CWA ICWA Foundation : Business Mathematics and Statistics Fundamentals - December 2007

CWA ICWA Foundation • Business Mathematics and Statistics Fundamentals • December 2007

This Paper has 61 answerable questions with 0 answered.

C—4(BMS) Revised Syllabus
Time Allowed : 3 Hours	Full Marks : 100

Answer all questions. The figures in the margin on the right side indicate full marks.
Section A
ARITHMETIC (15 marks)
Marks

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

Section A

ARITHMETIC (15 marks)

Marks

(a)

Answer any three of the following:
Choose the correct option showing the proper reason.

1x3

(i)

2x – 3y

2x + 3y

, then x : y is

(A)

2 : 7

(B)

7 : 2

(C)

2 : 5,

(D)

none of these

(0)

(ii)

The duplicate ratio of 2 : 3 is

(A)

√2 : √3.

(B)

4 : 6

(C)

4 : 9

(D)

none of these

(0)

(iii)

Compounded ratio of 3 : 7, 21 : 25 is

(A)

25 : 9,

(B)

7 : 21

(C)

3 : 25

(D)

none of these

(0)

(iv)

A person takes loan Rs. 3,000/– at 11% per annum from a bank. He repays the loan after 2 years. Then he pays

(A)

Rs. 3,300/–

(B)

Rs. 3,660/–

(C)

Rs. 4,000/–

(D)

none of these

(0)

(b)

Answer any two of the following:

2x2

(i)

The ages of 5 boys are 5, 8, 10, 13 and 14 years. What is their average age?

(0)

(ii)

The true discount on a bill due in 9 months at 4% per annum is Rs. 60. Find the amount of the bill.

(0)

(iii)

√x – √y

√x + √y

, find

(0)

Answer any two of the following:

4x2

(a)

y + z

z + x

x + y

= K prove that x + y + z = 0 if K # ½

(0)

(b)

In an examination 20% of the candidates fail in subject A, 25% in subject B and 10% in both the subjects. Find the percentage of those who fail exactly in one subject. Also find the percentage of candidates passed in both the subjects.

(0)

(c)

A car travels a distance of 40 km at a speed of 20 km per hour, a second distance of 50 km at a speed of 25 km per hour and a third distance of 210 km at a speed of 35 km per hour. Find the average speed of the car driving the whole distance.

(0)

(d)

A shop owner sells his goods at 17.5% discount. What price should he mark on an article that costs him Rs. 1,200 to make a profit of 37.5% on his cost?

(0)

Section B
ALGEBRA (25 marks)

Section B

ALGEBRA (25 marks)

(a)

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

2x5

(i)

The conjugate complex number of 2 + 3i is

(A)

2 – 3i,

(B)

3 – 2i,

(C)

3 + 2i,

(D)

none of these.

(0)

(ii)

If α and β be the roots of the quadratic equation x² – 3x + 5 = 0, the the value of α² + β² is

(A)

(B)

–1,

(C)

(D)

none of these

(0)

(iii)

If ⁿP_x = 336 and ⁿC_x = 56, then (n, x) will be in order

(A)

(8, 3)

(B)

(3, 8)

(C)

(8, 8)

(D)

none of these

(0)

(iv)

The logarithm of 400 to the base 2√5 is

(A)

2.6020

(B)

(C)

(D)

none of these

(0)

(v)

The difference between simple interest (SI) and compound interest (CI) on Rs. 1,000 for 2 years at 4% p.a. payable quarter is

(A)

Rs. 15,

(B)

Rs. 81,

(C)

Rs. 2,

(D)

none of these

(0)

(vi)

If A = {1, 2, 3, 4, 5} and B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} then A Δ B ( = (A ∼ B) ∪ (B ∼ A))is

(A)

{4, 2, 3, 5}

(B)

{6, 7, 8, 9}

(C)

{0, 6, 7, 8, 9}

(D)

none of these

(0)

