Date & Time: 8th October 2015, 4:00 pm

Duration: 3h

(1) All questions are compulsory.

(2) Answer to every question must be written on a new page.

(3) Log table will be provided on demand.

(4) Answer to Section-I and Section-II should be written in two separate answer books.

(5) Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.

(6) Graph paper ls necessary for L.P.P.

(7) Figure to the right Indicate full marks.

Find x and y if `x + y = [(7,0),(2,5)] , x - y[(3,0),(0,3)]`

Chapter: [0.012] Matrices [0.02] Matrices

Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`

Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation

**Use the quantifiers to convert the following open sentence defined on N into true statement**

5x - 3 < 10

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

**Use the quantifiers to convert the following open sentence defined on N into true statement:**x

^{2}≥ 1

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

**Examine the continuity of the following function :**

`{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4`

Chapter: [0.03] Continuity

Find the adjoint of the matrix `"A" = [(2,-3),(3,5)]`

Chapter: [0.02] Matrices

Find the elasticity of demand if the marginal revenue is ₹ 50 and price is ₹ 75.

Chapter: [0.06] Indefinite Integration

Evaluate:`int(tansqrtx)/sqrtxdx`

Chapter: [0.06] Indefinite Integration

Evaluate : `int 1/("x"^2 + 8"x" + 20)` dx

Chapter: [0.06] Indefinite Integration

Write converse, inverse contrapositive of the statement "If two triangles are not congruent then their areas are not equal.

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Examine the continuity of the followin function :

`{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`

Chapter: [0.03] Continuity

Find `"dy"/"dx"` if y = x^{x} + 5^{x}

Chapter: [0.04] Differentiation

Solve the following equations by reduction method :

x + 2y + z = 8

2x+ 3y - z = 11

3x - y - 2z = 5

Chapter: [0.012] Matrices [0.02] Matrices

Find the volume of the solid obtained by revolving about the X-axis, the region bounded by the curve `"x"^2/4 - "y"^2/9 = 1` and the lines x = 2 , x = 4.

Chapter: [0.06] Indefinite Integration

If the demand function is D = 50 - 3p - p^{2}, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.

Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative

By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or . contingency. (p → q) ∧ (p ∧ ~ q ).

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

If f (x) = `(1 - "sin x")/(pi - "2x")^2` , for x ≠ `pi/2` is continuous at x = `pi/4` , then find `"f"(pi/2) .`

Chapter: [0.03] Continuity

If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`

Chapter: [0.04] Differentiation

Cost of assembling x wallclocks is `( x^3/3 - 40x^2)` and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.

Chapter: [0.05] Applications of Derivative

Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx

Chapter: [0.06] Indefinite Integration

Evaluate : ∫ log (1 + x^{2}) dx

Chapter: [0.016] Definite Integration [0.07] Definite Integrals

Alex spends 20% of his income on food items and 12% on conveyance. If for the month of June 2010, he spent ~ 900 on conveyance, find his expenditure on food items during he same month.

Chapter: [0.08] Ratio, Proportion and Partnership

Find the age · specific Death Rates for the following data :

Age group (in years) |
Population |
number of deaths |

0 - 20 | 7000 | 140 |

20 - 45 | 20000 | 180 |

45 - 65 | 10000 | 120 |

65 and above | 4000 | 160 |

Chapter: [0.11] Demography

Compute the coefficient of correlation for the following data :

n = 100 ; `bar x` = 62 , ` bar y` = 53 . σx = 10 ; σ y = 12 ,

`sum ("x"_1 - bar x) ("y"_1 - bar y) = 8000`

Chapter: [0.12] Bivariate Data and Correlation

A building is insured for 80% of its value. The annual premium at 70 paise percent amounts to ₹ 2,800. Fire damaged the building to the extent of 60% of its value. How much amount for damage can: be claimed under the policy?

Chapter: [0.1] Insurance and Annuity

A random variable X has the following probability distribution :

X = x |
-2 | -1 | 0 | 1 | 2 | 3 |

P(x) |
0.1 | k | 0.2 | 2k | 0.3 | k |

Find the value of k and calculate mean.

Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution

A, B and C are in partnership. A's capital was ₹ 65,000, C's capital was ₹ 50,000. The total profit is ₹ 38,000, out of which B's profit is ₹ 15,000. What was B's capital ?

