Date & Time: 2nd March 2019, 11:00 am
Duration: 3h
- All questions are compulsory.
- Answer to every question must be written on a new page.
- Log lable will be provided on demand.
- Answer to Section-I and Section-II should be written in two separate answer books.
- Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.
- Graph paper is necessary for L.P.P.
- Figure to the right indicate full marks.
Write converse and inverse of the following statement:
“If a man is a bachelor then he is unhappy.”
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Discuss the continuity of f at x = 1
Where f(X) = `[ 3 - sqrt ( 2x + 7 ) / ( x - 1 )]` For x ≠ 1
= `-1/3` For x = 1
Chapter: [0.03] Continuity
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
Chapter: [0.03] Continuity
Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.
Chapter: [0.013999999999999999] Applications of Derivatives
Find `dy/dx if x^3 + y^2 + xy = 7`
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
Find the area bounded by the curve y = x^{4}, x-axis and lines x = 1 and x = 5.
Chapter: [0.07] Definite Integrals
Evaluate : `int_-2^3 dx/(x+5)`
Chapter: [0.06] Indefinite Integration
Evaluate
`int 1/(16 - 9x^2) dx`
Chapter: [0.04] Differentiation
Prove that the following statement pattern is a tautology : ( q → p ) v ( p → q )
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Find `dy/dx` if y = x^{x }+ 5^{x}
Chapter: [0.04] Differentiation
Evaluate :
`int xcos^-1x dx`
Chapter: [0.06] Indefinite Integration
Find the inverse of the matrix `[ (1, 2, 3), (1, 1, 5), (2, 4, 7)]` by using the adjoint method.
`
Chapter: [0.02] Matrices
If f is continuous at x = 0 then find f(0) where f(x) = `[5^x + 5^-x - 2]/x^2`, x ≠ 0
Chapter: [0.03] Continuity
A manufacturer can sell x items (x > 0) at a price of Rs.(280 - x) each. The cost of producing x items is Rs. (x^{2} + 40x + 35). Find the number of items to be sold so that the manufacturer can make maximum profit.
Chapter: [0.09] Commission, Brokerage and Discount
If p and q are true statements and r and s are false statements, find the truth value of the following :
( p ∧ ∼ r ) ∧ ( ∼ q ∧ s )
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Differentiate e^{4x + 5} w.r..t.e^{3x}
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
Evaluate :
`int [e^x ( 1 + x )]/[ cos^2 (xe^x)] dx`
Chapter: [0.06] Indefinite Integration
If A = `[(2, 3), (1, 2)], B = [(1, 0),(3, 1)]`, Find (AB)^{-1}
Chapter: [0.02] Matrices
For manufacturing x units, labour cost is 150 - 4x and processing cost is x^{2}. Price of each unit is p = 10,800 - 4x^{2}. Find the values of x for which :
(a) Total cost is decreasing
(b) Revenue is increasing
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Evaluate : `int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]`
Chapter: [0.07] Definite Integrals
Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
If the correlation coefficient between X and Y is 0.8, what is the correlation
coefficient between.
(a) X and 3Y
(b) X - 5 and Y - 3
Chapter: [0.12] Bivariate Data and Correlation
Find the premium on property worth Rs.12,50,000 at 3% if the property is insured to the extent of 80% of its value.
Chapter: [0.1] Insurance and Annuity
If the sum of squares of differences of ranks for 10 pairs of observations is 66, find the rank correlation coefficient.
Chapter: [0.12] Bivariate Data and Correlation
If the present worth of a bill due 6 months hence is Rs. 2,500 at 10% per annum, what is the true discount ?
Chapter: [0.09] Commission, Brokerage and Discount
From the following table find q_{0} :
x | 0 | 1 | 2 | 3 | 4 | 5 |
l_{x} | 1000 | 940 | 780 | 590 | 25 | 0 |
Chapter: [0.11] Demography
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.
Chapter: [0.15] Management Mathematics
What must be subtracted from each of the numbers 5, 7 and 10, so that the
resulting numbers are in continued proportion?
Chapter: [0.08] Ratio, Proportion and Partnership
An article is marked at Rs. 1500. A trader allows a discount at 3% and still gains 20% on the cost. Find the cost price of the article.
Chapter: [0.09] Commission, Brokerage and Discount
For a Binomial distribution n = 6 and p = 0.3, find the probability of getting
exactly 3 successes.
Chapter: [0.14] Random Variable and Probability Distribution
Diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1500 calories Two foods F_{1 }and F_{2} cost Rs. 50 and Rs. 75 per unit respectively. Each unit of food F_{1} contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas each unit of food F_{2} contains 100 units of vitamins, 2 units of minerals and 30 calories. Formulate the above problem as L.P.P. to satisfy the sick person's requirements at minimum cost.
Chapter: [0.15] Management Mathematics
Two samples from bivariate populations have 15 observations each. The sample mean of X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from means are 136 and 148. The sum of product of deviations from respective means is 122. Obtain the equation of line of regression of X on Y.
Chapter: [0.13] Regression Analysis Introduction
Suggest optimum solution to the following assignment. Problem, also find the total minimum service time.
Service Time ( in hrs.)
Counters | Salesmen | |||
A | B | C | D | |
W | 41 | 72 | 39 | 52 |
X | 22 | 29 | 49 | 65 |
Y | 27 | 39 | 60 | 51 |
Z | 45 | 50 | 48 | 52 |
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
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
For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is 3y - 5x + 180 = 0. The mean marks in accountancy is 44 and the variance of marks in statisstics is `(9/16)th` of the variance of marks in accountancy. Find the mean marks in statistics and the
correlation coefficient between marks in the two subjects.
Chapter: [0.13] Regression Analysis Introduction
The p.d.f. of a random variable X is given by :
f(x) = 2x, 0 ≤ x ≤ 1
= 0, otherwise
Find `P(1/3 < x < 1/2)`
Chapter: [0.14] Random Variable and Probability Distribution
Find the sequence that minimizes the total elapsed time required to complete the following task. The table below gives the processing time in hours. Also, find the minimum elapsed time and idle times for both machines.
Jobs | 1 | 2 | 3 | 4 | 5 |
M_{1} | 3 | 7 | 4 | 5 | 7 |
M_{2} | 6 | 2 | 7 | 3 | 4 |
Chapter: [0.15] Management Mathematics
A bill of Rs.7,500 was discounted for Rs. 7,290 at a bank on 28^{th} October 2006. If the rate of interest was 14% p.a., what is the legal due date ?
Chapter: [0.09] Commission, Brokerage and Discount
The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.
Also prepare marginal distributions :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )
Chapter: [0.14] Random Variable and Probability Distribution
Find the feasible solution for the following system of linear inequations:
0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
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