CWA ICWA Foundation : Business Mathematics and Statistics Fundamentals - December 2008

CWA ICWA Foundation • Business Mathematics and Statistics Fundamentals • December 2008

This Paper has 83 answerable questions with 5 answered.

P—4(BMS) Syllabus 2008
Time Allowed : 3 Hours	Full Marks : 100

Notations and symbols have usual meanings.
The figures in the margin on the right side indicate full marks.
Answer all* questions.*
SECTION I (Arithmetic — 10 Marks)
Marks

Notations and symbols have usual meanings.

The figures in the margin on the right side indicate full marks.

Answer all questions.

SECTION I (Arithmetic — 10 Marks)

Marks

Answer any three of the following.
Choose the correct option showing the proper reasons/calculations.

2x3

(a)

Two numbers are in the ratio 3 : 4. If 10 is subtracted from both of them the ratio will be 1 : 2. So the members are

(A)

15 and 20,

(B)

12 and 16,

(C)

30 and 40,

(D)

none of them

(1)

(b)

The mean of age of 5 men is 40 years. Three of them are of same age and they are excluded. The mean of the remaining two is 25. Age of one of the excluded persons in year is

(A)

20,

(B)

25,

(C)

40,

(D)

none of the above.

(1)

(c)

A man bought three qualities of tea in the ratio 5 : 4 : 3 with prices per kg. Rs. 390, Rs. 375 and Rs. 450 respectively and mixed them together. The cost price of the mixture per kg. in Rs. is

(A)

395,

(B)

420,

(C)

400,

(D)

none of them.

(1)

(d)

Ram lends Hari Rs. 1,000 and Hari repays Rs. 1300 to Ram at the end of 3 years in simple interest fully. The rate of interest Ram charged to Hari per annum for repayment of loan is

(A)

13%,

(B)

12%,

(C)

10%,

(D)

none of them.

(1)

(e)

A bill of Rs. 1020 is due in 6 months. True discount in rupees at interest rate 4% per annum is

(A)

25,

(B)

20,

(C)

20.4,

(D)

none of them.

(1)

Answer any one of the following:

4x1

(a)

The proportion of liquid I and liquid II in four samples are 2 : 1, 3 : 2, 5 : 3 and 7 : 5. A mixture is prepared by taking equal quantities of the four samples. Find the ratio of liquid I to liquid II in the final mixture.

(0)

(b)

If the difference between true discount and banker’s discount on a sum due in 3 months at 4% per annum is Rs. 20, find the amount of bill.

(0)

SECTION II (Algebra—15 marks)

SECTION II (Algebra—15 marks)

(a)

Answer any three of the following:
Choose the correct option showing necessary reasons/calculations.

2x3

(i)

After arranging 5, 3√3, 2√6 in decending order they are

(A)

3√3, 5, 2√6

(B)

2√6, 3√3, 5,

(C)

3√3, 2√6, 5,

(D)

none of them

(0)

(ii)

If y ∝

x³

and x = 2 when y = 3, then for x = 3 the value of y is

(A)

(B)

(C)

(D)

none of them

(0)

(iii)

The number of ways in which the letters of the word COLLEGE can be arranged is

(A)

240,

(B)

2520,

(C)

5040,

(D)

none of them

(0)

(iv)

The number of digits in 2⁴⁰ is (given log₁₀ 2 = 0.30103)

(A)

12,

(B)

11,

(C)

13,

(D)

none of them

(0)

(v)

Correct statement among 1 ⊂ {1, 3, 4}, {1, 3} ∈ {1, 3, 4} and {1, 4} ⊂ {1, 3, 4} is

(A)

1 ⊂ {1, 3, 4}

(B)

{1, 4} ⊂ {1, 3, 4},

(C)

{1, 3} ∈ {1, 3, 4},

(D)

none of them

(0)

(b)

Answer any three of the following:

1x3

(i)

Find the value of

(

) ^{– 2/5}

(0)

(ii)

Evaluate modulus of 3 – 2i.

