# Business Mathematics and Statistics Fundamentals Dec 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 P =
4
5
Q and Q = 2
1
2
R, then P:R is
(i) 1:2 (ii) 2:1 (iii) 4:5 (iv) none of the these
(0)
(b) A person drove his car 40 km at an average speed of 20 km per hour and next 60 km at an average speed of 30 km per hour. Then his average speed in his whole journey of 100 km is
(i) 25 km/h (ii) 20 km/h (iii) 30 km/h (iv) none of the these
(1)
(c) Time in which Rs. 5000/- will be the amount Rs. 6000/- at simple interest @ 5% p.a. is
(i) 2 years (ii) 5 years (iii) 4 years (iv) none of the these
(1)
2. Answer any one of the following: 4×1
(a) A dealer mixed two varieties of tea having cost Rs. 1200 and Rs. 2500 per kg each in such a way that he can gain 20% by selling the resultant mixture at Rs. 1800 per kg. Find the proportion in which the two types of teas are mixed. (0)
(b) A bill for Rs. 5400 is due in 2 years. Find Banker’s gain at the rate of 4% interest per annum. (0)
SECTION II (Algebra—15 marks)
3. Answer any three of the following:
Choose the correct option showing necessary reasons/calculations: 3×3
(a) The number of ways in which letters of the word ‘ALGEBRA’ can be arranged so that the two A’s will not remain together is
(i) 1600 (ii) 1800 (iii) 2000 (iv) none of the these
(0)
(b) Let p be ‘It is hot’ and q be ‘It is dry’. Then the statement ‘It is not hot and it is not dry’ can be written in symbolic form as
(i) ~ p ∨ q (ii) ~ p ∧ ~ q (iii) ~ p ∨ q (iv) p ∨ q
(0)
(c) The number of zeros between decimal point and the first significant digit in (0-5)20, given log102 = 0-30103, is
(i) 8 (ii) 7 (iii) 5 (iv) none of these
(0)
(d) If A = {1, 3, 5}, B = {1, 3, 6}, ∪ = {1, 2, 3, 4, 5, 6} then the set (A – B) ∩ Bc equals to
(i) {1} (ii) {5} (iii) {1, 5} (iv) none of these
(0)
(e)
If xa = yb = zc and xyz = 1 then the value of
1
a
+
1
b
+
1
c
is
(i) 1 (ii) 3 (iii) 0 (iv)
1
5
(0)
4. Answer any two of the following: 3×2
(a) The expense of a boarding house are partly fixed and partly varies with the number of boarders. The charge is Rs. 70 per head when there are 20 boarders and Rs. 60 per head when there are 40 boarders. Find the charge per head when there are 50 boarders. (0)
(b) If w be an imaginary cube root of unity find out the value of (1 – w) (1 – w2) (1 – w4) (1 – w8). (0)
(c) Draw the graph of x ≤ 2y – I in (x, y) coordinate system. (0)
SECTION III (Mensuration—15 marks)
5. Answer any three of the following:
Choose the correct option showing necessary reasons/calculations:

3×3
(a) The length, breadth and height of a box are 12 m, 4 m and 3 m respectively. The length of the largest rod that can be placed in the box is
(i) 15 m (ii) 13 m (iii) 12 m (iv) none of these
(0)
(b) If the hypotenuse of a right angled isosceles triangle is 4 cm then the area of the triangle is
(i) 12 sq.cm (ii) 8 sq.cm (iii) 4 sq.cm (iv) none of these
(0)
(c) The diameter of a sphere is 6 cm. The surface area of its hemisphere is
(i) 27 sq.cm (ii) 27 π sq.cm (iii) 18 π sq.cm (iv) none of these
(0)
(d) A right prism has a triangular base whose sides are respectively 14 cm, 21 cm and 21 cm. If the altitude of the prism is 10 √2 cm then the volume of the prism is
(i) 1200 cc (ii) 1134 cc (iii) 1000 cc (iv) none of these
(0)
(e) The curved surface of a right circular cone having base diameter 6 cm and height 4 cm is
(i) 10 π sq.cm (ii) 25 π sq.cm (iii) 15 π sq.cm (iv) none of these
(0)
6. Answer any two of the following: 3×2
(a)
If the area of a semicircle is 77 sq.cm then find the perimeter of the semicircle. ( π =
22
7
)
(0)
(b) A right pyramid with height 8 cm stands on a base which is a triangle with sides of lengths 3 cm, 4 cm and 5 cm. Find the volume of the pyramid. (0)
(c) The base radius of a cylindrical pipe is 3-5 cm and its height is 18 cm. Find the ratio of its curved surface to its total surface.
( π =
22
7
)
(0)
SECTION IV (Coordinate Geometry—10 marks)
SECTION IV (Coordinate Geomentry—10 marks)
7. Answer any two of the following:
Choose the correct option showing necessary reasons/calculations:

