This Paper has

**50**answerable questions with**0**answered.CAT—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 the questions. |

Notations and symbols have usual meanings |

SECTION I (Arithmetic — 10 marks) |

Marks |

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

(a) | If x is the mean proportional between x – 2 and x + 6 then the value of x is
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(b) | Of the five numbers the average of first four numbers is 8 and the average of the last four numbers is 6. Then the difference of the first and the fifth number is
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(c) | The true discount on a bill due in 6 months at 8% p.a. is Rs. 40. Then the amount of the bill is
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2. | Answer any one of the following. | 4x1 | ||||||||||||

(a) | Divide Rs. 6,200 in 3 parts such that the interest for the three parts for 2, 3 and 5 years respectively at 5% simple interest p.a. are same. | (0) | ||||||||||||

(b) | A dealer mixes two varieties of teas costing Rs. 100 per kg. and Rs. 160 per kg. in the proportion 5:1. He sold the 6 kg. mixture at the rate of Rs. 120 per kg. Find his profit. | (0) |

SECTION II (Algebra — 15 marks) |

Marks |

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

(a) | In a class of 80 students 52 read Mathematics, 36 read Statistics and 20 read both Mathematics and Statistics. The number of students who read neither Mathematics nor Statistics is
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(b) | If logarithm of a number to the base √2 is 4, then the logarithm of the same number to the base 2√2 is
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(c) | The number of ways in which letters of the word MONDAY be arranged beginning with the letter O and ending with the letter Y is
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(d) | If p and q be two logical statements then (p∨q) ∨ ∼ p is
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(e) | The area of a circle varies directly with square of its diameter. Area of the circle is 38.5 sq.cm when diameter is 7 cm. If diameter of the circle is 1 cm then area of the circle in sq.cm is
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4. | Answer any two of the following: | 3x2 | ||||||||||||||||||

(a) | If w be an imaginary cube root of unity then find the value of (1 – w + w^{2}) (1 + w – w^{2}). | (0) | ||||||||||||||||||

(b) | Simple interest and compound interest in 2 years for some principal are Rs. 200 and Rs. 210 at the same rate of interest per annum. Find the principal amount. | (0) | ||||||||||||||||||

(c) | The volume of a gas varies directly as the absolute temperature and inversely as pressure. When the pressure is is 15 units and the temperature is 260 units the volume is 200 units. What will be the volume when the pressure is 18 units and the temperature is 195 units? | (0) |

SECTION III (Mensuration — 15 marks) |

Marks |

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

(a) | The perimeter of an equilateral triangle is 36 cm. Then the area of the triangle is
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(b) | The sum of the interior angles and each interior angle of a pentagon is
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(c) | The sides of a cuboid are 40 cm, 20 cm and 10 cm. It is melted to form a new cube. The surface area of the new cube in sq.cm is
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(d) | The volume of hollow right circular cylinder of height 14 cm with internal and external radii of base 8 cm and 10 cm respectively, has the volume in cu.cm as
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(e) | A right prism has triangular base whose sides are 13 cm, 20 cm and 21 cm. If the altitude of the prism is 9 cm then the volume of the prism is
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6. | Answer any two of the following: | 3x2 | |||||||||||||||||

(a) | A right pyramid stands on a base 16 cm square and its height is 15 cm. Find the slant surface and volume of the pyramid. | (0) | |||||||||||||||||

(b) | A road of one meter wide is developed around a circular garden with diameter 20 m @ Rs. 100 per sq.m. Find the cost of development of the road.
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(c) | The volume of two spheres are in the ratio 64 : 27. Find their radii if the sum of their radii is 21 cm.
| (0) |

SECTION IV (Coordinate Geometry — 10 marks) |

Marks |

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

(a) | If the three points (1, 2), (2, 4) and (x, 6) are collinear then the value of x is
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(b) | If the line joining the points (2, –2) and (6, 4) are parallel to the line joining the points
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(c) | The three points A(a, 0), B(–a, 0), C(c, 0) and p is a point such that PB then the locus of ^{2} + PC^{2} = 2PA^{2}P is
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(d) | The centre of a circle 3(x^{2} + y^{2}) = 6x + 6y – 5 is
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8. | Answer any one of the following: | 4x1 | |||||||||||||||||

(a) | Find the equation of the parabola having vertex (3, 1) and focus (1, 1). | (0) | |||||||||||||||||

(b) | For the hyperbola 9x^{2} – 16y^{2} – 36x –108 = 0 find the coordinates of the centre and its latus rectum. | (0) |

SECTION V (Calculus — 15 marks) |

Marks |

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

(a) | f(x) = log_{e} (x – 3) (x – 5) is undefined in the region
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(b) |
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(c) |
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(d) | The differentiation of x^{4} with respect to x^{3} is
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(e) |
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10. | Answer any two of the following: | 3x2 | |||||||||||||||||||||||||||||||||

(a) | If y = log (x + √x^{2} + a^{2}) then prove that (a^{2} + x^{2}) y_{2} + xy_{1} = 0. | (0) | |||||||||||||||||||||||||||||||||

(b) |
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(c) | Find the area of the region bounded by curves y^{2} = x and y = x. | (0) |

SECTION VI (Statistical Methods — 35 marks) |

Marks |

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

(a) | The arithmetic mean of 4 observations is 8 and that of 10 observations including those 4 is 11. Then arithmetic mean of remaining 6 observations is
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(b) | Geometric mean of 10 observations 2, 2, 4, 4, 8, 8, 16, 16, 32, 32 is
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(c) | Harmonic mean of 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5 is
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(d) | Number of peas of 50 peapods are as follows:
Median of number of peas is
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(e) | The number of members in 30 families are as follows: 1, 3, 1, 3, 4, 5, 3, 3, 1, 3, 3, 4, 5, 4, 2, 3, 3, 2, 2, 5, 2, 4, 2, 2, 3, 2, 4, 2, 4, 4 Then mode of number of members in a family is
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(f) |
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(g) | The mean deviation about 12 of the following distribution
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(h) | If variance of 10 values is 9 and sum of deviation of those ten values about 3 is 60 then mean of squares of deviations of those 10 values about 5 is
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(i) | If runs of two players A and B in 10 cricket matches are such that player A has mean 50 and variance 36 and player B has mean 60 and variance 81 of runs then the player more consistence in runs is
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(j) | For a distribution with A.M. = 50, coefficient of skewness –0.4 and s.d. 20, value of mode is
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12. | (a) | Answer any two of the following: | 5x2 | ||||||||||||||||||||

(i) | Find the mean and standard deviation of following frequency distribution of ages:
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(ii) | Find median and mode of the following distribution:
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(iii) | If the first of two samples has 100 items with mean 15 and variance 9 and the second has 150 items with mean 16 and variance 16, find the mean and variance of the combined sample. | (0) | |||||||||||||||||||||

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

(i) | Central tendency of data; | (0) | |||||||||||||||||||||

(ii) | Ogive less than type. | (0) |