This Paper has

**83**answerable questions with**5**answered.P—4(BMS)Syllabus 2008 | |

Time Allowed : 3 Hours | Full Marks : 100 |

Notations and symbols have usual meanings. |

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

Answer all questions. |

SECTION I (Arithmetic — 10 Marks) |

Marks |

1. | Answer any three of the following. Choose the correct option showing the proper reasons/calculations. | 2x3 | ||||||||||||

(a) | Two numbers are in the ratio 3 : 4. If 10 is subtracted from both of them the ratio will be 1 : 2. So the members are
| (1) | ||||||||||||

(b) | The mean of age of 5 men is 40 years. Three of them are of same age and they are excluded. The mean of the remaining two is 25. Age of one of the excluded persons in year is
| (1) | ||||||||||||

(c) | A man bought three qualities of tea in the ratio 5 : 4 : 3 with prices per kg. Rs. 390, Rs. 375 and Rs. 450 respectively and mixed them together. The cost price of the mixture per kg. in Rs. is
| (1) | ||||||||||||

(d) | Ram lends Hari Rs. 1,000 and Hari repays Rs. 1300 to Ram at the end of 3 years in simple interest fully. The rate of interest Ram charged to Hari per annum for repayment of loan is
| (1) | ||||||||||||

(e) | A bill of Rs. 1020 is due in 6 months. True discount in rupees at interest rate 4% per annum is
| (1) | ||||||||||||

2. | Answer any one of the following: | 4x1 | ||||||||||||

(a) | The proportion of liquid I and liquid II in four samples are 2 : 1, 3 : 2, 5 : 3 and 7 : 5. A mixture is prepared by taking equal quantities of the four samples. Find the ratio of liquid I to liquid II in the final mixture. | (0) | ||||||||||||

(b) | If the difference between true discount and banker’s discount on a sum due in 3 months at 4% per annum is Rs. 20, find the amount of bill. | (0) |

SECTION II (Algebra—15 marks) |

3. | (a) | Answer any three of the following: Choose the correct option showing necessary reasons/calculations. | 2x3 | ||||||||||||||||||||||

(i) | After arranging 5, 3√3, 2√6 in decending order they are
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(ii) |
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(iii) | The number of ways in which the letters of the word COLLEGE can be arranged is
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(iv) | The number of digits in 2^{40} is (given log_{10} 2 = 0.30103)
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(v) | Correct statement among 1 ⊂ {1, 3, 4}, {1, 3} ∈ {1, 3, 4} and {1, 4} ⊂ {1, 3, 4} is
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(b) | Answer any three of the following: | 1x3 | |||||||||||||||||||||||

(i) |
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(ii) | Evaluate modulus of 3 – 2i. | (0) | |||||||||||||||||||||||

(iii) | Determine the quadratic equation whose roots are 3 and – 2. | (0) | |||||||||||||||||||||||

(iv) | Draw the graph of x ≤ – 3 in XOY plane. | (0) | |||||||||||||||||||||||

(v) |
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4. | Answer any two of the following: | 3x2 | |||||||||||||||||||||||

(a) |
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(b) |
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(c) | The volume of gas varies directly as the absolute temperature and inversely as pressure. When the pressure is 10 units and the temperature is 200 units, the volume is 160 units. What will be the volume when pressure is 12 units and temperature is 480 units. | (0) | |||||||||||||||||||||||

(d) | From 7 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done to include atleast one lady? | (0) |

SECTION III (Mensuration —15 marks) |

5. | (a) | Answer any three of the following: Choose the correct option showing necessary reasons/calculations. | 2x3 | ||||||||||||||||

(i) | The area of the triangle with sides of length 3 cm, 4 cm and 5 cm, (in sq. cm) is
| (0) | |||||||||||||||||

(ii) | The perimeter in cm of a semicircle of diameter 14 cm is
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(iii) | The volume in cu. ft of a right pyramid having altitude 6 ft and square base with length of a side 4 ft is
| (0) | |||||||||||||||||