(0)

(iii)

Determine the quadratic equation whose roots are 3 and – 2.

(0)

(iv)

Draw the graph of x ≤ – 3 in XOY plane.

(0)

(v)

If log p – log q = log (p – q), show that p =

q²

q – 1

(0)

Answer any two of the following:

3x2

(a)

Solve: 2^5x, 4^{4x – 2} =

8^3x–8

16^–3x

(0)

(b)

If x = 7 + 4√3 find the value of √x +

√x

(0)

(c)

The volume of gas varies directly as the absolute temperature and inversely as pressure. When the pressure is 10 units and the temperature is 200 units, the volume is 160 units. What will be the volume when pressure is 12 units and temperature is 480 units.

(0)

(d)

From 7 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done to include atleast one lady?

(0)

SECTION III (Mensuration —15 marks)

SECTION III (Mensuration —15 marks)

(a)

Answer any three of the following:

Choose the correct option showing necessary reasons/calculations.

2x3

(i)

The area of the triangle with sides of length 3 cm, 4 cm and 5 cm, (in sq. cm) is

(A)

12,

(B)

(C)

24,

(D)

none of them

(0)

(ii)

The perimeter in cm of a semicircle of diameter 14 cm is

(A)

25,

(B)

44,

(C)

36,

(D)

none of them

(0)

(iii)

The volume in cu. ft of a right pyramid having altitude 6 ft and square base with length of a side 4 ft is

(A)

32,

(B)

96,

(C)

48,

(D)

none of them

(0)

(iv)

A path of 5 ft wide is to be laid just outside round the square garden with length of a side 50 ft. The area of the path in sq. ft would be

(A)

1000,

(B)

2500,

(C)

1200,

(D)

none of them

(0)

(v)

The volume of a solid sphere is

π cu. cm. Surface area of the sphere in sq. cm is

(A)

12 π,

(B)

8 π

(C)

16 π

(D)

none of them

(0)

(b)

Answer any three of the following

1x3

(i)

Find the total surface area of a cube whose volume is 64 cu. cm.

(0)

(ii)

Find the hypotenuse of a right angled isosceles triangle having area 9/2 sq. ft.

(0)

(iii)

A bicycle wheel of radius 35 cm makes 5000 revolutions to cover x km. Find the volume of x.

(0)

(iv)

Two solid spheres of radil 3 cm and 2 cm are melted and by them another solid sphere is formed. Find the volume of the new sphere.

(0)

(v)

A solid right circular cone have 7 cm height and 3 cm radius of base. Find its volume.

(0)

(vi)

Find the length of a side of rhombus having diagonals 6 cm and 8 cm.

(0)

Answer any two of the following:

3x2

(a)

One diagonal of a rectangle is 10 ft. If the perimeter is 28 ft, find its length and breadth.

(0)

(b)

A circle of radius 7 cm is inscribed within a square touching the sides. Find the area of one fillet thus formed.

(0)

(c)

Find the volume and surface area of a hollow cylinder with height 7 inches internal and external radil of base 5 inches and 3 inches respectively.

(0)

(d)

A conical tent is required to accommodate 11 people. Each person must have 14 sq. ft of space on the ground and 140 cu. ft of air to breathe. Find the height, slant height and the width of the tent.

(0)

SECTION IV (Coordinate Geometry — 10 marks)

SECTION IV (Coordinate Geometry — 10 marks)

Answer any three of the following:

Choose the correct option showing necessary reasons/calculations.

