# CWA ICWA question papers Foundation

## Business Mathematics and Statistics Fundamentals June 2009

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)
Marks
1. Answer any two of the following:
Choose the correct option showing the proper reasons/calculations. 3×2
(a) Let marks obtained by Ram, Rahim and Jadu be A, B and C respectively. Given A : B = 1 : 2, B : C = 3 : 4. The combined ratio A:B:C is
(A) 1 : 2 : 4, (B) 3 : 6 : 8, (C) 1 : 6 : 8, (D) none of these
(0)
(b)
If
√a + √b
√a – √b
=
2
1
Then
a + b
a – b
is equal to
(A) 5/4, (B) 4/5 (C) 3 (D) none of them
(0)
(c) The time, in which the discount on amount Rs. 550 due is Rs. 50 at 4% per annum, is
(A) 2 yrs., (B) 3 yrs., (C) 2.5 yrs., (D) none of them
(0)
2. Answer any one of the following. 4×1
(a)
If
x
b+c
=
y
c+a
=
z
a–b
then show that (b – c) (x – a) + (c – a) (y – b) + (a – b) (z – c) = 0.
(0)
(b) A person borrowed Rs. 10,000 at some simple interest rate for 2 years. After expiry of one year he borrowed another Rs. 20,000 at 1% lower interest rate for 1 year. At the end he paid fully Rs. 33,000. Find the rate of interest at which he borrowed first. (0)
SECTION II (Algebra — 15 marks)
Marks
3. Answer any three of the following:
Choose the correct option showing necessary reasons/calculations. 3×3
(a)
After rationalization
√3 + √2i
√3 – √2i
will be
(i) 1 + 2√6i, (ii)
5 + 2√6i
5
(iii) 1 – 2 √6i, (iv)
1 + 2√6i
5
(0)
(b)
(2n+1) + (2n+2)
(2n+2) – 2 (½)1–n
simplifies to
(i) 4, (ii) 2, (iii) 8, (iv) 20
(0)
(c) The value of log2 log2 log3 81 is
(i) 1, (ii) 4, (iii) 3, (iv) 2
(0)
(d)
The value of x satisfying the equations √
x
1 – x
+ √
1 – x
x
=
13
6
is
(i) (
2
3
,
3
2
) (ii) (
4
9
,
9
4
) (iii) (4, 9), (iv) None of these
(0)
(e) Ifmc6 : m–3c3 = 91 : 4, then the value of m is
(i) 13, (ii) 15, (iii) 14, (iv) none of these
(0)
4. Answer any two of the following: 3×2
(a) Find the square root of x + √x2 – y2. (0)
(b) The total expenses of a boarding house varies partly with the number of boarders and partly fixed. The total expenses are Rs. 10,000 for 25 boarders and Rs. 11,500 for 30 boarders. Find the fixed expenses. (0)
(c) Of 50 students appearing in an examination 20 failed in Mathematics, 25 failed in English and 10 failed in both. Find the number of students of those 50 students who passed in bothMathematics and English. Write the formula you use completely. (0)
SECTION III (Mensuration — 15 marks)
Marks
5. Answer any three of the following:
Choose the correct option showing necessary reasons/calculations. 3×3
(a) The area of the equilateral triangle with a side of length 2 cm is
(i)
√3
2
sq.cm,
(ii) √3 sq.cm, iii)
√3
4
sq.cm,
(iv) none of these
(0)
(b) A circular garden having diameter 60 ft. has a path of width 10 ft. surrounding outside the garden. Area of the path is
(i) 2200 sq.ft., (ii)
28600
7
sq.ft.,
(iii)
1600
7
sq.ft.,
(iv) none of these
(0)
(c) A rectangular parallelopiped has length 20 cm, breadth 10 cm, and height 5cm. the total surface area of it is
(i) 350 sq.cm, (ii) 1000 sq.cm, (iii) 700 sq.cm (iv) none of these
(0)
(d) For a solid right circular cylinder of height 9 cm and radius of 7 cm the total surface area is
(i) 352 sq.cm, (ii) 550 sq.cm, (iii) 704 sq.cm (iv) none of these
(0)
(e) The diameter of the base of a conical tent is 14 ft, and height of the tent is 15 ft. The volume of the space covered by the tent is
(i) 2130 cu.ft., (ii) 8520 cu.ft., (iii) 700 cu.ft., (iv) none of these
(0)
6. Answer any two of the following: 3×2
(a) The area of a rectangle is 96 sq.cm and its perimeter is 40 cm, what are its length and breadth? (0)
(b) Find the quantity of water in litre flowing out of a pipe of cross–section area 5 cm2 in 1 minute if the speed of the water in the pipe is 30 cm/sec. (0)
(c) The volumes of two spheres are in the ratio 8 : 27 and the difference of their radii is 3 cm. Find the radii of both the spheres. (0)
SECTION IV (Coordinate Geometry — 10 marks)
Marks
7. Answer any two of the following:
Choose the correct option showing necessary reasons/calculations. 3×2
(a) The equation of a straight line passing through origin and perpendicular to the line 2x + 3y + 1 = 0 is
(i) 3y + 2x = 0, (ii) 2y + 3x = 0 (iii) 2y = 3x, (iv) 3y = 2x
(0)
(b) Eccentricity of the hyperbola 9×2 – 16y2 = 36 is
(i)
5
4
(ii)
4
5
(iii)
25
16
(iv)
16
25
(0)
(c) The equation of a parabola whose vertex and axis are (0, 0) and y = 0 respectively, passing through (5, 4), is
(i) 4y2 = 5x, (ii) 5×2 = 16 y, (iii) 5y2 = 4x, (iv) 5y2 = 16x
(0)
8. Answer any one of the following: 4×1
(a)
Prove that the points (x, 0), (0, y) and (1, 1) are collinear if
1
x
+
1
y
= 1.
(0)
(b) Find the equation of the circle which is concentric to the circle x2 + y2 + 8x – 10y + 5 = 0 and passes through the point (4, 11). (0)
SECTION V (Calculus — 15 marks)
Marks
9. Answer any three of the following:
Choose the correct option showing necessary reasons/calculations. 3×3
(a)
If f(x) =
x + 1
x – 1
, f (f(x)) for x ≠ 1 is
(i) 1, (ii) 2, (iii) x, (iv)
x + 1
x – 1
(0)
(b)
lim
x→1
(x2 – 1)2x
2×2 – 3x + 1
is evaluated as
(i) 1, (ii) 2, (iii) 3, (iv) 4
(0)
(c)
If y = (x2 + 5)2 then
dy
dx
at x = 2 is
(i) 18, (ii) 72, (iii) 81, (iv) 36
(0)
(d)
If f(x, y) = x3 + y3 then x
∂f
∂x
+ y
∂f
∂y
is
(i) f(x, y), (ii) 3f(x, y), (iii) 3, (iv) none of these
(0)
(e)
2