(iv) | A path of 5 ft wide is to be laid just outside round the square garden with length of a side 50 ft. The area of the path in sq. ft would be
| (0) | |||||||||||||||||

(v) |
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(b) | Answer any three of the following | 1x3 | |||||||||||||||||

(i) | Find the total surface area of a cube whose volume is 64 cu. cm. | (0) | |||||||||||||||||

(ii) | Find the hypotenuse of a right angled isosceles triangle having area 9/2 sq. ft. | (0) | |||||||||||||||||

(iii) | A bicycle wheel of radius 35 cm makes 5000 revolutions to cover x km. Find the volume of x. | (0) | |||||||||||||||||

(iv) | Two solid spheres of radil 3 cm and 2 cm are melted and by them another solid sphere is formed. Find the volume of the new sphere. | (0) | |||||||||||||||||

(v) | A solid right circular cone have 7 cm height and 3 cm radius of base. Find its volume. | (0) | |||||||||||||||||

(vi) | Find the length of a side of rhombus having diagonals 6 cm and 8 cm. | (0) | |||||||||||||||||

6. | Answer any two of the following: | 3x2 | |||||||||||||||||

(a) | One diagonal of a rectangle is 10 ft. If the perimeter is 28 ft, find its length and breadth. | (0) | |||||||||||||||||

(b) | A circle of radius 7 cm is inscribed within a square touching the sides. Find the area of one fillet thus formed. | (0) | |||||||||||||||||

(c) | Find the volume and surface area of a hollow cylinder with height 7 inches internal and external radil of base 5 inches and 3 inches respectively. | (0) | |||||||||||||||||

(d) | A conical tent is required to accommodate 11 people. Each person must have 14 sq. ft of space on the ground and 140 cu. ft of air to breathe. Find the height, slant height and the width of the tent. | (0) |

SECTION IV (Coordinate Geometry — 10 marks) |

7. | Answer any three of the following: Choose the correct option showing necessary reasons/calculations. | 2x3 | ||||||||||||||||

(a) | Equation of a line passing through (2, 4) and having y–intercept 2 (on the positive side) is
| (0) | ||||||||||||||||

(b) | The coordinates of the point which divide the line joining (3, 6) and (12, 9) internally in the ratio 1 : 2 is
| (0) | ||||||||||||||||

(c) | Perpendicular distance of the line 3x + 4y = 1 from the point (4, 1) is
| (0) | ||||||||||||||||

(d) | The radius of the circle 2x^{2} + 2y^{2} + 12y = 8x + 6 is
| (0) | ||||||||||||||||

(e) | Eccentricity of the elipse 5x^{2} + 9y^{2} = 405 is
| (0) | ||||||||||||||||

8. | Answer any one of the following: | 4x1 | ||||||||||||||||

(a) | Find the equation of a straight line passing through the point (2, 3) and perpendicular to the line x + 2y = 5. | (0) | ||||||||||||||||

(b) | Find the equation of the parabola whose focus is (1, 1) and directrix is x + y = 1. Find also the length of the latus rectum. | (0) |

SECTION V (Calculus — 15 marks) |

9. | (a) | Answer any three of the following: Choose the correct option showing necessary reasons/calculations. | 2x3 | |||||||||||||||||||||||||||

(i) |
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(ii) |
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(iii) |
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(iv) | The value of x for which x (12 – x^{2}) is maximum is
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(v) |
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(vi) | If the total cost function C = x^{3} – 2x^{2} + 5x, then the marginal cost is equal to
| (0) | ||||||||||||||||||||||||||||

(b) | Answer any three of the following: | 1x3 | ||||||||||||||||||||||||||||

(i) |
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(ii) |
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(iii) |
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(iv) |
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(v) |
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10. | Answer any two of the following: | 3x2 | ||||||||||||||||||||||||||||

(i) | A function (fx) is defined as
Examine whether f(x) is continuous at x = 1. | (0) | ||||||||||||||||||||||||||||