2x3

(a)

Equation of a line passing through (2, 4) and having y–intercept 2 (on the positive side) is

(i)

y = x – 2,

(ii)

y = x + 2

(iii)

y + x = 2

(iv)

none of them

(0)

(b)

The coordinates of the point which divide the line joining (3, 6) and (12, 9) internally in the ratio 1 : 2 is

(i)

(7, 6),

(ii)

(9, 3),

(iii)

(6, 7),

(iv)

none of them

(0)

(c)

Perpendicular distance of the line 3x + 4y = 1 from the point (4, 1) is

(i)

3 units,

(ii)

4 units,

(iii)

units,

(iv)

none of them

(0)

(d)

The radius of the circle 2x² + 2y² + 12y = 8x + 6 is

(i)

3 units,

(ii)

4 units,

(iii)

5 units,

(iv)

none of them

(0)

(e)

Eccentricity of the elipse 5x² + 9y² = 405 is

(i)

(ii)

(iii)

(iv)

none of them

(0)

Answer any one of the following:

4x1

(a)

Find the equation of a straight line passing through the point (2, 3) and perpendicular to the line x + 2y = 5.

(0)

(b)

Find the equation of the parabola whose focus is (1, 1) and directrix is x + y = 1. Find also the length of the latus rectum.

(0)

SECTION V (Calculus — 15 marks)

SECTION V (Calculus — 15 marks)

(a)

Answer any three of the following:
Choose the correct option showing necessary reasons/calculations.

2x3

(i)

If f(x) = e^2x–3 then

f(x + y)

f(x) f (y)

(A)

e³,

(B)

e^–3,

(C)

(D)

none of them

(0)

(ii)

The value of

lim
x→0

3^x – 2^x

(A)

log_e (

)

(B)

log₁₀ (

)

(C)

(D)

none of them

(0)

(iii)

If y = 4^x then

d²y

dx²

(A)

4^x,

(B)

4^x log_e4,

(C)

log_e4

(D)

none of them

(0)

(iv)

The value of x for which x (12 – x²) is maximum is

(A)

(B)

–2,

(C)

(D)

none of them

(0)

(v)

The value of

1
∫
0

e^xdx

1 + e^x

(A)

log_e (1 + e),

(B)

log_e (

1 + e

(C)

2 log_e (1 + e),

(D)

none of them

(0)

(vi)

If the total cost function C = x³ – 2x² + 5x, then the marginal cost is equal to

(A)

x² – 4x + 5,

(B)

3x² – 4x + 5,

(C)

3x² – 4x,

(D)

none of them

(0)

(b)

Answer any three of the following:

1x3

(i)

Find the domain of definition of the function

2x – 5

√x² – 9

(0)

(ii)

If y = f(x) =

x + 1

x – 1

Prove that f(y) = x.

(0)

(iii)

Evaluate

lim
x→2

x² – 6x + 8

x² – 5x + 6

(0)

(iv)

If y = 10^x + x¹⁰ find

(0)

(v)

= x² – 3x + 1 and y = 2 when x = 1 then show that y =

x³ –

x² + x +

(0)

10.

Answer any two of the following:

3x2

(i)

A function (fx) is defined as

f(x) = {	x + 1 when x ≤ 1
	3 – x² when x > 1

Examine whether f(x) is continuous at x = 1.

(0)

(ii)

Verify Euler’s theorem for the function ax² + 2bxy + cy².

(0)

(iii)

Evaluate

5
∫
3

3x² – 3x – 6

(0)

SECTION VI (Statistical Methods — 35 marks)

11.

(a)

Answer any nine of the following:
Choose the correct option showing necessary reasons/calculations.

2x9

(i)

The arithmetic mean of first 9 counting numbers occurring with same frequency has its value

(A)

45,

(B)

190,

(C)

(D)

none of them

(0)

(ii)

If 2 occurs 4 times, 4 occurs 3 times, 8 occurs twice and 16 occurs once then geometric mean of them is

(A)

(B)

(C)

(D)

none of them

(0)

(iii)

If a person travels first 2 km @ km/hr., next 3 km @ 3 km/hr and another 5 km @ 5 km/hr, his average speed during his journey is

(A)

3 km/hr,

(B)

km/hr,

(C)

km/hr,

(D)

none of them

(0)

(iv)

