HSC MATHS PRELIMINARY PAPER

SECTION - 1
Q1. Attempt any 6 of the following: [12]

(i) If y = (sinx)x, find dydx.  

(ii) If A = then show that A2 - 2A is a scalar matrix.

(iii) Write the negation of the following statements:
  1. y N, y2 + 3 7.
  2. If the lines are parallel then their slopes are equal.

(iv) Find dydx, if y = cos-1(sin 5x).

(v) Write the negation of the following.
  1. Economic growth and per capita income is more in America.
  2. is an irrational number.

(vi) State which of the following sentences are statements. In case of statement, write down the truth value:
  1. Every quadratic equation has only real roots.
  2. -4 is a rational number.

(vii) Express the following statements in symbolic form.
  1. Milk is white or grass is green.
  2. Shobha stays at home, while Dhanashree and Prashant go for a movie.

(viii) If A = {2, 3, 4, 5, 6, 7, 8 }, determine the truth value of each of the following:
  1. x A, Such that x + 5 = 8
  2. x A, x + 1 10.



Q2. (A) Attempt any 2 of the following: [6]

(i) Determine whether the following matrices are singular or nonsingular.
  1.       (b)

(ii) If A = , prove that A3 - 6A2 + 7A + 2I = 0.

(iii) If A = , then reduce it to I3 by using row transformations.


Q2. (B) Attempt any 2 of the following: [8]

(i) Are the following functions continuous on the set of real numbers? Justify your answers:
  1. h(x) = 5x2 + 7x + 2x3+ x2+ x + 3
  2. f(x) = 32x - 15x

(ii) Examine the continuity of the following function at the given point.
f(x) = sin 5x x, for x 0
     = 1   ,        for x = 0,    at x = 0.

(iii) Discuss the continuity of the following function.
f(x) = a2x-1x, for x 0 ; a > 0, a 1
     = 2 log a ,        for x = 0,    at x = 0.

Q3. (A) Attempt any 2 of the following: [6]

(i) If f is continuous at x = 0, then find f(0),
f(x) = 5x + 5-x - 2  x2 , x = 0

(ii) If  f(x) = 1 - sinx (π - 2x )2 , for x π2is continuous at x = π2, find f(π2).

(iii) Find the rate of change of demand (x) of a commodity with respect to its price (y), if y = 5 + x2e-x + 2x.

Q3. (B) Attempt any 2 of the following: [8]
(i) Find the marginal demand of a commodity where demand is x and price is y, if
Y = xex + x logx.

(ii) Differentiate: y = 5x + xx

(iii) Differentiate: y = (sinx) x + x. Sinx.

SECTION - 2
Q4. Attempt any 6 of the following: [12]

(i) Compute CDR using the information given below:

Age Group
(years)
0 - 15
15 - 35
35 - 65
65 and above
Population
9000
25000
32000
9000
Total number of deaths in a year is given to be 900.

(ii) Identify random variables as either discrete or continuous in each of the following situations. Also write the range wherever it is possible.
  1. Number of attempts required by a candidate to clear I.A.S. examination.
  2. Height of a skyscraper.

(iii) Verify whether the function can be regarded as p.m.f. For the given values of x.


x
2
4
6
8
P(X = x)
0.2
0.4
0.6
0.8

(iv) Determine K such that the following function is a p.m.f.

(v) Obtain the expected value and variance of x for the following probability distribution.

x
-2
-1
0
1
2
P(X = x)
0.2
0.3
0.1
0.15
0.25

(vi) Draw scatter diagram for the following data and interpret it:

x
10
20
30
40
50
60
70
y
32
20
24
36
40
28
38

(vii) If d2 = 66 and n = 10 then find the rank correlation coefficient.

(viii) From the two regression equation
Y = 4x - 5  and 3x = 2y + 5 find x and y.

Q5. (A) Attempt any 2 of the following: [6]

(i) A fair coin is tossed 12 times. Find the probability of getting
  1. Exactly 7 heads.
  2. At least 2 heads.

(ii) Compute T40 given that e400=31, l40=550.

(iii) In a complete life table l25=50,000 and L25=49,865. Find the value of q25.


Q5. (B) Attempt any 2 of the following: [8]
(i) In the following data one of the values of y is missing. Arithmetic means of x and y series are 6 and 8 respectively.

x
6
2
10
4
8
y
9
11
?
8
7

  1. Estimate missing observation.
  2. Calculate correlation.

(ii) If the rank correlation coefficient is 23and d2 = 55, then find the number of pairs of observations. (Assume that no rank is repeated)

(iii) The following table gives the apritude test scores and productivity indices of 10 workers selected at random.

Aptitude Score (x)
60
62
65
70
72
48
53
73
65
82
Productivity Index (y)
68
60
62
80
85
40
52
62
60
81
Obtain the two regression equations and estimate:
  1. The productivity index of a worker whose test score is 92.
  2. The test score when productivity index is 75.

Q6. (A) Attempt any 2 of the following: [6]

(i) Given the following data, obtain the linear equations estimate of x for y = 10.
x = 7.6 and y= 14.8, x = 3.2, y = 16.  r = 0.7.  

(ii) The equations of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.  Find
  1. Correlation coefficient.
  2. x y

Q6. (B) Attempt any 2 of the following: [8]
(i) A pharmaceutical company has four branches one at each city A, B, C and D. A branch manager is to be appointed one at each city out of four candidates P, Q, R and S. The monthly business depends upon the city and effectiveness of the branch manager in that city.

Branch Manager

           City



A
B
C
D

Monthly
Business
(in Lakhs of Rs. )

P
11
11
9
9
Q
13
16
11
10
R
12
17
13
8
S
16
14
16
12



(ii) Find the sequence that minimize the total elapsed time (in hours) required to complete the following jobs on the three machines M1, M2 , M3 ,  in the order M1, M2 , M3. Also find the minimum total elapsed time and idle time for three machines.

Job
A
B
C
D
E
M1
5
7
6
9
5
M2
2
1
4
5
3
M3
3
7
5
6
7
(iii) A Chartered Accountants’ firm has accepted ‘five’ new cases. The estimated number of days required by each of their ‘five’ employees for each case are given below, where ‘-‘means that the particular employee cannot be assigned the particular case. Determine he optimal assignment of cases to the employees so that the total number of days required completing these ‘five’ cases will be minimum. Also find the minimum number of days.


Employees

Cases




I
II
III
IV
V
E1
5
2
4
2
6
E2
3
4
-
5
7
E3
6
3
4
1
2
E4
4
2
2
3
5
E5
3
6
4
7
3