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(ii) | Show that mean deviation about mean and s.d. of two observations x_{1} and x_{2} are same. | (0) | |||||||||||||||||||||||||||||||

(iii) | Find the variance of the following frequency distribution:
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(b) | Write a short note on any one of the following: | 4x1 | |||||||||||||||||||||||||||||||

(i) | Tabulation; | (0) | |||||||||||||||||||||||||||||||

(ii) | Central tendency of data. | (0) |