This Paper has 65 answerable questions with 3 answered.
C—4(BMS)
Revised Syllabus |
| Time Allowed : 3 Hours | Full Marks : 100 |
The figures in the Margin on the right side indicate full marks.
Answer all questions. |
| Section A |
| ARITHMETIC (15 marks) |
| Marks |
| 1. | (a) | Answer any three of the following: Choose the correct option showing the proper reasons/calculations. | 1x3 | |
| | | (i) | If x be added to the numbers 10 and 30, then they will be in the ratio 3 : 4. So x is | (A) | 20 | (B) | 30 | (C) | 50 | (D) | none of these |
| | (2) |
| | | (ii) | Subtriplicate ratio of 8 : 27 is | (A) | 2 : 3, | (B) | 512 : 19683 | (C) | 2√2 : 3√3 | (D) | none of these |
| | (0) |
| | | (iii) | Rs. 20 is decreased at the rate of 5% to | (A) | Rs. 21 | (B) | Rs. 19 | (C) | Rs. 15 | (D) | none of these |
| | (1) |
| | | (iv) | The mean of 4 numbers is 10. If one number is excluded the mean will be 8. The excluded number is | (A) | 10 | (B) | 8 | (C) | 9 | (D) | none of these |
| | (0) |
| | (b) | Answer any two of the following: | 2x2 | |
| | | (i) | | If | | = | | = | | then find the value of | | where none of a, b, c is zero. |
| | (0) |
| | | (ii) | A bill for Rs. 816 is due in 6 months. Find the true discount if the rate of interest is 4% per annum. | | (0) |
| | | (iii) | A man mixed 3 litres of kerosene oil, purchased at Rs. 2 per litre and 2 litres of kerosene oil, purchased at Rs. 4.50 per litre. Find the cost price of the mixture per litre. | | (0) |
| 2. | Answer any two of the following. | 4x2 | |
| | (a) | A man spent 20% of his money and Rs. 50 after it. Then he spent 20% of the remainder. If he had Rs. 1980 left, what was his original money? | | (0) |
| | (b) | A class has 3 divisions. Average marks of the students of the class, first division, second division and third division are 47, 44, 50 and 45 respectively in Mathematics. If first two divisions have 30 and 40 students, find the number of students in third division when all the students of the class have Mathematics as a subject. | | (0) |
| | (c) | Two vessels contain mixtures of milk and water in the proportion 1 : 2 and 3 : 2 respectively. In what proportion should the two mixtures be mixed together so as to form a new mixture containing equal quantity of milk and water? | | (0) |
| | (d) | If the banker’s gain on a bill, due for 6 months at the rate of 4% per annum be Rs. 10 find the bill value and the present value. | | (0) |
| Section B |
| ALGEBRA (25 marks) |
| 3. | (a) | Answer any five of the following: Choose the correct option showing necessary reasons/calculations. | 2x5 | |
| | | (i) | The value of 2 —√— 16 + √— 4 is | (A) | — 1 | (B) | 0 | (C) | — (2 + i) | (D) | none of these |
| | (1) |
| | | (ii) | I w is an imaginary cube root of unity the value of w(1 + w – w2) is | (A) | 1 | (B) | 0 | (C) | 2 | (D) | none of these |
| | (0) |
| | | (iii) | | If both a and b are rational numbers the value of (a, b) in | | = a + b + √3 is |
| (A) | (2, –1) | (B) | (–2, 1) | (C) | (2, 1) | (D) | none of these |
| | (0) |
| | | (iv) | Quadratic equation having one root 2 + √3 is | (A) | x2 + 1 = 4x | (B) | x 2 + 4x + 1 = 0 | (C) | x 2 + 4x = 1 | (D) | none of these |
| | (0) |
| | | (v) | The value of x in 10Cx = 10 x 9Cx–1 is | (A) | 0 | (B) | 1 | (C) | 2 | (D) | none of these |
| | (0) |
| | | (vi) | The logarithm of 25 to the base √5 is | (A) | 2 | (B) | 3 | (C) | 4 | (D) | none of these |
| | (0) |
| | | (vii) | The number of digits in 220, where log10 2 = 0.30103, is | (A) | 7 | (B) | 6 | (C) | 5 | (D) | none of these |
| | (0) |
| | | (viii) | If M = {1, 2, 3, 5}, N = {2, 3, 4, 5}, P = {3, 5, 6} then M ∩ (N – P) is | (A) | {5} | (B) | {3} | (C) | {2, 4} | (D) | none of these |
| | (0) |
| | (b) | Answer any three of the following: | 1x3 | |
| | | (i) | Simplify: 3√48 –2√75 | | (0) |
| | | (ii) | Write the set {x : x is an integer, –2 < x < 2} in Roster form. | | (0) |
| | | (ii) | If 5 is one root of the quadratic equation x2 – ax = 15 find the value of a. | | (0) |
| | | (iv) | The following statements are given:
p : A number is an odd number
q : A number is greater than 5.
