This Paper has 50 answerable questions with 1 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) |
| Marks |
| 1. | Answer any two of the following:
Choose the correct option showing the proper reasons/calculations. | 3x2 | |
| | (a) | The number to be added to each term of the ratio 3 : 7 to make it 1 : 2 is | (A) | 2, | (B) | 1, | (C) | 3, | (D) | none of these |
| | (1) |
| | (b) | The average of 7 numbers is 27. If one number is included, the average becomes 25. The included number is | (A) | 11, | (B) | 10, | (C) | 12, | (D) | none of these |
| | (0) |
| | (c) | The time in which a sum of money becomes double at 10% p.a., simple interest is | (A) | 8 years | (B) | 10 years, | (C) | 12 years, | (D) | none of these |
| | (0) |
| 2. | Answer any one of the following. | 4x1 | |
| | (a) | A bill for Rs. 2060 is due in 6 months. Calculate the difference between True Discount (TD) and Banker’s Discount (BD), the rate of interest being 6%. | | (0) |
| | (b) | In a liquid mixture 20% is water and in another mixture water is 25%. These two mixtures are mixed in the ratio 5:3. Find the percentage of water in the final mixture. | | (0) |
| SECTION II (Algebra — 15 marks) |
| Marks |
| 3. | Answer any three of the following: Choose the correct option showing necessary reasons/calculations. | 3x3 | |
| | (a) | | If x = 2 + √3 then the value of x4 + | | is |
| (i) | 98, | (ii) | 196, | (iii) | 194, | (iv) | None of these. |
| | (0) |
| | (b) | ncr + ncr-1 is equal to | (i) | n–1cr‘ | (ii) | n+1cr‘ | (iii) | ncr+1‘ | (iv) | none of these. |
| | (0) |
| | (c) | Given a varies as bx +c, Value of a is 3 when b = 1, c= 2 and is 5 when b = 2, c = 3. The value of x would be | (i) | –1, | (ii) | 2, | (iii) | 3, | (iv) | none of these |
| | (0) |
| | (d) | If one root of the equation x2 - bx + K = 0 is twice the other root then the value of K is | (i) | –8, | (ii) | 8, | (iii) | 4, | (iv) | None of these |
| | (0) |
| | (e) | | If | | = | | = | | then the value of zyz is |
| (i) | 1, | (ii) | 0, | (iii) | –1, | (iv) | none of these |
| | (0) |
| 4. | Answer any two of the following: | 3x2 | |
| | (a) | If universal set is {1, 2, 3, 4, 5, 6}, A = {2, 4, 5}, B = {1, 3, 5}, C = {5, 6} then find | (i) | B – A, | (ii) | A ∪ B – c, | (iii) | (A ∪ B ∪ C)’ where A’ represent complement of A. |
| | (0) |
| | (b) | Find the square root of 16 ‘ 30i. | | (0) |
| | (c) | The number of handshakes in a party was counted as 66. Determine the number of guests attending the party, assuming all guests shake hands with each other. | | (0) |
| SECTION III (Mensuration — 15 marks) |
| Marks |
| 5. | Answer any three of the following: Choose the correct option showing necessary reasons/calculations. | 3x3 | |
| | (a) | If two adjacent sides of right angle of a right–angled triangle are such that the length of one side is twice the other and the hypotenuse is 5 cm then area of the triangle in sq. cm is | (i) | 10, | (ii) | 5, | (iii) | 2.5, | (iv) | none of these |
| | (0) |
| | (b) | If the parameter of a semicircle is 36 cm then area of that semicircle in sq.cm is| (i) | 144, | (ii) | 22, | (iii) | 77, | (iv) | none of these |
| | (0) |
| | (c) | Surface of a cube of volume 125 cu.ft. are painted with black colour at cost of Rs. 10 per sq.ft. The amount required to paint the outer surface of the cube in Rs. is | (i) | 1500, | (ii) | 1250, | (iii) | 1,000, | (iv) | none of these |
| | (0) |
| | (d) | If 3 solid spheres of radii 3 ft., 4 ft, and 5 ft. of iron are melted to form a new sphere, the surface area of the new sphere in square feet is| (i) | | (ii) | | (iii) | | (iv) | none of these |
| | (0) |
| | (e) | A right pyramid stands on a base of 12 cm square and its height is 8 cm. Then its total surface area in sq. cm is | (i) | 240, | (ii) | 384, | (iii) | 624, | (iv) | none of these |
| | (0) |
| 6. | Answer any two of the following: | 3x2 | |
| | (a) | The area of three adjacent sides of a cuboid are 15 sq. cm, 10 sq. cm and 6 sq. cm. Find the volume of the cuboid. | | (0) |
| | (b) | The height and slant height of a right circular cone is 24 cm and 25 cm respectively. Find the area of the curved surface and volume. | | (0) |
| | (c) | The sum of length, breadth and height of a rectangular parallelopiped is 24 cm and its diagonal is 15 cm. Find the area of the whole surface of the parallelopiped. | | (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) | The area of the triangle formed by points (0, 0), (5, 0) and (0, 6) is | (i) | 11 sq. unit, | (ii) | 30 sq. unit, | (iii) | 15 sq. unit, | (iv) | 7.5 sq. unit. |
| | (0) |
| | (b) | The equation of a straight line passing through the point (5, 5) and is perpendicular to the line x = y is | (i) | x - y = 5, | (ii) | x + y = 5, | (iii) | x + y = 10 | (iv) | None of these. |
