# CWA ICWA Exam Papers Foundation

## Business Mathematics and Statistics Fundamentals June 2010

CAT—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) If x is the mean proportional between x – 2 and x + 6 then the value of x is
(A) 4, (B) 3, (C) 2, (D) none of these
(0)
(b) Of the five numbers the average of first four numbers is 8 and the average of the last four numbers is 6. Then the difference of the first and the fifth number is
(A) 6, (B) 8, (C) 10, (D) none of these
(0)
(c) The true discount on a bill due in 6 months at 8% p.a. is Rs. 40. Then the amount of the bill is
(A) Rs. 1000 (B) Rs. 1200 (C) Rs. 1040 (D) none of these
(0)
2. Answer any one of the following. 4×1
(a) Divide Rs. 6,200 in 3 parts such that the interest for the three parts for 2, 3 and 5 years respectively at 5% simple interest p.a. are same. (0)
(b) A dealer mixes two varieties of teas costing Rs. 100 per kg. and Rs. 160 per kg. in the proportion 5:1. He sold the 6 kg. mixture at the rate of Rs. 120 per kg. Find his profit. (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) In a class of 80 students 52 read Mathematics, 36 read Statistics and 20 read bothMathematics and Statistics. The number of students who read neither Mathematics nor Statistics is
(i) 60, (ii) 8, (iii) 12, (iv) None of these.
(0)
(b) If logarithm of a number to the base √2 is 4, then the logarithm of the same number to the base 2√2 is
(i)
4
3
(ii) 4 (iii) 8 (iv) none of these.
(0)
(c) The number of ways in which letters of the word MONDAY be arranged beginning with the letter O and ending with the letter Y is
(i) 120, (ii) 24, (iii) 96, (iv) none of these
(0)
(d) If p and q be two logical statements then (p∨q) ∨ ∼ p is
(i) (FFFF) (ii) (TFFT), (iii) (FTTT), (iv) None of these
(0)
(e) The area of a circle varies directly with square of its diameter. Area of the circle is 38.5 sq.cm when diameter is 7 cm. If diameter of the circle is 1 cm then area of the circle in sq.cm is
(i)
5.5
7
(ii)
11
7
(iii)
22
7
(iv) none of these
(0)
4. Answer any two of the following: 3×2
(a) If w be an imaginary cube root of unity then find the value of (1 – w + w2) (1 + w – w2). (0)
(b) Simple interest and compound interest in 2 years for some principal are Rs. 200 and Rs. 210 at the same rate of interest per annum. Find the principal amount. (0)
(c) The volume of a gas varies directly as the absolute temperature and inversely as pressure. When the pressure is is 15 units and the temperature is 260 units the volume is 200 units. What will be the volume when the pressure is 18 units and the temperature is 195 units? (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 perimeter of an equilateral triangle is 36 cm. Then the area of the triangle is
(i) 30√3 sq.cm, (ii) 36.√3 sq.cm, (iii) 36.√2 sq.cm, (iv) none of these
(0)
(b) The sum of the interior angles and each interior angle of a pentagon is
(i) (5400, 1080) (ii) (4500, 900), (iii) (7200, 1440), (iv) none of these
(0)
(c) The sides of a cuboid are 40 cm, 20 cm and 10 cm. It is melted to form a new cube. The surface area of the new cube in sq.cm is
(i) 1400, (ii) 2800, (iii) 4,000, (iv) none of these
(0)
(d) The volume of hollow right circular cylinder of height 14 cm with internal and external radii of base 8 cm and 10 cm respectively, has the volume in cu.cm as
(Take π =
22
7
)
(i) 4400 (ii) 1584 (iii) 88 (iv) none of these
(0)
(e) A right prism has triangular base whose sides are 13 cm, 20 cm and 21 cm. If the altitude of the prism is 9 cm then the volume of the prism is
(i) 1134 c.c, (ii) 1200 c.c, (iii) 1000 c.c, (iv) none of these
(0)
6. Answer any two of the following: 3×2
(a) A right pyramid stands on a base 16 cm square and its height is 15 cm. Find the slant surface and volume of the pyramid. (0)
(b) A road of one meter wide is developed around a circular garden with diameter 20 m @ Rs. 100 per sq.m. Find the cost of development of the road.
(Take π =
22
7
)
(0)
(c) The volume of two spheres are in the ratio 64 : 27. Find their radii if the sum of their radii is 21 cm.
(Take π =
22
7
)
(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) If the three points (1, 2), (2, 4) and (x, 6) are collinear then the value of x is
(i) 1 (ii) 2 (iii) 3 (iv) 4.
(0)
(b) If the line joining the points (2, –2) and (6, 4) are parallel to the line joining the points
(
10
3
, 9) and (x, 7) then x is
(i) 1 (ii) 2 (iii) –3 (iv) ½
(0)
(c) The three points A(a, 0), B(–a, 0), C(c, 0) and p is a point such that PB2 + PC2 = 2PA2 then the locus of P is
(i) (a – c) x + a2 = c2 (ii) (3a – c) x = a2 – c2
(iii) (6a – 2c) x + a2 – c2 = 0 (iv) (6a –2c) x + c2 – a2 = 0
(0)
(d) The centre of a circle 3(x2 + y2) = 6x + 6y – 5 is
(i) (2, 2) (ii) (1, 1) (iii) (3, 3) (iv) none of these.
(0)
8. Answer any one of the following: 4×1
(a) Find the equation of the parabola having vertex (3, 1) and focus (1, 1). (0)
(b) For the hyperbola 9×2 – 16y2 – 36x –108 = 0 find the coordinates of the centre and its latus rectum. (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) f(x) = loge (x – 3) (x – 5) is undefined in the region
(i) x < 3 (ii) x > 5 (iii) 3 ≤ x ≤ 5 (iv) none of these.
(0)
(b)
The value of lim
x→2
2×2 – 10x + 12
2×2 + 4x – 16
is
(i)
1
2
(ii) 1 (iii)
1
6
(iv) none of these
(0)
(c)
If y = x +
1
x
then x2
dy
dx
– xy is
(i) 0 (ii) 1 (iii) 2 (iv) –2.
(0)
(d) The differentiation of x4 with respect to x3 is
(i)
4
3
(ii)
4x
3
(iii)
3x
4
(iv) None of these.
(0)
(e)
The value of ∫1
0
ex
ex + 3
dx is
(i)
loge
(e + 3)
4
(ii)
loge
(e + 3)
3
(iii)
e
e + 3