(ii) | Verify Euler’s theorem for the function ax^{2} + 2bxy + cy^{2}. | (0) | ||||||||||||||||||||||||||||

(iii) |
| (0) |

SECTION VI (Statistical Methods — 35 marks) |

11. | (a) | Answer any nine of the following: Choose the correct option showing necessary reasons/calculations. | 2x9 | |||||||||||||||||

(i) | The arithmetic mean of first 9 counting numbers occurring with same frequency has its value
| (0) | ||||||||||||||||||

(ii) | If 2 occurs 4 times, 4 occurs 3 times, 8 occurs twice and 16 occurs once then geometric mean of them is
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(iii) | If a person travels first 2 km @ km/hr., next 3 km @ 3 km/hr and another 5 km @ 5 km/hr, his average speed during his journey is
| (0) | ||||||||||||||||||

(iv) | The median of marks 55, 60, 50, 40, 57, 45, 58, 65, 57, 48 of 10 students is
| (0) | ||||||||||||||||||

(v) | If the relation between two variables x and y is 3x – 2y = 5 and mode of x is 5 then mode of y is
| (0) | ||||||||||||||||||

(vi) | If the two variables x and y are related by the equation 3x = 2y + 4 and mean deviation of x about its means is 4 then mean deviation of y about its mean is
| (0) | ||||||||||||||||||

(vii) |
x
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(viii) | If the relation between two variables x and y is 2x + 3y = 5 and standard deviation of y is 10 then the standard deviation of x is
| (0) | ||||||||||||||||||

(ix) | If mean, mode and standard deviation of 10 observations are 65, 80 and 25 respectively then type of skewness of the data is
| (0) | ||||||||||||||||||

(x) | If the mean of 50 observations is 50 and one observation 94 us wrongly recorded there as 49 then correct mean will be
| (0) | ||||||||||||||||||

(xi) | If for two observations arithmetic mean is 80 and harmonic mean is 5 then geometric mean of them is
| (0) | ||||||||||||||||||

(xii) | For moderately skewed distribution A.M. = 110, Mode = 104, then median is
| (0) | ||||||||||||||||||

(xiii) | If the maximum and minimum values of 10 observations are 40 and 10 then coefficient of range is
| (0) | ||||||||||||||||||

(xiv) | The standard deviation (SD) of a variable x is 10, then the SD of the variable 2x + 10 is
| (0) | ||||||||||||||||||

(b) | Answer any three of the following: | 1x3 | ||||||||||||||||||

(i) | 12 observations are 2, 4, 6, 3, 3, 5, 6, 8, 4, 3, 5, 4. If I is subtracted from each of them find the range. | (0) | ||||||||||||||||||

(ii) |
| (0) | ||||||||||||||||||

(iii) | If the means of two groups of m and n observations are 50 and 60 respectively and combined mean is 54. Find the ratio m : n. | (0) | ||||||||||||||||||

(iv) | The mean deviation about mean 40 is 20, find the coefficient of mean deviation about mean. | (0) | ||||||||||||||||||

(v) | If mean and standard deviation of runs scored by a batsman in 10 tests are 50 and 4 respectively, find the coefficient of variation of runs. | (0) | ||||||||||||||||||

(vi) |
| (0) | ||||||||||||||||||

12. | (a) | Answer any two of the following: | 5x2 | |||||||||||||||||

(i) | Calculate mean and variance of the following grouped frequency distribution:
| (0) | ||||||||||||||||||

(ii) | Prove that the standard deviation of n observations is independent of change of origin but is dependent on the change of scale. | (0) | ||||||||||||||||||

(iii) | Two samples of sizes 100 and 150 have means 45 and 55 and standard deviations 7 and 12 respectively. Find the mean and standard deviation of the combined sample. | (0) | ||||||||||||||||||

(iv) | Find median and mode from the following grouped frequency distribution:
| (0) | ||||||||||||||||||

(b) | Write short note on any one of the following | 4x1 | ||||||||||||||||||

(i) | Histogram, | (0) | ||||||||||||||||||

(ii) | Pic Chart. | (0) |