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 (i) | 9 and 12 | (ii) | 12 and 16 | (iii) | 15 and 20 | (iv) | none of these | | | (2) |
| (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 (i) | 40 km/h | (ii) | 25 km/h | (iii) | 10 km/h | (iv) | none of these | | | (1) |
| (c) | For a sum of money to become 2¼ times of itself in 5 years, the rate of interest is (i) | 25% | (ii) | 30% | (iii) | 35% | (iv) | none of these | | | (0) |
2. | Answer any one of the following. | 4x1 | |
| (a) | If | | = | | = | | then prove that α + β + γ = 0 =pα + qβ + rγ. | | | (0) |
| (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 (i) | x = 1 | (ii) | x = 3 | (iii) | x = 4 | (iv) | None of these. | | | (0) |
| (b) | The number of ways can the letters of the word MONDAY be arranged to end with Y but not begin with M is (i) | 24 | (ii) | 96 | (iii) | 600 | (iv) | none of these. | | | (0) |
| (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. (i) | 100 | (ii) | 200 | (iii) | 250 | (iv) | none of these | | | (0) |
| (d) | If A = (1, 2, 3, 4), B = (2, 3, 5, 6) and C = (3, 4, 6, 7) then (A − B) ∩ (A − C) is (i) | (1) | (ii) | (1, 2) | (iii) | (1, 2, 3) | (iv) | None of these | | | (0) |
| (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 (i) | p ∨ q | (ii) | p ∧ q | (iii) | p ∧ ∼ q | (iv) | ∼ p ∧ ∼ q | | | (0) |
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) | Evaluate : | log 3√3 + log √8 − log √125 | log 6 − log 5 |
| | | (0) |
| (c) | lf w be an imaginary cube root of unity then show that (1 + w − w2 ) (1 − w + w2) = 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 (i) | √3 cm | (ii) | | cm | (iii) | | cm | (iv) | none of these | | | (0) |
| (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? (i) | 600 | (ii) | 625 | (iii) | 650 | (iv) | none of these | | | (0) |
| (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 (i) | 594 | (ii) | 595 | (iii) | 596 | (iv) | none of these | | | (0) |
| (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 (i) | 4cm | (ii) | 9cm | (iii) | 12cm | (iv) | none of these | | | (0) |
| (e) | The volumes of two cones having equal radius of their bases are in the ratio 1 : 2. The ratio of their heights is (i) | 1 : 3 | (ii) | 3 : 1 | (iii) | 2 : 1 | (iv) | none of these | | | (0) |
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 (i) | 1 : 2 | (ii) | 2 : 1 | (iii) | 1 : 3 | (iv) | none of these | | | (0) |
| (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 (i) | 3x + 2y = 1 | (ii) | 2x − 3y = 1 | (iii) | 3x + 2y = 7 | (iv) | none of these | | | (0) |
| (c) | The centre and radius of the circle (x − 2) (x − 4) + (y − 3) (y − 5) = 0 are (i) | (3, − 4); 2 | (ii) | (3, 4); √2 | (iii) | ( − 3, 4); 4 | (iv) | none of these | | | (0) |
| (d) | The eccentricity of the ellipse 4x2 − 24x + 9y2 + 36y + 36 = 0 is (i) | | (ii) | | (iii) | | (iv) | none of these | | | (0) |
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 Find the equation of the hyperbola. | | (0) |
SECTION V (Calculus — 15 marks) |
9. | Answer any three of the following: Choose the correct option showing proper reasons/calculations. | 3x3 | |
| (a) | (i) | x | (ii) | | (iii) | | (iv) | none of these | | | (0) |
| (b) | The value of k for which f(x) = x + 2 for x ≤ 2 = k – x2 for x > 2 is continuous at x = 2 is (i) | 8 | (ii) | 6 | (iii) | 4 | (iv) | none of these | | | (0) |
| (c) | If y = x3,then the value of 1 + ( | | )2 when x = − 1 is |
(i) | − 37 | (ii) | 37 | (iii) | 35 | (iv) | none of these | | | (0) |
| (d) | If u = x2 + y2 + z2, the value of xux + yuy + zuz is (i) | 2u | (ii) | 2 | (iii) | −2u | (iv) | None of these | | | (0) |
| (e) | The value of | ∫1 0 | 52x dx is |
(i) | 12 loge 5 | (ii) | 12 log5e | (iii) | 2 loge5 | (iv) | None of these | | | (0) |
10. | Answer any two of the following: | 3x2 | |
| (a) | If y = x2 loge x, show that x2 | | + 4y = 3x | | | | (0) |
| (b) | Show that x3 - 6x2 + 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) | The harmonic mean of the numbers 1, | | , | | , ....., | | is |
(i) | | (ii) | | (iii) | | (iv) | none of these | | | (0) |
| (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 (i) | 64 | (ii) | 32 | (iii) | 128 | (iv) | none of these | | | (0) |
| (c) | The median of the following frequency distribution of x x: frequency: is | 1 11 | 2 20 | 3 29 | 4 25 | 5 13 | 6 2 |
(i) | 2.5 | (ii) | 3.5 | (iii) | 4.5 | (iv) | none of these | | | (0) |
| (d) | For a group of 10 items Σx = 60, Σx2 = 850 and mode = 5. Then the Pearson’s coefficient of skewness is (i) | | (ii) | | (iii) | | (iv) | none of these | | | (0) |
| (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 (i) | 12 | (ii) | 10 | (iii) | 15 | (iv) | none of these | | | (0) |
| (f) | Mean deviation about median of 13, 84, 68, 24, 96, 139,84,27 is (i) | 33.88 | (ii) | 34.88 | (iii) | 35.88 | (iv) | none of these | | | (0) |
| (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 (i) | 1 | (ii) | 1.5 | (iii) | 2 | (iv) | none of these | | | (0) |
| (h) | If | 5 Σ i=1 | (xi – 2) = 15, | 5 Σ i = 1 | (xi— 3)2 = 50, then variance of x1, x2, x3, x4, and x5 is |
(i) | 2 | (ii) | 4 | (iii) | 6 | (iv) | none of these | | | (0) |
| (i) | If the variance of the first n natural numbers is 14, then the value of n is (i) | 12 | (ii) | 11 | (iii) | 13 | (iv) | none of these | | | (0) |
| (j) | Arithmetic mean of a series of observations is 6 and its coefficient of variation is 50%, then the variance of the observations is (i) | 10 | (ii) | 9 | (iii) | 8 | (iv) | none of these | | | (0) |
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: Year Number of students | : : | 1970 20 | 1971 30 | 1972 40 | 1973 35 | | | (0) |
| | (ii) | Show that mean deviation about mean and s.d. of two observations x1 and x2 are same. | | (0) |
| | (iii) | Find the variance of the following frequency distribution: class interval frequency | : : | 5 – 10 5 | 10 – 15 9 | 15 – 20 16 | 20 – 25 14 | 25 – 30 6 | | | (0) |
| (b) | Write a short note on any one of the following: | 4x1 | |
| | (i) | Tabulation; | | (0) |
| | (ii) | Central tendency of data. | | (0) |