State the truth value of the conjunction p ∧ q. | | (0) |
| 4. | Answer any three of the following: | 4x3 | |
| | (a) | Find the number of years at the end of which Rs. 100 will become Rs. 1000 allowing compound interest at 4% per annum. [Given log 104 = 2.0170333]. | | (0) |
| | (b) | Solve : XY = 6, yz = 2, zx = 3. | | (0) |
| | (c) | The monthly expenses of a boarding house are partly fixed and partly varied with the number of boarders. The monthly charges are Rs. 100 per head where there are 25 boarders and Rs. 80 per head when there are 15 boarders. If the monthly charge per head is Rs. 70 find the number of boarders. | | (0) |
| | (d) | Form a quadratic equation whose roots are cube of the roots of 2x2 + 4 = 6x. | | (0) |
| | (e) | Out of 5 different consonants and 4 different vowels how many different words can be formed each containing 3 consonants and 2 vowels? | | (0) |
| | (f) | Prove that xlog y – logz ylog z – logx zlog x – logy = 1. | | (0) |
| Section C |
| MENSURATION (30 marks) |
| 5. | Answer any seven of the following: Choose the correct option showing necessary reasons/calculations. | 2x7 | |
| | (a) | If a circle’s area and length of circumference are numerically equal then the length of circumference is | (A) | 2π units | (B) | π units | (C) | 4π units | (D) | none of these |
| | (0) |
| | (b) | If a rectangle’s length is twice its breadth and length of one diagonal is 5√5 cm, then the area of the rectangle is | (A) | 20 sq.cm | (B) | 10 sq.cm | (C) | 5 sq.cm | (D) | none of these |
| | (0) |
| | (c) | If total surface area of a cube is 54 sq.cm, then length of one diagonal of the cube is | (A) | 3√3 cm | (B) | 6√3 cm | (C) | 9√3 cm | (D) | none of these |
| | (0) |
| | (d) | The volume of a hollow cylinder of height 10 cm with internal and external radii of base 4 cm and 5 cm is | (A) | 250π cu.cm | (B) | 160π cu.cm | (C) | 90π cu.cm | (D) | none of these |
| | (0) |
| | (e) | If the volume of a right circular cone, having same radius of its base and height, is 9π cu cm, then diameter of the base is | (A) | 6 cm | (B) | 12 cm | (C) | 18 cm | (D) | none of these |
| | (0) |
| | (f) | If the point P divides the line segment joining (4, 5) and (7, 2) internally in the ratio 1 : 2 then the co-ordinates of P are | (A) | (5.5, 3.5) | (B) | (6, 3) | (C) | (5, 4) | (D) | none of these |
| | (0) |
| | (g) | The equation of a line passing through the point (4, 5) and having gradient 2 is | (A) | y + 3 = 2x | (B) | x + 6 = 2y | (C) | x + 2y = 14 | (D) | none of these |
| | (0) |
| | (h) | The equation of a circle touching the x–axis at origin and having radius unity on the positive side of y–axis, is | (A) | x2 + y2 = 2y + 2 | (B) | x2 + y2 = 2y | (C) | x2 + y2 = 2x | (D) | none of these |
| | (0) |
| | (i) | A parabola y2 = px passing through (4, –2) has the directrix | (A) | x + 1 = 0 | (B) | x = ¼ | (C) | x + ¼ = 0 | (D) | none of these |
| | (0) |
| | (j) | The (eccentricity, length of latus rectum) of a hyperbola 9x2 – 16y2 = 144 is | (A) | ( | | , | | ) | (B) | ( | | , | | ) | (C) | ( | | , | | ) | (D) | None of these |
| | (0) |
| | (k) | The area of a road, 7 ft wide around a circular park, having circumference 88 ft outside, is | (A) | 770 sq.ft | (B) | 220 sq.ft | (C) | 154 sq.ft | (D) | none of these |
| | (0) |
| 6. | Answer any four of the following: | 4x4 | |
| | (a) | One side of a rhombus is 5 ft and one of its diagonal is 6 ft. Find the other diagonal and the area of the rhombus. | | (0) |
| | (b) | A solid cylindrical wire of length 50.82 metre, radius 1.5 cm is melted into a cube. What is the length of a side of the cube? | | (0) |