(vii)

If p and q be the propositions, then De Morgan’s Laws are stated as

(A)	p ∨ q = p ∧ q	}	(B)	∼ (p ∨ q) = ∼ p ∧ ∼ q	}
	∼ p ∧ q =∼ p ∨ q	}		∼ (p ∧ q) = ∼ p∨ ∼ q	}
(C)	p ∨ ∼ q = p ∧ ∼ q	}	(D)	None of these
	∼ p ∧ q = ∼ (p ∨ q)	}

(0)

(b)

Answer any three of the following:

1x3

(i)

Express ³√108 as mixed surd.

(0)

(ii)

Find the value of

(

) ^{– ¾} X (

) ^{– 1/3}

(0)

(iii)

If a varies as b then show that a + b varies as a – b

(0)

(iv)

Represent the following by Venn diagram (A ∩ B) ∪ (A ∩ C)

(0)

Answer any three of the following:

4x3

(a)

Find the square roots of 17 + 12√2.

(0)

(b)

If w be an imaginary cube root of 1, show that (1 + w – w²) (1 – w + w²) = 4.

(0)

(c)

In how many ways the letters of the word BALLOON be arranged so that two L’s do not come together?

(0)

(d)

If α β be the roots of the equation x² + px + 7 = 0 and α² + β² = 22, find the value of p.

(0)

(e)

In a certain population the annual birth and death rates per 1000 are 39.4 and 19.4 respectively. Find the number of years in which the population will be double assuming that there is no immigration or emmigration.

(0)

(f)

Solve using Crammer’s rule x + y + z = –6, x – y + z = 2, 2x + y – z = 1.

(0)

Section C
MENSURATION (30 marks)

Section C

MENSURATION (30 marks)

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

2x7

(a)

The area of an equilateral triangle of side x cm is

(i)

3x²

sq.cm,

(ii)

x² √

sq.cm,

(iii)

√3

x ² sq.cm,

(iv)

None of these

(0)

(b)

The curved surface of the cone having base radius 5 cm. and slant height 7 cm, is

(i)

100 sq. cm,

(ii)

111 sq. com,

(iii)

121 sq. cm,

(iv)

none of these

(0)

(c)

The distance between two points (1, 3) and (4, 7) is

(i)

7 units,

(ii)

5 units,

(iii)

25 units,

(iv)

none of these

(0)

(d)

The coordinates of the centroid of a triangle having vertices (1, 4), (5, 3) and (6, 2) is

(i)

(4, 3),

(ii)

(3, 4),

(iii)

(12, 9),

(iv)

none of these

(0)

(e)

The equation of the line passing through the points (3, –4) and (1, 2) is

(i)

x + 3y – 5 = 0

(ii)

3x + y – 5 = 0

(iii)

x + y – 5 = 0

(iv)

none of these

(0)

(f)

The radius of the circle x² + y² + 16x + 14y – 8 = 0 is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(g)

The circumference of a circle is 44 cm. The area of the circle is

(i)

154 sq. cm

(ii)

616 sq. cm

(iii)

44 sq. cm

(iv)

none of these

(0)

(h)

If in a rectangular garden of length 40 ft and breadth 30 ft, a man travelled along perimeter and one diagonal of the garden, then the distance he travelled is

(i)

120 ft,

(ii)

190 ft,

(iii)

140 ft,

(iv)

240 ft.

(0)

(i)

The coordinates of vertex and the length of the latus rectum of the parabola y² – 4x – 2y – 7 = 0 are given in pairwise as

(i)

(– 2, 1); 2 units,

(ii)

(– 2, 1); 8 units,

(iii)

(1, – 2); 4 units

(iv)

(– 2, 1); 4 units.

(0)

(j)

The eccentricity and foci of the ellipse

x²

100

y²

= 1 are

(i)

; (0 ± 8),

(ii)

( ±

, 0)

(iii)

; (± 8, 0)

(iv)

None of these

(0)

Answer any four of the following:

4x4

(a)

Three glass balls of radii x cm, 6 cm, and 8 cm are melted into a solid spherical ball of radius 12 cm. Determine the value of x.