Chapter: [0.08] Ratio, Proportion and Partnership

If the Crude Death Rate (CDR) for the following data is 13.4 per thousand, find x:

Age groups (in years) |
Population | Number of deaths |

0 - 20 | 40,000 | 350 |

20 - 65 | 65,000 | 650 |

65 and above | 15,000 | x |

Chapter: [0.09] Commission, Brokerage and Discount

Draw scatter diagram for the following data and identify the type of correlation.

capital (₹ in crore) |
2 | 3 | 4 | 5 | 6 | 8 | 9 |

Profit (₹ in lakh) |
6 | 5 | 7 | 7 | 8 | 12 | 11 |

Chapter: [0.12] Bivariate Data and Correlation

Mrs. Menon plans to save for her daughter's marriage. She wants to accumulate a sum of ₹ 4,00,000 at the end of 4 years. How much should she invest at the end of each year from now, if she can get interest compounded at 10% p.a. ?

[Given (1.1)^{4} = 1.4641]

Chapter: [0.1] Insurance and Annuity

A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .

Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution

A fair coin is tossed 12 times. Find the probability of getting at least 2 heads .

Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution

For the following problem find the sequence that minimizes total elapsed time (in hrs) required to complete the jobs on 2 machines M_{1} and M_{2} in the order M_{1} - M_{2} · Also find the minimum elapsed time T.

Job |
A |
B |
C |
D |
E |
F |

M_{1} |
4 | 8 | 3 | 6 | 7 | 5 |

M_{2} |
6 | 3 | 7 | 2 | 8 | 4 |

Chapter: [0.15] Management Mathematics

A bill of ₹ 2,000 drawn on 15th February 2003 for 10 months was discounted on 13th May 2003 at `3 3/4` % p.a. Calculate the banker's discount.

Chapter: [0.09] Commission, Brokerage and Discount

From the following table which relates to the number of animals of a certain

species at age x. complete the life table :

x |
0 |
1 |
2 |
3 |
4 |
5 |

l_{x} |
1000 | 850 | 760 | 360 | 25 | 0 |

Chapter: [0.11] Demography

A person makes two types of gift items A and B requiring the services of a cutter and a finisher. Gift item A requires 4 hours of the cutter's time and 2 hours of finisher's time. Fifth item B requires 2 hours of the cutter's time and 4 hours of finisher's time. The cutter and finisher have 208 hours and 152 hours available time respectively every month. The profit on one gift item of type A is ₹ 75 and on one gift item of type, B is ₹ 125. Assuming that the person can sell all the gift items produced, determine how many gift items of each type should he make every month to obtain the best returns?

Chapter: [0.15] Management Mathematics

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible :

Variable of X = 9

Regression equations : 8x - 10g + 66 = 0 and 40x- 18g = 214

Find Mean values of X and Y on the basis of the above information .

Chapter: [0.12] Bivariate Data and Correlation

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible :

Variable of X = 9

Regression equations : 8x - 10g + 66 = 0 and 40x- 18g = 214

Find Correlation coefficient between X and Y on the basis of the above information.

Chapter: [0.12] Bivariate Data and Correlation

Mr. Rajesh has ₹ 1800 to spend on fruits for a meeting. Grapes cost ₹ 160/kg llnd peaches ₹ 200/kg. Let x and g represent the number of kilogrames of grapes and peaches he can buy. Write the graph of an inequation to model the amounts of grapes and peaches he can buy within his budget.

Chapter: [0.15] Management Mathematics

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:

(a) the mean and

(b) P(X = 0). [Given: e^{-2} = 0.1353)

Chapter: [0.14] Random Variable and Probability Distribution

Ranking of 8 trainees at the beginning (X) and at the end (Y), of a certain course are given below :

Trainees |
A | B | C | D | E | F | G | H |

X |
1 | 2 | 4 | 5 | 6 | 8 | 3 | 7 |

Y |
2 | 4 | 3 | 7 | 8 | 1 | 5 | 6 |

Calculate Spearman's rank correlation coefficient.

Chapter: [0.12] Bivariate Data and Correlation

A computers centre has four expert programmers . The centre needs four application programmes to be developed. The head of the computer centre , after stying carefully the programmes to be developed , estimates the computer time in minutes required by the respective experts to develop the application programmes as follows :

Programmes |
||||

Programmes |
1 | 2 | 3 | 4 |

(Times in minutes) | ||||

A | 120 | 100 | 80 | 90 |

B | 80 | 90 | 110 | 70 |

C | 110 | 140 | 120 | 100 |

D | 90 | 90 | 80 | 90 |

How should the head of the computer centre assign the programmes to the programmers so that the total time required is minimum ?

Chapter: [0.012] Matrices [0.02] Matrices

From the data 7 pairs of observations on X and Y following results are obtained :

∑(x_{i} - 70) = - 38 ; ∑ (y_{i} - 60) = - 5 ;

∑ (x_{i} - 70)2 = 2990 ; ∑ (y_{i} - 60)2 = 475 ;

∑ (x_{i} - 70) (y_{i} - 60) = 1063

Obtain :

(a) The line of regression of Y on X.

(b) The line of regression of X on Y.

Chapter: [0.13] Regression Analysis Introduction

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