| | (c) | The radii of two ends of a frustum of a rectangular cone are 6 cm and 3 cm respectively. Its height is 4 cm. Find the total surface area of the frustum. | | (0) |
| | (d) | Find the equation of a straight line passing through the point (3, 5) and perpendicular to 3x + 5y = 7. | | (0) |
| | (e) | Show that the two circles x2 + y2 – 2x – 2y = 2 and x2 + y2 – 10x + 4y + 20 = 0 touch each other externally. Find the point of contact. | | (0) |
| | (f) | Find the vertex, focus and length of latus rectum of the curve y2 – 6y – 4x = 3. | | (0) |
| | (g) | | Find the equation of the ellipse passing through (2, 1) and having eccentricity √ | | and centre (0, 0). |
| | (0) |
| Section D |
| ELEMENTARY STATISTICS (30 marks) |
| 7. | Answer any nine of the following: | 2x9 | |
| | (a) | The mean of the first 10 odd numbers | (A) | 11 | (B) | 10 | (C) | 12 | (D) | none of these |
| | (0) |
| | (b) | The geometric mean of 3, 6, 24 and 48 is | (A) | 16 | (B) | 12 | (C) | 10 | (D) | none of these |
| | (0) |
| | (c) | A.M. and H.M. of two observations are 25 and 9 respectively. Their G.M. is | (A) | 15 | (B) | 17 | (C) | 8 | (D) | none of these |
| | (0) |
| | (d) | The mean of 20 observations is 16.5. If by mistake one observation was copied 12 instead of 21. Then the correct mean is | (A) | 16 | (B) | 17 | (C) | 16.94 | (D) | 17.80 |
| | (0) |
| | (e) | If the number of observations of the two groups G1 and G2 are in the ratio 1 : 2 and their arithmetic means (a.m.) are 16 and 10 respectively then a.m. of the combined group is | (A) | 13 | (B) | 12 | (C) | 14 | (D) | none of these |
| | (0) |
| | (f) | Let h be the harmonic mean (h.m.) of n positive observations. If each of the observations are repeated once more then h.m. of those 2n observations is | (A) | h | (B) | 2 h | (C) | ½h | (D) | none of these |
| | (0) |
| | (g) | | For 10 values of x it is given that Σu = 4, Σu2 = 20 where u = | | , then Σx2 is |
| (A) | 360 | (B) | 600 | (C) | 1100 | (D) | none of these |
| | (0) |
| | (h) | If the relation between two variables x and y is 5x + 2y = 6 and the mean deviation (M.D.) of x about its mean is 6 then the M.D. of y about its mean is | (A) | 6 | (B) | 15 | (C) | 18 | (D) | none of these |
| | (0) |
| | (i) | If two variables x and y are such that 2y + 5 = 3x and quartile deviation (q.d) of x is 8, the (q.d.) of y is | (A) | 2 | (B) | 8 | (C) | | (D) | none of these |
| | (0) |
| | (j) | | If for 10 observations x1, x2,..............x10, | 10
Σ
i=1 | (xi − 4) = 30 and | 10
Σ
i=1 | (xi − 4)2 = 100 |
variance of the observations is | (A) | 1 | (B) | 5 | (C) | 10 | (D) | none of these |
| | (0) |
| | (k) | For two observations a and b standard deviation is | (A) | | (B) | | (C) | √ | | (iv) | None of these |
| | (0) |
| | (l) | If the coefficient of variation and the variance of a series of values are 80% and 256 respectively then mean of the series is | (A) | 12.8 | (B) | 320 | (C) | 3.2 | (D) | none of these |
| | (0) |
| 8. | Answer any three of the following: | 4x3 | |
| | (a) | Write short note on ogive. | | (0) |
| | (b) | What is central tendency of data? Write most important measure of it. Describe why it is most important. | | (0) |
| | (c) | Find the mean deviation about mean of the arithmetic progression: a, a + d, a + 2d, ......... a + 2nd. | | (0) |
| | (d) | Show that for the two groups of observations the grouped mean always lies between two group means. | | (0) |
| | (e) | The means, of two samples of sizes 10 and 20 are 8 and 5, and variances are 4 and 9 respectively. Obtain mean and variance of the combined sample. | | (0) |
| | (f) | Calculate mean and standard deviation of the following distribution: Class interval
Frequency | :
: | 0—10
1 | 10—20
4 | 20—30
6 | 30—40
4 | 40—50
1 | Total
16 |
| | (0) |
