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INDIAN SCHOOL CERTIFICATE BOARD EXAMINATION

ISC 2008 Question Paper

BUSINESS MATHEMATICS

(Reading Time: 15 minutes) 

(Examination Time: Three Hours)

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PART I

Answer all questions


Question 1.
(i) Using Bionomial Theorem, find the value of :

(√2 + 1)6 + (√2 − 1)6


Answer:
(ii) Use the principle of mathematical induction to prove that :

4 + 8 + 12 + … + 4n = 2n (n + 1)


Answer:
(iii) In how many ways can three prizes be distributed among four boys when no boy gets more than one prize?
Answer:

(iv) Use Cramer's rule to solve the following system of equations :

2xy = 17, 3x + 5y = 6


Answer:
(v) If A =


3   −2
4   −2

is a 2 × 2 matrix, find the scalar k such that A2 = k A − 2 I.
Answer:
(vi) If the minor axis of an ellipse is equal to the distance between its focii, prove that its eccentricity is 1/√2.
Answer:
(vii) Differentiate √(sin x3) w.r.t. x.
Answer:
(viii) Evaluate :
tan (sin−1x)

√(1 − x2)
dx


Answer:
(ix) Calculate the mean deviation from the mean of the following data :
x:     2    4   6    8    10
f:      1   4   6    4    1

Answer:
(x) Three dice are thrown together. Find the probability of getting a sum of at least 5.
Answer:


PART II


SECTION A 

Answer any four questions

Question 2.
(a) Find the tenth term in the expansion of (2x2 + 1/x)12.
Answer:
(b) Out of 5 men and 2 women, a committee of 3 is to be formed. In how many ways can it be done so as to include at least 1 woman?
Answer:
Question 3.
(a) Using properties of determinants, prove that :
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1      b + c      b2 + c2

1      c + a      c2 + a2

1      a + b      a2 + b2
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= (bc) (ca) (ab).

Answer:
(b) Find the coordinates of vertex, focus and the length of latus rectum of the parabola y2 = 6(x + y).
Answer:
Question 4.
If A =
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0       1       2

1      2       3

3       1       1

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(i) Find A−1.

Answer:
(ii) Hence, solve the following system of equations :
y + 2z + 8 = 0, x + 2y + 3z + 14 = 0, 3x + y + z +8 = 0

Answer:
Question 5.
(a) If y = [x + √(x2 + a2)]n, prove that
dy

dx
=        ny

√(x2 + a2)

Answer:
(b) If y = etan−1x, show that
(1 + x2) d2y

dx2
+ (2x − 1) dy

dx
= 0.


Answer:
Question 6.
Evaluate the following :
(a)
             cos x

√(sin2 x + 2 sin x + 3)
dx


Answer:
(b)
   x + 3

√(2 − x2)
dx


Answer:
Question 7.
(a) Show that the height of a closed cylinder of given surface area and maximum volume is equal to the diameter of its base.
Answer:
(b) Differentiate (sin x)cos−1x w.r.t. x.
Answer:


SECTION B

Answer any two questions
Question 8.
Evaluate the following :
(a)
1

0
sin−1x dx


Answer:
(b)
π/2

0
         sin3/2x

sin3/2x + cos3/2x
  dx 


Answer:
Question 9.
Find the area enclosed by the curve y = 3x x2 and the x-axis. If this area is rotated through four right angles about the x-axis, find the volume of the solid generated.
Answer:
Question 10.
Solve the following differential equations:
(a)
x dy

dx
 −3y = x2

Answer:
(b)
dy

dx
 = 1 − x + y xy 


Answer:


SECTION C

Answer any two questions
Question 11.
(a) Calculate the standard deviation of the following data :
 Class :   1 - 3
3 - 5
5 - 7
7 - 9
9 - 11
11 - 13 13 - 15
 Frequency :    3
9
25
35
17
10
1

Answer:
(b) Using 2000 as base year, the index numbers for the price of a commodity in 2001 and 2002 are 118 and 125 respectively.  Calculate the index numbers for 2000 and 2002, if 2001 is taken as base year.
Answer:
Question 12.
Calculate Karl Pearson's coefficient of correlation between the marks in English and Hindi obtained by 9 students :
 Marks in English (x) :   25
17
21
23
12
18 22
 15
19
 Marks in Hindi (y) :    22
20
24
25
17
11
18
25
16
[Take the assumed mean for x as 12 and that for y as 17].
Answer:
Question 13.
(a) A problem is given to three students whose chances of solving it are 1/2, 1/3 and 1/4 respectively. What is the probability that the problem will not be solved by any of them?
Answer:
(b) If 10 unbiased coins are tossed, what is the probability of getting not more than 3 heads?
Answer:


SECTION D

Answer any two questions
Question 14.
(a) Calculate the banker's gain on Rs. 2500 due in 6 months at 5% per annum.
Answer:
(b) A bill for Rs. 1000 drawn on May 7 for 6 months was discounted on August 29 of the same year, for a cash payment of Rs. 988.  Find the rate of interest charged by the bank.
Answer:
Question 15.
(a) A person borrowed a sum of money and returned it in three equal quarterly instalments of Rs. 4630.50 each. If the rate of interest charged was 20% per annum, find:
(i) the sum of money borrowed
Answer:
(ii) the total interest charged.
Answer:
(b) A retailer asks his supplier to make out a single bill in lieu of the following bills:
Rs. 2528 payable on January 6, 2007, Rs. 3860 on February 13, 2007, Rs. 5000 on February 26, 2007 and Rs. 1650 on March 9, 2007.  What would be the due date of the single bill?
Answer:
Question 16.
(a) The fixed cost of a new product is Rs. 30000 and the variable cost per unit is Rs. 800.  If the demand function is given by p(x) = 4500 − 100x, find the break-even points.
Answer:
(b) The cost function of a firm is given by C(x) = 300x − 10x2 + x3/3. Calculate the output x at which the marginal cost is minimum.
Answer:


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