| | (0) |
| | (c) | A point p having coordinate (x, y) moves such that its distance from the points (1, 3) and (2, –3) are equal. Then locus of p is | (i) | 2x = 12y + 3, | (ii) | 2x = 6y - 3, | (iii) | 6x = y + 3, | (iv) | none of these. |
| | (0) |
| | (d) | The directrix of the parabola x2 = 4x + 3y + 5 is | (i) | 4y = 15, | (ii) | 4y + 15 = 0, | (iii) | 4x + 15 = 0, | (iv) | none of these. |
| | (0) |
| 8. | Answer any one of the following: | 4x1 | |
| | (a) | Find the equation of a circle which touches both the axes in the first quadrant at a distance of 5 units from the origin. | | (0) |
| | (b) | Find the co-ordinate of the centre and eccentricity of the ellipse 4x2+ 9y2 + 18y = 16x + 11. | | (0) |
| SECTION V (Calculus — 15 marks) |
| Marks |
| 9. | Answer any three of the following:
Choose the correct option showing necessary reasons/calculations. | 3x3 | |
| | (a) | | If h = g(x) = | | , then g(h) is equal to |
| (i) | q, | (ii) | x, | (iii) | p, | (iv) | none of these. |
| | (0) |
| | (b) | | (i) | q –p, | (ii) | | (iii) | p –q, | (iv) | none of these |
| | (0) |
| | (c) | | If y = x√ 1+x2 then | | at x = √3 is |
| (i) | | (ii) | | (iii) | 5, | (iv) | None of these. |
| | (0) |
| | (d) | | (i) | | (ii) | | (iii) | | (iv) | None of these. |
| | (0) |
| | (e) | | If f(x, y) = 3x3 – 5x2y + 2y3 then x | | + y | | is |
| (i) | f(x, y), | (ii) | 3, | (iii) | 3f(x, y), | (iv) | none of these |
| | (0) |
| 10. | Answer any two of the following: | 3x2 | |
| | (a) | | If xayb = (x + y)a + b show that | | = | | where a and b are independent of x and y. |
| | (0) |
| | (b) | If y = Aemx + Be–mx show that y2 – m2y = 0. | | (0) |
| | (c) | Find the area of the region lying in the first quadrant bounded by the parabola y2 = 4x, the x–axis and the ordinate x = 4. | | (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) | First 10 odd counting numbers each occurring twice has arithmetic mean | (i) | 40, | (ii) | 10, | (iii) | 20, | (iv) | none of these |
| | (0) |
| | (b) | Geometric mean (G.M) of six numbers is 16. If G.M. of first four of them is 8 then G.M. of other two is | (i) | 8, | (ii) | 16, | (iii) | 32, | (iv) | none of these |
| | (0) |
| | (c) | Two positive observations have arithmetic mean 3 and geometric mean 2√2. If each observation is multiplied by 2 then harmonic mean will be | (i) | | (ii) | | (iii) | 12, | (iv) | none of these |
| | (0) |
| | (d) | If the sum of deviations of a number of observations about 4 and that about 3 are 40 and 50 respectively then arithmetic mean of the observations is | (i) | 4 | (ii) | 6 | (iii) | 10 | (iv) | none of these |
| | (0) |
| | (e) | If the relation between 2 variable x and y is xy = 2 and arithmetic mean of variable x is 10, then harmonic mean of variable y is | (i) | | (ii) | | (iii) | | (iv) | none of these |
| | (0) |
| | (f) | | If the variable x takes 20 values x1, x2,.........x10–x1, –x2,.........–x10 such that | | x12 |
= 40 then standard deviation of x is | (i) | 1, | (ii) | 2, | (iii) | 4, | (iv) | none of these |
| | (0) |
| | (g) | If relation between 2 variables x and y is 2x + 3y = 5 and mean deviation of x values about mean is 9 for 10 observations, then sum of absolute deviations of corresponding 10 y–values about mean is | (i) | 90, | (ii) | 30, | (iii) | 6, | (iv) | none of these |
| | (0) |
| | (h) | If for 10 values of x sum of deviations about 5 us 10 and sum of squares of deviations about 4 is 100 then variance of x is | (i) | 4, | (ii) | 6, | (iii) | 10, | (iv) | none of these |
| | (0) |
| | (i) | If two samples of sizes 4 and 5 have same mean but different standard deviations 1 and 3 respectively then the standard deviation of the combined sample is | (i) | √5, | (ii) | | (iii) | | (iv) | none of these |
| | (0) |
| | (j) | If the mode, variance and coefficient of skewness of a frequency distribution are 100, 16 and 6 respectively then mean of the distribution is | (i) | 124, | (ii) | 76, | (iii) | 108, | (iv) | none of these |
| | (0) |
| 12. | (a) | Answer any two of the following: | 5x2 | |
| | | (i) | Find the mean and the mean deviation about mean of the following frequency distribution: Weight (in kg):
No. of persons | 50
1 | 55
4 | 60
2 | 65
2 | 70
1 | Total
10 |
| | (0) |
| | | (ii) | Show that the combined arithmetic mean x of two groups lies between the arithmetic means x1 and x2 of the two groups. | | (0) |
| | | (iii) | The arithmetic mean and geometric mean of two observations are 20 and 12 respectively. Find the observations and harmonic mean of them. | | (0) |
| | (b) | Write short note on any one of the following | 4x1 | |
| | | (i) | Histogram, | | (0) |
| | | (ii) | Skewness and its two important measures. | | (0) |
