# Business Mathematics and Statistics Fundamentals Dec 2011

P—4(BMS)
Syllabus 2008
Time Allowed : 3 Hours Full Marks : 100
The figures in the margin on the right side indicate full marks.
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. 3×2
(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
(0)
(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
(0)
(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. 4×1
(a)
If
α
q − r
=
β
r − p
=
γ
p − q
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. 3×3
(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: 3×2
(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. 3×3
(a) Altitude of an equilateral triangle having a base of length 2 cm is
(i) √3 cm (ii)
√ 3
2
cm (iii)
√ 3
4
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?
( π =
22
7
)
(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: 3×2
(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.
( π =
22
7
).
(0)
SECTION IV (Co–ordinate Geometry — 10 marks)
7. Answer any two of the following:
Choose the correct option showing the proper reasons/calculations. 3×2
(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 4×2 − 24x + 9y2 + 36y + 36 = 0 is
(i)

5
3
(ii)
√5
3
(iii)
5
3
(iv) none of these
(0)
8. Answer any one of the following: 4×1
(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
√13
2
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. 3×3
(a)
If f(x) =
x − 1
x + 1
Then f (
x − 1
x + 1
) is
(i) x (ii)
1
x
(iii)

1
x
(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 + (
d2y
dx2
)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: 3×2
(a)
If y = x2 loge x, show that x2
d2y
dx2
+ 4y = 3x
dy
dx
(0)
(b) Show that x3 – 6×2 + 9x – 10 is maximum at x = 1 but is minimum at x = 3. (0)
(c)
Evaluate: ∫
dx
√x+2 – √x + 3
(0)
SECTION VI (Statistical Methods — 35 marks)
11. Answer any seven of the following:
Choose the correct option showing proper reasons/calculations. 3×7
(a)
The harmonic mean of the numbers 1,
1
2
,
1
3
, …..,
1
n
is
(i)
1
n + 1
(ii)
2
n + 1
(iii)
3
n + 1
(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)
1
7
(ii)
1
8
(iii)
1
9
(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: 5×2
(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: 4×1
(i) Tabulation; (0)
(ii) Central tendency of data. (0)