Maths and Stats

SELECTION MENU

TIME : 3 hours                     Marks : 100

NB: 1. Attempt any TWO questions from Section I and any THREE questions from Section II.
2. Use of simple non-programmable calculator is allowed.
3. Graph papers will be provided on request,
4. Answers to both the sections should be written in the same answer book
5. Figures to the right indicate marks allotted to the sub questions.

SECTION I

(a) A merchant advises his agent to buy 300 umbrellas and sell them at 20 % profit. The agent charges 1 % commission on the purchase and 2 % on the sales. On the whole the agent gets a commission of Rsl,020/-. Find the.selling price of each umbrella. (5)

(b) Find the equation of a line passing through the origin and parallel to the line with equation
3y = -5x + 7. (5)

(c) Solve the following L.P.P. graphically: 
Maximise Z= 3x + 5y 
subject to 0 < x <= 300 , x-y=0, 0 < y <400 (6)

(d) The line y = mx + x passes through the point of intersection of the lines y = 2x - 4 and y = 5x - 7. Find m. (4)

(a) Find dy/dx ( any TWO ) (5)
(1) y = log(3x² - 4x + 2) 
(2) y = (x + I )^(2x+5)  
(3) y=e^x (x² + 5x -7)
(4) y=(x² -3x + 5) / 2x + 1

(b) If the slope of the tangent to the curve y = ax - bx + 1 at x = 2 is ;3. Show that 8a - 2b + 3 = 0. (5)

(c) Divide SO into two parts so that their product is maximum. (6)

(d) The demand function is given by D = 20 - p - p2 where D = demand and p = price. Find the elasticity of demand w.r. to price when price is 2. (4)

(a) (I) Write an expression in BASIC for the following: (3)
(1) [-b+sq(b²-4ac)]/2a
(2) [2 + 5(3+7)]/16
(3) sqrt[S(S-A)(S-B)(S-C)]

(H) Write statements in BASIC. (2)
(1) Repeat a set of instructions 20 times.
(2) To increase the value of A by 5 units and to decrease the value of B by 7 units.
(b) 
(1) Give one example of each to explain three control transfer statements in BASIC. (3) 
(2) State the difference between END and STOP statements in BASIC. (2)

(c) State whether the following statements are true or false in BASIC: (6)
1. Every READ statement must have its own DATA statement.
2. There can be many REM statements in a program.
3. The GO TO statement is an example of conditional transfer of control.
4. A string variable name can be used in a FOR - TO statement.
5. Using semi-colons, more than 5 values can be printed on a line.
6. AND, OR are examples of relational operators.

(d) Find the output when the following program is executed. (4)
10 A=4
20 FOR K = I to 3
30 A = A+K
40 B = K î 2
50 PRINT K ; A; B
60 NEXT K
70 END

Q.4. (a) Prepare forward difference table for the function given by

Show that the function f(x) is a polynomial of degree 2. (5) 

(b) Using Lagrange's formula calculate f(5) for the following data:

(c ) Using Newton's interpolation formula, find the number of companies earning profit
below Rs 85, 000. (6)

(d) The total revenue function R is given by R=8 - 3x + x where x is the quantity demanded Find AR. MR at x-4. (4)

SECTION II

Q.5. (a) Tabulate the following information: (6)
In 1995, out of the total 3,000 workers in a factory 2, 300 were skilled workers. The number of women employed was 300 out of which 250 were unskilled. In 1996, the number of skilled workers was 2, 750 of which 2, 500 were men. The number of unskilled workers was 760 of which 300 were women.

(b) The arithmetic mean of runs scored by three batsmen Sachin, Kambly and Jadeja in the same series of 10 innings are 50, 48 and 12 respectively. The standard deviation of their runs are respectively 15, 12 and 2. Who is the most consistent of the three ? If one of the three is to be selected for the Pakistan tour, who will be selected ? (7)

(c) The median of the following data is 36.8. Find the missing frequency. (7)

Q.6. (a) Calculate Karl Pearson's correlation from the following data.  (6)

(b) Coefficient of correlation between debenture prices and share prices is found to be 0.143. If the sum of the squares of differences in ranks is given to be 48, find the number of pairs.(6)
(c) If the two regressions 4x - 5y + 30 = 0 and 20x - 9y - 106 = 0, find the mean values of x and y and correlation coefficient between x and y. Also find standard deviation of y if standard deviation of x is 3.(7) 

Q.7.(a) The following data is given for marks of 10 students in two subjects. Economics and Mathematics

Coefficient of correlation is 0. 75
Estimate: 
(1) Marks in economics of students securing 37 marks in mathematics.
(2) Marks in mathematics of a student with 60 marks in Economics. (6)

(b) The following are figures of circulation of Hindi dailies according to an Agency. Draw a suitable diagram to represent the data:

(c ) Calculate the cost of living index number for the following data which give the group index with their respective weights for 1999 ( with base 1995 ).

Q.8.(a) Define an Index number and mention its limitations. (6)
(b) State and prove addition theorem for the events of Probability. (7)
(c) A husband and wife appeared in an interview for two vacancies in an office. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. Find the probability that 
(1)both of them are selected (2) only one of them is selected (7)


Q.9.(a) Write a probability distribution of X if X represents number of heads appearing when three coins arc tossed simultaneously. Also find (he mean and variance of the distribution. (6) 

(b) Three percent of a given lot of manufactured pans are defective. What is the probability that in a sample of four items
(1) none will be defective
(2) At least one will be defective.
(3) Exactly two are defective. (7)

(c) The distribution of marks obtained by a group of 500 students is normally distributed with means 60 and standard deviation 5. Find
(1) The number of students who secured 65 or more marks
(2) The number of students who scored at most 67 marks 
Given that f(1) = 0.84 and f(1.4) = 0.92.