The median of marks 55, 60, 50, 40, 57, 45, 58, 65, 57, 48 of 10 students is

(A)

55,

(B)

57,

(C)

52.5,

(D)

none of them

(0)

(v)

If the relation between two variables x and y is 3x – 2y = 5 and mode of x is 5 then mode of y is

(A)

(B)

7.5,

(C)

10,

(D)

none of them

(0)

(vi)

If the two variables x and y are related by the equation 3x = 2y + 4 and mean deviation of x about its means is 4 then mean deviation of y about its mean is

(A)

(B)

(C)

(D)

none of them

(0)

(vii)

10
Σ
i=1

(x_i − 3) = 10 and

10
Σ
i =1

(x_i— 3)² = 100 then standard deviation of 10 observations

x₁, x₂, ...... x₁₀ is

(A)

(B)

(C)

(D)

none of these

(0)

(viii)

If the relation between two variables x and y is 2x + 3y = 5 and standard deviation of y is 10 then the standard deviation of x is

(A)

15,

(B)

10,

(C)

(D)

none of them

(0)

(ix)

If mean, mode and standard deviation of 10 observations are 65, 80 and 25 respectively then type of skewness of the data is

(A)

Symmetric,

(B)

Positively skewed,

(C)

Negatively skewed,

(D)

none of them.

(0)

(x)

If the mean of 50 observations is 50 and one observation 94 us wrongly recorded there as 49 then correct mean will be

(A)

49.1

(B)

50,

(C)

50.9

(D)

none of them

(0)

(xi)

If for two observations arithmetic mean is 80 and harmonic mean is 5 then geometric mean of them is

(A)

20,

(B)

400,

(C)

16,

(D)

none of them.

(0)

(xii)

For moderately skewed distribution A.M. = 110, Mode = 104, then median is

(A)

112,

(B)

108,

(C)

104,

(D)

none of them.

(0)

(xiii)

If the maximum and minimum values of 10 observations are 40 and 10 then coefficient of range is

(A)

(B)

(C)

(D)

none of them

(0)

(xiv)

The standard deviation (SD) of a variable x is 10, then the SD of the variable 2x + 10 is

(A)

10,

(B)

–10

(C)

20,

(D)

none of these

(0)

(b)

Answer any three of the following:

1x3

(i)

12 observations are 2, 4, 6, 3, 3, 5, 6, 8, 4, 3, 5, 4. If I is subtracted from each of them find the range.

(0)

(ii)

n
Σ
i=1

(x_i − 4) = 10 and

n
Σ
i =1

(x_i— 3) = 15, find n.

(0)

(iii)

If the means of two groups of m and n observations are 50 and 60 respectively and combined mean is 54. Find the ratio m : n.

(0)

(iv)

The mean deviation about mean 40 is 20, find the coefficient of mean deviation about mean.

(0)

(v)

If mean and standard deviation of runs scored by a batsman in 10 tests are 50 and 4 respectively, find the coefficient of variation of runs.

(0)

(vi)

Show that the standard deviation of x for its two values x₁ and x₂ is

|x₁ – x₂|

(0)

12.

(a)

Answer any two of the following:

5x2

(i)

Calculate mean and variance of the following grouped frequency distribution:

Score
No. of Students

:
:

0–10
10

10–20
30

20–30
60

30–40
60

40–50
30

50–60
10

Total
200

(0)

(ii)

Prove that the standard deviation of n observations is independent of change of origin but is dependent on the change of scale.

(0)

(iii)

Two samples of sizes 100 and 150 have means 45 and 55 and standard deviations 7 and 12 respectively. Find the mean and standard deviation of the combined sample.

(0)

(iv)

Find median and mode from the following grouped frequency distribution:

Class
Frequency

:
:

0–5
5

5–10
15

10–15
25

15–20
30

20–25
20

25–30
5

Total
100

(0)

(b)

Write short note on any one of the following

4x1

(i)

Histogram,

(0)

(ii)

Pic Chart.

(0)

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