1
dx
√ x – 1
is evaluated as
(i) 2, (ii) 2√2 (iii) –2 (iv) –2√2
(0)
10. Answer any two of the following: 3×2
(a)
If y =
x
√ 1 – x2
show that x
dy
dx
= y (y2 + 1).
(0)
(b) If y = (x + √1 + x2)m, show that (1 + x2)y2 + xy1 = m2y. (0)
(c)
Evaluate 1

2 x logex dx.
(0)
SECTION VI (Statistical Methods — 35 marks)
Marks
11. Answer any seven of the following:
Choose the correct option showing necessary reasons/calculations. 3×7
(a) Arithmetic mean of 5 observations is 8. After calculation it was noted that observations 10 and 20 have been wrongly taken in place of correct values 15 and 25 respectively. The correct mean is
(i) 18, (ii) 9, (iii) 13, (iv) none of these
(0)
(b) Two groups of 10 and 15 observations have means 10 and 20 respectively. Then grouped mean is
(i) 15, (ii) 16, (iii) 14, (iv) none of these
(0)
(c) Geometric mean of first group of 5 observations is 8 and that of second group of 4 observations is 128√2. Then grouped geometric mean is
(i) 64, (ii) 32√2, (iii) 32, (iv) none of these
(0)
(d)
If two groups with 2 and 3 observations have harmonic means
2
5
and
1
5
respectively, then combined harmonic mean of 5 observations
(i)
1
2
(ii)
1
4
(iii)
1
3
(iv) none of these
(0)
(e) If the two observations have harmonic mean and geometric mean 9 and 15 respectively, then arithmetic mean of the two observations
(i) 12, (ii) 25, (iii) √135, (iv) none of these
(0)
(f) If the two variables x and y are related by 2x + 3y = 12 and standard deviation of x is 6 then standard deviation of y is
(i) 2, (ii) 10, (iii) 4, (iv) none of these
(0)
(g)
For 10 values of variable x it is given that Σx = 13, Σx2 = 400 and u =
x – 5
2
. Then Σu2 is
(i) 100, (ii) 520, (iii) 260, (iv) none of these
(0)
(h) Mean deviation about mean is 5.8. Coefficient of mean deviation about mean is 0.2. Then mean is
(i) 1.16, (ii) 2.9, (iii) 29, (iv) none of these
(0)
(i) For a group of 10 observations, ∑x = 452, ∑x2 = 24270 and mode 43.7 the coefficient of skewness is
(i) 0.8, (ii) 0.08, (iii) 8, (iv) none of these
(0)
(j) The mean and coefficient of variation of runs made by a batsman in 10 innings are 40 and 125% respectively. Then s.d. of the runs made by the batsman is
(i) 50, (ii) 40, (iii) 20, (iv) none of these
(0)
12. (a) Answer any two of the following: 5×2
(i) Find the median and mode of the following grouped frequency distribution:
Salaries (in Rs.) per hour
No. of persons 5–9
10 10–14
20 15–19
30 20–24
25 25–29
15 Total
100
(0)
(ii) Prove that for any two positive real quantities AM ≥ GM ≥ HM. (0)
(iii) The following are the sizes of 50 families in a village:
2, 3, 4, 5, 4, 3, 2, 6, 1, 3, 5, 3, 5, 5, 4, 2, 4, 4, 3, 3, 3, 2, 4, 4, 3,

3, 2, 3, 4, 3, 5, 4, 2, 3, 4, 4, 2, 4, 4, 2, 6, 4, 3, 5, 4, 3, 2, 3, 3, 1.

Obtain the frequency distribution of the family size and calculate the mean deviation about mean of the family size.

(0)
(b) Write short note on any one of the following 4×1
(i) Pie Chart, (0)
(ii) Primary data. (0)