2×3
(a) If the point (x, 0) is equidistant from the points (-1,3) and (6, 4) then value of x is
(i) 1 (ii) 3 (iii) 4 (iv) none of these
(0)
(b) The equation of a circle with (1,0) and (0, 1) as end points of its diameter is
(i) x2 + y2 – x – y = 0 (ii) x2 + y2 – 2x – 2y = 0 (iii) x2 + y2 – 2x – 2y + 1= 0 (iv) none of these
(0)
(c) Focus of the parabola x2 + 8y – 6x + 1 = 0 is
(i) (0, -2) (ii) (-1, 3) (iii) (3, -1) (iv) none of these
(0)
(d) The eccentricity of the ellipse x2 + 4y2 – 2x + 8y + 1 = 0 is
(i)
1
2
(ii)
√3
2
(iii)
√3
4
(iv) none of these
(0)
8. Answer any one of the following: 4×1
(a) Prove that the two circles x2 + y2 + 2x – 6y + 5 = 0 and x2 +y2 + 10x – 2y + 21 =0 touch each other externally. (0)
(b) Find the equation of a straight line passing through the point of intersection of the lines 2x- + y = 4 and x – y + 1 =0 and is perpendicular to the line 6x – 7y + 8 = 0. (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|
x
then for c≠0, |f(c) – f(-c)| will be
(i) 1 (ii) 2 (iii) 0 (iv) none of these
(0)
(b)
The value of lim
x→0
1- √1 – x2
x2
is
(i)
1
2
(ii)
1
3
(iii) 0 (iv) none of these
(0)
(c)
When x = 4t – t2, y = t2 + 3,
dy
dx
at t = 1 is
(i) 0 (ii) -1 (iii) 2 (iv) none of these
(0)

(d)
1
The value of ∫
0
dx
√x+1 – √x
is
(i)
4√2
3
(ii) 3√2 (iii)
2√2
3
(iv) none of these
(0)
(e)
If f(x, y) = 2×3 – 11x2y + 3 y3 then x
δf
δx
+ y
δf
δy
is
(i) 2f(x, y) (ii) 3f(x, y) (iii) 4f(x, y) (iv) none of these
(0)
10. Answer any two of the following: 3×2
(a) If y = Aemx + Be-mx show that y2 – m2y = 0. (0)
(b) If sum of two values is 8 find the maximum value of their product. (0)
(c)
If f'(x) =
e4x + e6x
ex + e–x
and f(0)=
1
5
then find f(x).
(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) If the relation between x and y is x = 2y + 5 and the median of x is 25 then the median of y is
(i) 20 (ii) 10 (iii) 12.5 (iv) none of these
(0)
(b) Geometric mean of 10 observations is 8. If geometric mean of first six observations is 4 then geometric mean of last four observations is
(i) 16√2 (ii) 8√2 (iii) 16 (iv) none of these
(0)
(c)
If harmonic mean of first 5 observations is
5
2
and harmonic mean of first 5 observations is
9
2
then harmonic mean of all 10 observations is
(i) 7 (ii)
45
14
(iii)
101
36
(iv) none of these
(0)
(d) Out of 100 observations 25 observations have the value 1 and rest of the observations are zero. The standard deviation of 100 observations is
(i)
√3
2
(ii)
3
2
(iii)
√3
4
(iv) none of these
(0)
(e) If the sum of deviations of a number of observations about 4 is 30 and that about 3 is 40. Then mean of the observations is
(i) 7 (ii) 10 (iii) 11 (iv) none of these
(0)
(f) Variance of first 5 positive integers is
(i) 3 (ii) 2 (iii) 1 (iv) none of these
(0)
(g) Mean deviation of first 5 positive integers about median is
(i) 0 (ii) 1.7 (iii) 1.2 (iv) none of these
(0)
(h) The mean and variance of n values of a variable x are 0 and σ2 respectively. If the variable y = x2, the mean of y is
(i) σ (ii) σ2 (iii) 1 (iv) none of these
(0)

(i)
For 5 values of a variable x,
5
Σ
i = 1
xi = 25 and
5
Σ
i = 1
(xi – 5)2 = 30, the variable of x is
(i) 2 (ii) 4 (iii) 6 (iv) none of these
(0)
(j) If group G, has a.m = 20, mode = 25, s.d = 10 and group G2 has a.m =18, median = 18, s.d = 9 then
(i) G1 is more skewed than G2 (ii) G1 and G2 are equally skewed
(iii) G1 is less skewed than G2 (iv) none of these
(0)
12. (a) Answer any two of the following: 5×2
(i) Draw a pie chart to represent the following data of expenditure of a family:
Item : Food Rent Clothing Education Fuel & Electricity Others
Expenditure : 300
(in Rs.) 200 150 125 75 100
(0)
(ii) Find the standard deviation from the following distribution:
Height (in inches) 59—61 61—63 63—65 65—67 67—69
No. of students 4 30 45 15 6
(0)
(iii) Prove that the standard deviation of two values x1, and x2 of a variable x is half of their difference. (0)
(b) Write short note on any one of the following: 4×1
(i) Histogram and its uses; (0)
(ii) Kurtosis. (0)