1
3
(iv) None of these.
(0)
10. Answer any two of the following: 3×2
(a) If y = log (x + √x2 + a2) then prove that (a2 + x2) y2 + xy1 = 0. (0)
(b)
If f(u, v) = u3 – v3 + 3u2v – 3uv2, then verify that u
∂f
∂u
+ v
∂f
∂v
= 3f (u, v).
(0)
(c) Find the area of the region bounded by curves y2 = x and y = x. (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) The arithmetic mean of 4 observations is 8 and that of 10 observations including those 4 is 11. Then arithmetic mean of remaining 6 observations is
(i) 13, (ii) 11, (iii) 10, (iv) none of these
(0)
(b) Geometric mean of 10 observations 2, 2, 4, 4, 8, 8, 16, 16, 32, 32 is
(i) 2, (ii) 4, (iii) 8, (iv) none of these
(0)
(c) Harmonic mean of 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5 is
(i)
1
5
(ii) 4 (iii) 5 (iv) none of these
(0)
(d) Number of peas of 50 peapods are as follows:
No. of peas
No. of peapods :
: 0
2 1
10 2
12 3
15 4
10 5
1 Total
50
Median of number of peas is

(i) 2.5 (ii) 3 (iii) 3.5 (iv) none of these
(0)
(e) The number of members in 30 families are as follows:
1, 3, 1, 3, 4, 5, 3, 3, 1, 3, 3, 4, 5, 4, 2, 3, 3, 2, 2, 5, 2, 4, 2, 2, 3, 2, 4, 2, 4, 4
Then mode of number of members in a family is
(i) 2.5 (ii) 4 (iii) 3.5 (iv) none of these
(0)
(f)
If arithmetic mean and harmonic mean of two positive numbers are 3 and
8
3
then the two numbers are
(i) 1 and 5 (ii) 2 and 4 (iii) 3 and 3 (iv) none of these
(0)
(g) The mean deviation about 12 of the following distribution
x
Frequency :
: 10
5 11
9 12
20 13
13 14
3 Total
50
is
(i) 0.66, (ii) 0.7, (iii) 0.76, (iv) none of these
(0)
(h) If variance of 10 values is 9 and sum of deviation of those ten values about 3 is 60 then mean of squares of deviations of those 10 values about 5 is
(i) 25, (ii) 16, (iii) 9, (iv none of these
(0)
(i) If runs of two players A and B in 10 cricket matches are such that player A has mean 50 and variance 36 and player B has mean 60 and variance 81 of runs then the player more consistence in runs is
(i) A (ii) B (iii) Both are equally consistent (iv) none of these
(0)
(j) For a distribution with A.M. = 50, coefficient of skewness –0.4 and s.d. 20, value of mode is
(i) 66 (ii) 42 (iii) 58 (iv) none of these
(0)
12. (a) Answer any two of the following: 5×2
(i) Find the mean and standard deviation of following frequency distribution of ages:
Class of age (yrs.)
No. of persons :
: 0–10
2 10–20
4 20–30
9 30–40
3 40–50
2 Total
20
(0)
(ii) Find median and mode of the following distribution:
Weekly wages (Rs.)
No. of workers :
: 50–59
6 60–69
14 70–79
16 80–89
13 90–99
3 Total
52
(0)
(iii) If the first of two samples has 100 items with mean 15 and variance 9 and the second has 150 items with mean 16 and variance 16, find the mean and variance of the combined sample. (0)
(b) Write short note on any one of the following 4×1
(i) Central tendency of data; (0)
(ii) Ogive less than type. (0)