Date: July 2018
- All questions are compulsory.
- Figures to the right lndicatefull marks.
- Graph paper Is necessary for L.P.P.
- Write answer of every question on separate page.
If p : It is a day time
q : It is warm
Give the verbal statement for the following symbolic statement :
(a) p ^ - q (b) P → q
Chapter: [1] Mathematical Logic
Express the truth of each of the following statements using Venn diagrams:
(a) No circles are polygons
(b) Some quadratic equations have equal roots
Chapter: [1] Mathematical Logic
Find the values of x and y if
`2 [(x,5),(7,y-3)] [(3,-4),(1,2)] = [(7,6),(15,14)]`
Chapter: [2] Matrices
Find `"dy"/"dx"` ; if x = sin^{3}θ , y = cos^{3}θ
Chapter: [4] Differentiation
Find `"dy"/"dx"` ; if y = cos^{-1} `("2x" sqrt (1 - "x"^2))`
Chapter: [4] Differentiation
Evaluate: ∫ x . log x dx
Chapter: [1] Mathematical Logic
The cost C of producing x articles is given as C = x^{3}-16x^{2 }+ 47x. For what values of x, with the average cost is decreasing'?
Chapter: [4] Differentiation
Evaluate : `int _0^(pi/4) 1/(1 + "x"^2) "dx"`
Chapter: [7] Definite Integrals
Solve the following equations by reduction method:
x+ y+z = 6,
3x-y+3z = 10
5x+ y-4z = 3
Chapter: [2] Matrices
Evaluate : `int (2"x" + 1)/(("x" + 1)("x"-2)) "dx"`
Chapter: [6] Indefinite Integration
Evaluate `int _0^l "x" (1 - "x")^(3/2) "dx"`
Chapter: [7] Definite Integrals
Using the rules of negation, write the negatlon of the following:
(a) p ∧ (q → r)
(b) ~P ∨ ~q
Chapter: [1] Mathematical Logic
If the function f is continuous at x = 2 and x = 4 then find the values of a and b.
Where f(x) = x^{2} + ax + b, x < 2
= 3x + 2, 2 ≤ x ≤ 4
= 2ax + 5b, 4 < x
Chapter: [6] Indefinite Integration
A manufacturing company produces x items at the total cost of Rs (180+4x). The demand function of this product is P=(240 - x). Find x for which profit is increasing.
Chapter: [6] Indefinite Integration
If A = `[(1,2),(3,-1)] , "B" = [(7,1),(2,5)]`
Verify that |AB| = |A|.|B|
Chapter: [2] Matrices
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Chapter: [7] Definite Integrals
Find the volume of solid generated by rotating the area bounded by x^{2}+y^{2} =36 and the lines x = 0, x = 3 about X -axis.
Chapter: [6] Indefinite Integration
If f is continuous at x = 0, then find f (0).
Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`
Chapter: [3] Continuity
If x^{7} . y^{9} = (x + y)^{16} then show that `"dy"/"dx" = "y"/"x"`
Chapter: [4] Differentiation
In a firm the cost function for output x is given as C = `"x"^2/3 - 20"x"^2 + 70 "x"`. Find the 3 output for which marginal cost (C_{m}) is minimum.
Chapter: [5] Applications of Derivative
The price of a T.V. set is Rs 17,000. An agent charges at 3% and earns Rs. 25,500. Find the number of T.V. sets sold by him.
Chapter: [9] Commission, Brokerage and Discount
Find the Age-Specific Death Rate (Age SDR) for the following data.
Age groups (In year) |
Population (in '000) |
Number of Dearths |
0 - 10 | 11 | 275 |
10 - 20 | 12 | 180 |
20 - 60 | 9 | 81 |
60 and above | 2 | 32 |
Chapter: [11] Demography
The regression equation of y on x is given by 3x + 2y - 26 = O. Find b_{yx}.
Chapter: [13] Regression Analysis Introduction
Verify whether the following function can be regarded as p.m.f. of the random variable X:
\[ P(X) = \begin{cases} \frac{x - 1}3 & \quad \text{x } = \text{ 1,2,3}\\ 0 & \quad \text{} , \text{otherwise} \end{cases} \]
Chapter: [14] Random Variable and Probability Distribution
If X has a binomial distribution with n = 20 , p = `1/10` , find E (X) and V (X).