(0)

(b)

A right pyramid of height 15 ft stands on a square base, whose length of a side is 16 ft. Find the area of total slant surface and volume of the pyramid.

(0)

(c)

The sides of an equilateral triangle are shortened by 3 cm, 4 cm and 5 cm respectively and a right angled triangle is obtained. Find the length of each side of the equilateral triangle.

(0)

(d)

Find the locus of a point (x, y) equidistant from the points (a, b) and (b, a).

(0)

(e)

Find the equation of the circle which passes through the points (3, 4) and (– 1, 6) and the centre of it lies on the line 3x + 5y = 28.

(0)

(f)

If the point R (6, 3) divides the line segment joining the points p (4, 5) and Q (x, y) in the ratio 2 : 5, find Q (x, y).

(0)

Section D

ELEMENTARY STATISTICS (30 marks)

Answer any nine of the following:

Choose the correct option showing necessary reasons/calculations.

2x9

(a)

The mean of first 10 even number is

(i)

5.5,

(ii)

55,

(iii)

11,

(iv)

none of these

(0)

(b)

The median of the marks 4, 12, 7, 9, 14, 17, 16, 21 of eight students is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(c)

If for a distribution mean = 22, median = 24 and s.d. = 10, then coefficient of skewness is

(i)

0.6

(ii)

– 0.6

(iii)

– 6

(iv)

none of these

(0)

(d)

For two positive observations x₁ and x₂ which one of the following is true?

(i)	(AM) (HM) = (GM)²	(ii)	(AM) (GM) = (HM)²
(iii)	(GM) (HM) = (AM)²	(iv)	None of the above.

(0)

(e)

Given

n
Σ
i=1

(x_i – 4) = 72 and

n
Σ
i =1

(x_i– 7) = – 3. Then arithmetic mean of x is

(i)

68.8

(ii)

6.88

(iii)

0.688

(iv)

none of these

(0)

(f)

The geometric mean of 4, x and 8 is 8. Then x is

(i)

(ii)

(iii)

(iv)

(0)

(g)

For a variable the mean is 10 and the coefficient of variation is 50%. Then the variance is

(i)

(ii)

(iii)

400

(iv)

(0)

(h)

For a set of 10 observations, mean is 50. One observation is wrongly recorded as 50 instead of 60. Then the correct mean is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(i)

The harmonic mean of

and

(i)

(ii)

(iii)

(iv)

(0)

(j)

Means and standard deviations of runs of 10 innings of two players Mr. X and Mr. Y are as follows:

X: mean = 60 std. dev. = 8
Y: mean = 50 std. dev. = 5
Which player is more consistent?

(i)

(ii)

(iii)

equally consistent

(iv)

no conclusion is possible.

(0)

(k)

10
Σ
i =1

(x – 4) = 50 then the value of

10
Σ
i =1

(

x₁– 1

) is

(i)

(ii)

(iii)

(iv)

(0)

(l)

The mode of the observations 40, 50, 30, 20, 25, 35, 30, 30, 20, 30, 30, 20, 25, 20 is

(i)

(ii)

(iii)

(iv)

(0)

Answer any three of the following:

4x3

(a)

What is a pie diagram? Describe how it can be drawn.

(0)

(b)

The following are the sizes of 40 families in a village:

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

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

Obtain the frequency distribution of family size and calculate the variance.

(0)

(c)

Prove that standard deviation calculated from two values is half of their difference and hence compute the variance of 37 and 67.

(0)

(d)

Find the mean deviation about the mean of the following observations:

x :
Frequency :

10
3

11
12

12
18

13
12

14
3

(0)

(e)

Write short note (any one):

(i) Bar diagram, (ii) Histogram

(0)

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