Chapter: [14] Random Variable and Probability Distribution
Bring out the inconsistency; if any: b_{yx} + b_{xy} = 1.30. and r = 0.75
Chapter: [13] Regression Analysis Introduction
A train travelled between two stations. The distance and time were recorded as below:
Distance (Km) | 80 | 120 | 160 | 200 | 240 |
Time (Hr) | 2 | 3 | 4 | 5 | 6 |
Draw scatter diagram and identify the type of correlation.
Chapter: [12] Bivariate Data and Correlation
If r = 0.5, σ_{x} = 1 and σ_{y} = 4, then find Cov.(X,Y).
Chapter: [12] Bivariate Data and Correlation
Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following:
Age x | 0 | 1 | 2 |
l_{x} | 1000 | 880 | 876 |
T_{x } | - | - | 3323 |
Chapter: [14] Random Variable and Probability Distribution
Calculate the CDR for District A and B and compare them:
Age group (in years) | District A | District B | ||
No.of. persons (in '000) |
No.of. deaths |
No.of. persons (in '000) |
No.of. deaths | |
0 - 15 | 1 | 20 | 2 | 50 |
15 - 60 | 3 | 30 | 7 | 70 |
60 and above | 2 | 40 | 1 | 25 |
Chapter: [15] Management Mathematics
The equation of the line of regression of Y on X is 3x + 2y = 26 and X on Y is 6x + y = 31. Find Var. (X) if Var. (Y) = 36.
Chapter: [13] Regression Analysis Introduction
Calculate Spearman's Rank Correlation Coefficient between the following marks given by 'two' judges (A) and (B) to •eight' contestants in the elocution competition:
Contestants | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Marks by A | 81 | 72 | 60 | 33 | 29 | 11 | 56 | 42 |
Marks byB | 75 | 56 | 42 | 15 | 30 | 20 | 60 | 80 |
Chapter: [12] Bivariate Data and Correlation
Solve the following assignment problem to minimize the cost:
Persons | Jobs | ||
I | II | III | |
A | 7 | 3 | 5 |
B | 2 | 7 | 4 |
C | 6 | 5 | 3 |
D | 3 | 4 | 7 |
Chapter: [5] Applications of Derivative
Find the sequence of the following five jobs to be processed on three machines M_{1} M_{2,} , M_{3} that will minimize the total elapsed time to complete the tasks . Each job is to be processed in the order M_{1} - M_{2} - M_{3} :
Jobs | 1 | 2 | 3 | 4 | 5 |
Machine M_{1} | 5 | 11 | 5 | 7 | 6 |
Machine M_{2} | 1 | 4 | 2 | 5 | 3 |
Machine M_{3 } | 1 | 5 | 2 | 3 | 4 |
Chapter: [15] Management Mathematics
Minimize: Z = 2x + y
Subject to: x + y ≤ 5
x + 2y ≤ 8
4x + 3y ≥ 12
x ≥ o, y ≥ o
Solve graphically.
Chapter: [15] Management Mathematics
Find the graphical solution for the following system of linear equations :
3x + 4y ≥ 12 , 4x + 7y ≤ 28 , y ≥ 1 , x ≥ 0 , y ≥ 0 ,
Hence find the co-ordinates of comer points of the common region.
Chapter: [15] Management Mathematics
A wholesaler allows 25% trade discount and 5% cash discount. Find the list price of an article, if it was sold for the net amount of Rs 1,140.
Chapter: [9] Commission, Brokerage and Discount
Find the accumulated value after 3 years of an immediate annuity of Rs 2,000 p.a. with interest compounded at 10% p.a. [Given (1.1)^{3} = 1.331]
Chapter: [10] Insurance and Annuity
If the difference between true discount and banker's discount on a sum due 4 months hence is Rs 20, find true discount, banker's discount and amount of the bill, the rate of simple interest charged being 5% p.a .
Chapter: [9] Commission, Brokerage and Discount
If a random variable X follows Poisson distribution such that P(X = l) =P(X = 2), then find P(X ≥ 1). [Use e^{-2} = 0.1353]
Chapter: [14] Random Variable and Probability Distribution
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